# Schedule

The plenary and session talks will take palce in the Solvay amphitheater.

The workshop will start on Monday at 09:20 am and end on Friday at 01:00 pm.

There will be a welcome beer party on monday at 05:00 pm. Here is the booklet. You will receive a printed copy the first day of the meeting.
Mon 7Tue 8Wed 9 Thu 10Fri 11
8:45Registration
9:20 Wei Dolbeault Loss Souplet Carrillo
10:10 Winkler Tanaka Dancer Felli Maris
10:40Coffee breakCoffee break Coffee breakCoffee break Coffee break
11:10 Hamel del Pino Quittner Pistoia Van Schaftingen
12:00 Reichel Fila Le Coz de Figueiredo Terracini
12:30 Sandwich buffet LunchLunch Lunch
13:00 Sandwich buffet LunchLunch Lunch Complimentary
14:00 Contributed talks Grossi Gazzola Farina closing lunch
14:50 6 parallel sessions Sirakov Ianni Stinner
15:30 Coffee break Coffee break Coffee break Van Phan
16:00 Soave d'Avenia Grunau Coffee break
17:00 Welcome party 16:50 Secchi Mederski

# Schedule of contributed talks

Forum EForum FForum G Forum HOF 2070OF 2072
14:00 Hernández Barile Román Balbus Pimenta Choi
14:15 Pomponio Maia Loiudice Salazar Majda Kalli
14:30 Zographopoulos Cozzi Ikoma Duzgun Benmakhlouf Yoo
14:45 Tentarelli Picon Azzollini Hasegawa Zappale Cieślak
15:00 Özsari Kulikov Coelho Buoso Fourti Gallouët
15:15 Siciliano Figueiredo

# Abstracts

▸ Click on the talk titles to see the corresponding abstracts.


## Plenary speakers

José Antonio Carrillo, Imperial College London
Minimizing Interaction Energies — Slides

I will start by reviewing some recent results on qualitative properties of local minimizers of the interaction energy to motivate the main topic of my talk: to discuss global minimizers. We will show the existence of compactly supported global miminizers under quite mild assumptions on the potential in the complementary set of classical H-stability in statistical mechanics. A strong connection with the classical obstacle problem appears very useful when the singularity is strong enough at zero. An approach from discrete to continuum is also quite nice under convexity assumptions on the potential. This is based on three works, one together with F. Patacchini, J.A. Caniizo, another one with M. Delgadino and A. Mellet, and finally with M. Chipot and Y. Huang.

Manuel del Pino, Universidad de Chile
Bubbling in the critical heat equation: the role of Green's function — Slides

We investigate the point-wise, infinite-time bubbling phenomenon for positive solutions of the semilinear heat equation at the critical exponent in a bounded domain. We build an invariant manifold for the flow which ends at $$k$$ bubbling points of the domain for any given $$k$$. The delicate role of dimension is described. This is joint work with C. Cortázar and M. Musso.

Jean Dolbeault, Université Paris Dauphine
Sharp functional inequalities and nonlinear diffusions — Slides

Nonlinear diffusion flows are very interesting tools for the study of sharp functional inequalities, the stability of the optimal functions and for improvements of the inequalities. The lecture will be devoted to a series of recent results which provide a unified picture of rigidity methods in nonlinear elliptic equations. They have been obtained in a series of papers in collaboration with Maria J. Esteban, Michal Kowalczyk, and Michael Loss. The method is based on entropy methods and related with the Bakry-Emery method and various qualitative techniques for functional inequalities in probability theory, information theory and nonlinear analysis. As an illustration, a computation based on a well chosen nonlinear diffusion flow solves a longstanding conjecture on symmetry breaking in Caffarelli-Kohn-Nirenberg inequalities. Results on spectral estimates in various settings can also be deduced.

Alberto Farina, Université de Picardie
On the classification of nonnegative solutions to nonlinear equations in half-spaces — Slides

We shall discuss the classification of nonnegative solutions to some semilinear elliptic problems on Euclidean half-spaces. Extensions and open questions will be also considered.

Filippo Gazzola, Politecnico di Milano
Torsional instability in suspension bridges: the Tacoma Narrows Bridge case — Slides

Most people have seen the videos of the spectacular collapse of the Tacoma Narrows Bridge (TNB). The torsional oscillations were considered the main cause of the collapse [AM,SC]. But the appearance of torsional oscillations is not an isolated event occurred only at the TNB, several suspension bridges manifested aerodynamic instability and uncontrolled oscillations leading to collapses.

Most attempts of explanations of the TNB collapse are based on aeroelastic effects such as the frequency of the vortex shedding, parametric resonance, and flutter theory. All these attempts received criticisms because the quantitative parameters do not fit the explanations and the experiments in wind tunnels do not confirm the underlying theory. In [AG] we gave an explanation in terms of a structural instability by modeling the bridge as a system of coupled oscillators.

In this talk we improve the results in [AG] from several points of view. We take into account the behavior of all the structural components of a suspension bridge: the deck, the sustaining cables, and the connecting hangers. We compute all the energies involved and we derive the differential equations from variational principles: the dynamics of a suspension bridge is described by a system of nonlinear nonlocal “hyperbolic'' equations. Then we introduce into the model the parameters of the TNB and we obtain quantitative responses; our numerical results confirm the behavior seen on the day of the collapse. Finally, we insert the obtained results into a suitable theoretical framework, both to justify the results and to explain the underlying phenomenon. The contents of the present talk are taken from two joint papers with Gianni Arioli (Milan).

#### References

[AM] O.H. Ammann, T. von Kármán, G.B. Woodruff, The failure of the Tacoma Narrows Bridge, Federal Works Agency, Washington D.C. (1941)

[AG] G. Arioli, F. Gazzola, A new mathematical explanation of what triggered the catastrophic torsional mode of the Tacoma Narrows Bridge collapse, Appl. Math. Modelling 39, 901–912 (2015)

[SC] R. Scott, In the wake of Tacoma. Suspension bridges and the quest for aerodynamic stability, ASCE Press (2001)

Massimo Grossi, Università di Roma Sapienza
Multi-layer radial solutions for a supercritical neumann problem — Slides

We consider the problem $\begin{cases} -\Delta u+u=u^p & \text{ in }B_1 \\ u > 0, & \text{ in }B_1 \\ \partial_\nu u=0 & \text{ on } \partial B_1, \end{cases}$ where $$B_1$$ is the unit ball and we show the existence of radial solutions with $$k$$ nodal zones for $$p$$ large enough.

Hans Christoph Grunau, Universität Magdeburg
Minimising a relaxed Willmore functional for graphs subject to Dirichlet boundary conditions — Slides

The talk is based on joint work with Klaus Deckelnick (Magdeburg) and Matthias Röger (Dortmund) [DGR].

For a bounded smooth domain $$\Omega\subset \mathbb{R}^2$$ and a smooth boundary datum $$\varphi:\overline{\Omega}\to \mathbb{R}$$, we consider the minimisation of the Willmore functional $W(u) := \frac{1}{4} \int_{\Omega} H^2 \; \sqrt{1+ | \nabla u |^2} \, dx$ for graphs $$u: \overline{\Omega}\to \mathbb{R}$$ with mean curvature $$H := \div\Bigl(\frac{\nabla u }{\sqrt{1+ |\nabla u|^2}}\Bigr)$$ subject to Dirichlet boundary conditions, i.e. in the class \begin{equation*} \mathcal{M} := \bigl\{ u\in H^2(\Omega): (u-\varphi) \in H^2_0(\Omega)\bigr\} . \end{equation*} Making use of a celebrated result by L. Simon [Si, Lemma 1.2] we first show that in this class, bounds for the Willmore energy imply area and diameter bounds. Examples show that stronger bounds in terms of the Willmore energy are not available. This means that $$L^\infty\cap BV(\Omega)$$ is the natural solution class where, however, the original Willmore functional is not defined. So, we need to consider its $$L^1$$-lower semicontinuous relaxation. Our main result states that this relaxation coincides on $$\mathcal{M}$$ with the original Willmore functional so that the relaxed functional is indeed its largest possible $$L^1$$-lower semicontinuous extension to $$BV(\Omega)$$. Moreover, finiteness of the relaxed energy encodes attainment of the Dirichlet boundary conditions in a suitable sense. Finally, we obtain the existence of a minimiser in $$L^\infty\cap BV(\Omega)$$ for the relaxed/extended energy.

The major benefit of our non-parametric approach is the validity of a-priori diameter and area bounds, which are not available in the general setting of R. Schätzle's work [Sc]. On the other hand we need to leave open most of the regularity issues.

#### References

[DGR] Klaus Deckelnick, Hans-Christoph Grunau, Matthias R\”oger, Minimising a relaxed Willmore functional for graphs subject to boundary conditions, Preprint 2015, arxiv:1503.01275.

[Sc] Reiner Schätzle, The Willmore boundary problem, Calc. Var. Partial Differential Equations, 37, 275–302, 2010.

[Si] Leon Simon, Existence of surfaces minimizing the {W}illmore functional, Comm. Anal. Geom. 1, 281–326, 1993.

Francois Hamel, Université d'Aix-Marseille & Institut Universitaire de France
Transition fronts for the Fisher-KPP equation — Slides

The standard notions of reaction-diffusion fronts can be viewed as examples of generalized transition fronts. These notions involve uniform limits, with respect to the geodesic distance, to a family of hypersurfaces which are parametrized by time. The existence of transition fronts has been proved in various contexts where the standard notions of fronts make no longer sense. Even for homogeneous equations, fronts with varying speeds are known to exist. In this talk, I will report on some recent existence results and qualitative properties of transition fronts for monostable homogeneous and heterogeneous one-dimensional equations. I will also discuss their asymptotic past and future speeds. The talk is based on some joint works with Luca Rossi [hr1,hr2].

#### References

[hr1] F. Hamel, L. Rossi, Transition fronts for the Fisher-KPP equation, Trans. Amer. Math. Soc., forthcoming.

[hr2] F. Hamel, L. Rossi, Admissible speeds of transition fronts for time-dependent KPP equations, SIAM J. Math. Anal., forthcoming.

Michael Loss, Georgia Institute of Technology
The phase diagram of the Caffarelli-Kohn-Nirenberg inequalities — Slides

The Caffarelli-Kohn-Nirenberg inequalities form a two parameter family of inequalities that interpolate between Sobolev's inequality and Hardy's inequality. The functional whose minimization yields the sharp constant is invariant under rotations. It has been known for some time that there is a region in parameter space where the optimizers that yield the sharp constant are not radial. In this talk I indicate a proof that in the remaining parameter region the optimizers are in fact radial. The proof will proceed via a well chosen flow that decrases the functional unless the function is a radial optimizer. This is joint work with Jean Dolbeault and Maria Esteban.

Angela Pistoia, Università di Roma Sapienza
Blowing-up solutions for Yamabe-type problems — Slides

The Yamabe equation is one of the most natural and well-studied second-order semilinear elliptic equations arising in geometric variational problems. The issue of the compactness of the set of solutions of the geometric Yamabe equation has been recently studied and it is strictly related to the existence of solutions blowing-up at one or more points in the manifold. In this lecture, I will review these results and present more recent works on the Yamabe problem, where solutions blowing-up at multiple (clustering and towering) points have been found.

Pavol Quittner, Comenius University
Liouville theorems for superlinear parabolic problems — Slides

Liouville theorems for scaling-invariant nonlinear parabolic problems (guaranteeing nonexistence of positive bounded entire solutions) imply universal estimates for solutions in general domains. We prove such Liouville theorems for two classes of problems. First, we consider several parabolic equations and systems with gradient structure and show that each positive bounded entire solution has to be time-independent, see [1]. Second, we consider a class of two-component parabolic systems without gradient structure and show that the components of any positive bounded entire solution have to be proportional, see [2].

It is known that the universal estimates guaranteed by our Liou­vil­le theorems yield optimal blow-up rate estimates and also imply boundedness of threshold solutions lying on the borderline between global existence and blow-up. We use these universal estimates to prove the existence of positive periodic solutions of strongly cooperative parabolic Lotka-Volterra systems with equal diffusion coefficients.

#### References

[1] Quittner P., “Liouville theorems for scaling invariant superlinear parabolic problems with gradient structure,'' Math. Ann., DOI 10.1007/s00208-015-1219-7.

[2] Quittner P., “Liouville theorems, universal estimates and periodic solutions for cooperative parabolic Lotka-Volterra systems,'' Preprint arXiv:1504.07031.

Philippe Souplet, Université Paris 13
Space profile of single-point gradient blow-up on the boundary for the diffusive Hamilton-Jacobi equation — Slides

We study the asymptotic behavior of gradient blow-up solutions for the diffusive Hamilton-Jacobi equation $$u_t-\Delta u=|\nabla u|^p$$ ($$p>2$$) in planar domains, with Dirichlet boundary conditions. We will present results on the concentration of singularities of the gradient at a single point of the boundary in finite time. In particular we obtain the precise final space profile near the singularity. Interestingly, unlike in other related problems, the profile turns out to be strongly non-isotropic, being more singular in the tangential direction than in the normal direction. Joint work with Alessio Porretta.

Susanna Terracini, Università di Torino
Existence and regularity of solutions to optimal partition problems involving Laplacian eigenvalues — Slides

Let $$\Omega\subset \mathbb{R}^N$$ be an open bounded domain and $$m\in \mathbb{N}$$. Given $$k_1,\ldots,\linebreak[2] k_m\in \mathbb{N}$$, we consider a wide class of optimal partition problems involving Dirichlet eigenvalues of elliptic operators, including the following $\inf\biggl\{ \Phi(\omega_1,\ldots,\omega_m) := \sum_{i=1}^m \lambda_{k_i}(\omega_i) :\ (\omega_1,\ldots, \omega_m)\in \mathcal{P}_m(\Omega) \biggr\},$ where $$\lambda_{k_i}(\omega_i)$$ denotes the $$k_i$$-th eigenvalue of $$\bigl(-\Delta,H^1_0(\omega_i) \bigr)$$ counting multiplicities, and $$\mathcal{P}_m(\Omega)$$ is the set of all open partitions of $$\Omega$$, namely $\mathcal{P}_m(\Omega)=\bigl\{(\omega_1,\ldots,\omega_m) :\ \omega_i\subset \Omega \text{ open},\ \forall i\neq j, \omega_i\cap \omega_j=\emptyset \bigr\}.$ We prove the existence of an open optimal partition $$(\omega_1,\ldots, \omega_m)$$, proving as well its regularity in the sense that the free boundary $$\bigcup_{i=1}^m \partial \omega_i\cap \Omega$$ is, up to a residual set, locally a $$C^{1,\alpha}$$ hypersurface.

In order to prove this result, we first treat some general optimal partition problems involving all eigenvalues up to a certain order. This class of problems includes the one with cost function $$\Phi_p(\omega_1,\ldots, \omega_m) := \sum_{i=1}^m \Bigl(\sum_{j=1}^{k_i} \bigl(\lambda_j(\omega_i)\bigr)^{p} \Bigr)^{1/p}$$, whose solutions approach the solutions of the original problem as $$p\to +\infty$$. The study of this new class of problems is done via a singular perturbation approach with a class of Schr\”odinger-type systems which models competition between different groups of possibly sign-changing components. An optimal partition appears in relation with the nodal set of the limiting components, as the competition parameter becomes large. This is a joint work with Miguel Ramos and Hugo Tavares.

Jean Van Schaftingen, Université catholique de Louvain
Nonlinear Schrödinger equation under a magnetic field kept in the foreground — Slides

I shall present results on standing waves for the nonlinear Schrö­din­ger equation in the strong magnetic field régime. In contrast with the well-studied mean magnetic field régime in which the behaviour of solutions in the semiclassical limit is governed by the electric field alone, the concentration of solutions depends on both the electric and magnetic fields and allows to recover a semiclassical formula for the Lorentz electromagnetic field. I shall explain how we have overcome the lack of knowledge of the solutions of the limiting problem.

I shall also describe the symmetry and decay properties of the solutions of the limiting problem when the magnetic field is not too large. We have obtained these properties by an asymptotic analysis that we could perform along a family of functional spaces that depend on the magnetic field.

#### References

[BNVS] D. Bonheure, M. Nys and J. Van Schaftingen, Properties of groundstates of nonlinear Schr\”odinger equations under a constant magnetic field, in preparation.

[DCV] J. Di Cosmo and J. Van Schaftingen, Semiclassical stationary states for nonlinear Schrödinger equations under a strong external magnetic field, J. Differential Equations 259 (2015), no. 2, 596–627.

Juncheng Wei, University of British Columbia & Chinese University of Hong Kong
On Type II Singularity Formation of Harmonic Map Flow — Slides

We consider the following harmonic map $u_t=\Delta u + |\nabla u|^2 u$ $u: \Omega \to S^2$ where $$\Omega$$ is a general two dimensional domain. We show that the Type II blow up rate $\frac{T-t}{\log^2 (T-t)}$ is generic and stable. We also discuss the multiple bubbling and reverse bubbling. We develop gluing techniques for parabolic equations. (Joint work with Manuel del Pino and Juan Davila).

## Session speakers

### Schrödinger equations

chaired by Silvia Cingolani, Louis Jeanjean, Marco Squassina
Riccardo Adami, Politecnico di Torino
Ground states for the NLS on quantum graphs: the $$L^2$$-critical case — Slides

The search for ground states (namely, minimizers of the energy with fixed $$L^2$$-norm) for the quintic NLS on the line has a particular feature: solutions exist only for a particular value of the $$L^2$$-norm. We consider the same problem on some quantum graphs and show a much richer structure of the set of ground states. Such a structure strongly depends on the topology of the graph. This is a joint work with E. Serra and P. Tilli.

Bound and Ground states to systems of coupled NLS-KdV equations — Slides

#### References

[BM] T. Bartsch, J. Mederski, Ground and bound state solutions of semilinear time-harmonic Maxwell equations in a bounded domain, Arch. Ration. Mech. Anal. 215 (1), (2015), 283–306.

[M] J. Mederski Ground states of time-harmonic semilinear Maxwell equations in $$\mathbb{R}^3$$ with vanishing permittivity, arXiv:1406.4535.

Wolfgang Reichel, Karlsruhe Institute of Technology
Standing waves for nonlinear curl-curl wave equations — Slides

We consider a class of nonlinear wave equations arising in the theory of nonlinear electromagnetics. The equations may be quasilinear $\nabla\times \nabla \times U + \partial_t^2 \bigl(V(x) U + \Gamma(x) |U|^{p-1}U\bigr) = 0$ or semilinear $\nabla\times \nabla \times U + V(x) \partial_t^2 U + \Gamma(x) |U|^{p-1}U = 0$ for given coefficients $$V,\Gamma: \mathbb{R}^3\to \mathbb{R}$$ and exponents $$p>1$$. We prove the existence of some classes of standing waves. Depending on the model the solutions $$U(x,t)$$ defined for $$(x,t)\in \mathbb{R}^3\times \mathbb{R}$$ may take either complex or real vector-values. In the case of monochromatic complex valued standing waves we use variational methods. In the case of real valued so-called breathers we use a specific, partly explicit construction based on a simple phase-plane argument.

Simone Secchi, Università di Milano Bicocca
Some recent results on pseudo-relativistic Hartree equations — Slides

In this talk I will present some results on nonlinear fractional field equations, and in particular on the pseudo-relativistic Hartree equation $\sqrt{-\varepsilon^2 \Delta+m^2} \; u + Vu = \left( I_\alpha * |u|^p \right) |u|^{p-2} u \quad \hbox{in $$\mathbb{R}^N$$},$ where $$V \colon \mathbb{R}^N \to \mathbb{R}$$ is an external scalar potential and $$I_\alpha(x) = |x|^{\alpha -N}$$ up to some multiplicative constant. This is a joint work with Silvia Cingolani.

Kazunaga Tanaka, Department of Mathematics, Waseda University
Singular perturbation problems for nonlinear elliptic problems

In this talk we study singular perturbation problems for a system of nonlinear Schrödinger equations and for a nonlinear elliptic problem in perturbed cylindrical domains. We show the existence of a family of solutions which concentrate at a prescribed part of domain — especially we consider the situation where the prescribed part is corresponding to “local maxima''.

### Higher order PDEs and systems

chaired by Ederson Moreira dos Santos, Benedetta Noris, Hugo Tavares
Elvise Berchio, Politecnico di Torino
Structural instability of nonlinear plates modelling suspension bridges — Slides

We analyze the structural instability arising in a nonlinear fourth order hyperbolic-like problem which describes the dynamics of suspension bridges. With a finite dimensional approximation, two different kinds of oscillating modes are found: longitudinal and torsional. We show the existence of an energy threshold where stability is lost and part of the energy is suddenly transferred from longitudinal to torsional modes giving rise to wide torsional oscillations. An explanation of the precise mechanism which governs the energy transfer is also provided. The results stated are contained in the joint papers [BFG] and [BGZ].

#### References

[BFG] E. Berchio, A. Ferrero, F. Gazzola, Structural instability of nonlinear plates modelling suspension bridges: mathematical answers to some long-standing questions, arXiv:1502.05851

[BGZ] E. Berchio, F. Gazzola, C. Zanini, Which residual mode captures the energy of the dominating mode in second order Hamiltonian systems?, arXiv:1410.2374

Norman Dancer, University of Sydney & Swansea University
Multiplicity of Solutions of Certain Systems — Slides

Suppose $$D$$ is a bounded domain in 3 dimensional space.

We discuss how the shape of $$D$$ can affect the multiplicity of solutions for large $$k$$ of \begin{equation*} \begin{cases} -\Delta u= a u -k u v^2 & \text{in } D, \\ -\Delta v= d v - k u^2 v & \text{in } D, \\ u,v\geq0 & \text{in } D, \\ u=v=0 & \text{on } \partial D. \end{cases} \end{equation*}

Djairo de Figueiredo, Universidade Estadual de Campinas
On semipositone problems for the p-Laplacian — Slides

The existence of positive solutions to a semipositone p-Laplacian problem is proved combining variational methods and a priori estimates.

Boyan Sirakov, PUC - Rio de Janeiro
A priori bounds for elliptic PDEs with natural growth in the gradient — Slides

We present new results and methods for proving a priori bounds and existence/multiplicity result for the Dirichlet problem for a general class of elliptic operators in which the first and the second order terms have the same scaling with respect to dilations.

Nicola Soave, Justus-Liebig-Universität Gießen
Geometric aspects and asymptotic analysis in phase separation of coupled elliptic equations — Slides

We consider a family of positive solutions to the system of $$k$$ components $-\Delta u_{i,\beta} = f(x, u_{i,\beta}) - \beta u_{i,\beta} \sum_{j \neq i} a_{ij} u_{j,\beta}^2 \qquad \text{in $$\Omega$$},$ where $$\Omega \subset \mathbb R^N$$ with $$N \ge 2$$. It is known that uniform bounds in $$L^\infty$$ of $$\{\mathbf{u}_{\beta}\}$$ imply convergence of the densities to a segregated configuration, as the competition parameter $$\beta$$ diverges to $$+\infty$$. In this paper we establish sharp quantitative point-wise estimates for the densities around the interface between different components, and we characterize the asymptotic profile of $$\mathbf{u}_\beta$$ in terms of entire solutions to the limit system $\Delta U_i = U_i \sum_{j\neq i} a_{ij} U_j^2.$ Moreover, we develop a uniform-in-$$\beta$$ regularity theory for the interfaces. This is a joint work with Alessandro Zilio (EHESS, Paris).

### Parabolic PDEs

chaired by Juraj Foldes, Alberto Saldaña
Marek Fila, Comenius University
Positive solutions of an elliptic equation with a dynamical boundary condition — Slides

We consider a semilinear elliptic equation in the half-space. On the boundary, a linear dynamical boundary condition is imposed. We present sharp results on existence and non-existence of positive solutions. We also study the large-time behavior of small solutions. This is a joint work with Kazuhiro Ishige and Tatsuki Kawakami.

#### References

[FIK1] Fila M., Ishige K., Kawakami T., “Large-time behavior of solutions of a semilinear elliptic equation with a dynamical boundary condition,'' Adv. Differential Equations 18, 69–100 (2013).

[FIK2] Fila M., Ishige K., Kawakami T., “Large-time behavior of small solutions of a two-dimensional semilinear elliptic equation with a dynamical boundary condition,'' Asymptotic Analysis 85, 107–123 (2013).

[FIK3] Fila M., Ishige K., Kawakami T., “Existence of positive solutions of a semilinear elliptic equation with a dynamical boundary condition,'' Calc. Var. Partial Differential Equations, DOI: 10.1007/s00526-015-0856-8.

[FIK4] Fila M., Ishige K., Kawakami T., “Minimal solutions of a semilinear elliptic equation with a dynamical boundary condition,'' preprint.

Isabella Ianni, Seconda Università degli Studi di Napoli
Existence of sign-changing solutions to the Lane-Emden problem via parabolic approach — Slides

We consider the semilinear elliptic Lane-Emden problem $$\label{problemAbstract} \begin{cases} -\Delta u= |u|^{p-1}u&\text{in }\Omega,\\ u=0 &\text{on }\partial \Omega, \end{cases} \tag{$$\mathcal E_p$$}$$ where $$p>1$$ and $$\Omega$$ is a smooth bounded simply connected domain of $$\mathbb R^2$$. By imposing some symmetry on the domain we show the existence of sign-changing solutions $$u_p$$ having two nodal regions and whose nodal line doesn't touch the boundary. The proof relies on a dynamical approach which combines the use of the parabolic flow associated to the problem together with some geometrical arguments.

We also discuss the blow-up in finite time of the solutions of the parabolic problem associated to \eqref{problemAbstract} having initial condition $$u_0=\lambda u_p$$ with $$\lambda\in \mathbb R$$.

The results presented are obtained in collaboration with F. De Mar­chis and F. Pacella (Università Sapienza, Roma).

Christian Stinner, Technische Universität Kaiserslautern
Finite time versus infinite time blowup for a fully parabolic Keller-Segel system — Slides

Several variants of the Keller-Segel model are used in mathematical biology to describe the evolution of cell populations due to both diffusion and chemotactic movement. In particular, the emergence of cell aggregation is related to blowup of the solution. Critical nonlinearities with respect to the occurrence of blowup had been identified for a quasilinear parabolic-parabolic Keller-Segel system, but it was not known whether the solution blows up in finite or infinite time. We show that indeed both blowup types appear and that the growth of the chemotactic sensitivity function is essential to distinguish between them. We provide conditions for the existence of each blowup type and discuss their optimality. An important ingredient of our proof is a detailed analysis of the Liapunov functional. This is a joint work with T. Cieślak (Warsaw).

Tuoc Van Phan, University of Tennessee
Gradient estimates and global existence of smooth solutions to a cross-diffusion system — Slides

We investigate the global time existence of smooth solutions for the Shigesada-Kawasaki-Teramoto system of cross-diffusion equations of two competing species in population dynamics. If there are self-diffusion in one species and no cross-diffusion in the other, we show that the system has a unique smooth solution for all time in bounded domains of any dimension. We obtain this result by deriving global $$W^{1,p}$$-estimates of Calderón-Zygmund type for a class of nonlinear reaction-diffusion equations with self-diffusion. These estimates are achieved by employing Caffarelli-Peral perturbation technique together with a new two-parameter scaling argument.

The talk is based on the joint work with L. Hoang (Texas Tech) and T. Nguyen (U. of Akron).

Michael Winkler, Universität Paderborn
How far can chemotactic cross-diffusion enforce exceeding carrying capacities? — Slides

We consider variants of the Keller-Segel system of chemotaxis which contain logistic-type source terms and thereby account for proliferation and death of cells. We briefly review results and open problems with regard to the fundamental question whether solutions \vadjust{\pagebreak[2]}exist globally in time or blow up. The primary focus will then be on the prototypical parabolic-elliptic system \begin{equation*} \begin{cases} u_t=\varepsilon u_{xx} - (uv_x)_x + ru - \mu u^2, \$1mm] 0= v_{xx}-v+u, \end{cases} \end{equation*} in bounded real intervals. The corresponding Neumann initial-boundary value problem, though known to possess global bounded solutions for any reasonably smooth initial data, is shown to have the property that the so-called carrying capacity $${r}/{\mu}$$ can be exceeded dynamically to an arbitrary extent during evolution in an appropriate sense, provided that $$\mu<1$$ and that $$\varepsilon>0$$ is sufficiently small. This is in stark contrast to the case of the corresponding Fisher-type equation obtained upon dropping the term $$-(uv_x)_x$$, and hence reflects a drastic peculiarity of destabilizing action due to chemotactic cross-diffusion, observable even in the simple spatially one-dimensional setting. Numerical simulations underline the challenge in the analytical derivation of this result by indicating that the phenomenon in question occurs at intermediate time scales only, and disappears in the large time asymptotics. # Communications The contributed talks will be scheduled on Monday September 7. • Antonio Azzollini, Positive radial solutions of a prescribed mean curvature equation in Lorentz-Minkowski space — Slides We are interested in finding radial solutions for the problem $$\begin{cases} \nabla \cdot \Bigl[\frac{\nabla u}{\sqrt{1-|\nabla u|^2}}\Bigr] + u^p = 0, \\ u(x)>0& \hspace{-6em}\text{in }\IR^N,\\ u(x) \to 0 ,& \hspace{-6em}\text{as } |x|\to \infty, \end{cases} \label{eq:mean}\tag{$${\cal P}$$}$$ where $$N\ge 3$$ and $$1<p$$. Such solutions are known as radial ground states. The equation at the first line is quasilinear and involves the so called mean curvature operator in the Lorentz-Minkowski space which has been object of investigation in some recent papers. Problem \eqref{eq:mean} was first studied in [BDD] where a finite energy solution was found in the supercritical case. Inspired by some questions left as open problems by [BDD], we will study existence and, in the case, multiplicity of radial ground states in both the subcritical and the supercritical case. Some considerations on the behaviour at infinity of solutions will be produced. #### References [BDD] D. Bonheure, A. Derlet, C. De Coster, Infinitely many radial solutions of a mean curvature equation in Lorentz-Minkowski space, Rend. Istit. Mat. Univ. Trieste 44 (2012), 259–284. • Joanna Balbus, Average conditions for permanence in nonautonomous competitive systems with nonlocal dispersal — Slides In this talk we consider the nonlinear evolution system \begin{equation*} \begin{split} \frac{\partial u_i}{\partial t} = \rho_i\int_{\Omega} K_i(x,y)u_i(t,y) \intd y + f_i(t,x,u_1\dots,u_N)u_i, \\ t > 0, x \in \overline{\Omega}, i = 1,\dotsc, N, \end{split} \end{equation*} where $$u_i(t,x)$$ is the density of the $$i$$th species at time $$t$$ and spatial location $$x\in \bar{\Omega}$$ and $$\Omega$$ is a compact spatial region, $$\rho_i>0$$ is the dispersal rates of the $$i$$th species, $$f_i(t,x,u_1, \dotsc, u_N)$$ is the local per capita growth rate of the $$i$$th species. Applying Ahmad and Lazers definitions of lower and upper averages of a function and using the sub- and supersolution methods for PDEs we give sufficient conditions for permanence in such models. Moreover we allow the intrinsic growth rates to be negative. • Sara Barile, Existence of constant-sign and changing-sign ground states for some classes of $$p\&q$$ elliptic problems with vanishing potentials and critical nonlinearities — Slides We aim to present recent existence results obtained in [BarileFigueiredo] to positive, negative and nodal ground state solutions to the following class of quasilinear elliptic problems \begin{equation*} \tag{$$P$$} -\div \bigl( a(|\nabla u|^{p})|\nabla u|^{p-2}\nabla u \bigr) + V(x) b(|u|^p) |u|^{p-2} u = K(x) f(u) \end{equation*} in $$\mathbb{R}^N$$ where $$N \geq 3$$, $$2 \leq p < N$$, $$a$$, $$b$$ and $$f$$ are regular real functions sati­sfying suitable growth and monotonicity conditions and $$V$$ and $$K$$ are positive continuous potentials belonging to a general class which includes potentials vanishing at infinity (“zero mass case''). The choice of this class allows us to exploit some compact embeddings in weighted Sobolev spaces (Hardy-type inequalities) involving $$V$$ and $$K$$ introduced in [OpicKufner] and to overcome the loss of compactness of the problem due to the unboundedness of the domain and the critical growth at infinity of the nonlinearity. By means of Mountain Pass Theorem, Hölder regularity and Harnack's inequality we get problem $$(P)$$ possesses a positive and a negative solution with energy levels equal to the Mountain Pass levels related to the energy functional $$J$$ associated to the problem, that is, a positive and a negative ground state solution to $$(P)$$. Furthermore, we prove the existence of a least energy solution $$w$$ of $$(P)$$ which is nodal or changing sign in $$\mathbb{R}^N$$, i.e. $$w= w^+ + w^-$$ with $$w^{+} = \max\{w, 0\} \neq 0$$, $$w^{-}= \min\{ w,0\} \neq 0$$ in $$\mathbb{R}^N$$ and the supports of $$w^+$$ and $$w^-$$ are disjoint. Indeed, by means of a minimization argument we show the existence of a $$w \in \mathcal{M}$$ such that \begin{equation*} J(w) = \min_{v \, \in \, \mathcal{M}}J(v), \qquad \mathcal{M}= \Bigl\{v\in \mathcal{N}: v^{\pm} \neq 0, \left\langle J'(v^{\pm}),v^{\pm}\right\rangle=0 \Bigl\} \end{equation*} with $$\mathcal{M}$$ the subset of the Nehari manifold $$\mathcal{N}$$ containing all changing sign solutions of $$(P)$$. The main difficulty facing this problem is due to the fact that $$\mathcal{M}$$ is not a submanifold of the functional space on which we work, thus we cannot talk about vector fields on $$\mathcal{M}$$ and deformations cannot be easily construct on $$\mathcal{M}$$. However, following the arguments in [Weth] based on a suitable quantitative deformation lemma (without Palais-Smale condition), we are able to prove that every minimizer on $$\mathcal{M}$$ of $$J|_{\mathcal{M}}$$ is a critical point of $$J$$. Finally we show that such solution has precisely two nodal domains or changes sign exactly once in $$\mathbb{R}^N$$. These results extend previous works to a larger class of $$p\&q$$ type problems that include $$- \Delta_p + V(x)$$ or $$- \Delta_p - \Delta_q + V(x)$$, $$2 \leq p < q < N$$, whose interest has increased considerably in the literature of the last years due to applications in physics and related sciences and to mathematical techniques used such as variational and topological arguments. #### References [BarileFigueiredo] S. Barile and G. M. Figueiredo, Existence of least energy positive, negative and nodal solutions for a class of $$p\&q$$-problems with potentials vanishing at infinity, J. Math. Anal. Appl. 427 (2015), 1205–1233. [Weth] T. Bartsch, T. Weth and M. Willem, Partial symmetry of least energy nodal solutions to some variational problems, J. Anal. Math. 96 (2005), 1–18. [OpicKufner] B. Opic and A. Kufner, Hardy-Type Inequalities, Pitman Res. Notes Math. Ser., vol. 219, Longman Scientific and Technical, Harlow, 1990. • Sonia Benmakhlouf, Existence and boundary behavior of solutions for a nonlinear Dirichlet problem in the annulus — Slides In this paper we mainly study the following semilinear Dirichlet problem $$-\Delta u=q(x)f(u),\;u>0,\;x\in \Omega ,$$ $$u_{|\partial \Omega }=0,$$ where $$\Omega$$ is an annulus in $$\mathbb{R}^{n},\;\left( n\geq 2\right) .$$ The function $$f$$ is nonnegative in $$\mathcal{C}^{1}(0,\infty )$$ and $$q\in\mathcal{C}_{loc}^{\gamma }(\Omega ),\;(0<\gamma <1),$$ is positive and satisfies some required hypotheses related to Karamata regular variation. We establish the existence of a positive classical solution to the problem. We also give the boundary behavior of such solution. • Davide Buoso, A new Steklov-type problem for the biharmonic operator — Slides In this talk we formulate a Steklov-type eigenvalue problem for the biharmonic operator. This problem should not be confused with the well-known problem considered e.g., in [BG,FGW]. We provide a physical motivation for our problem, highlighting the relations with the Neumann problem for the biharmonic operator via mass concentration arguments. Then we compute Hadamard-type formulas for the shape derivatives of the eigenvalues and use them to prove that balls are critical domains for the eigenvalues under volume constraint. Finally we prove a quantitative isoperimetric inequality and, as a corollary, that the ball maximizes the first non-zero eigenvalue among domains with the same volume. Based on the papers [BP1,BP2]. #### References [BG] D. Bucur, F. Gazzola, The first biharmonic Steklov eigenvalue: positivity preserving and shape optimization, Milan J. Math. 79 (2011), no. 1, 247-258. [BP1] D. Buoso, L. Provenzano, A few shape optimization results for a biharmonic Steklov problem, J. Differential Equation, doi:10.1016/j.jde.2015.03.013. [BP2] D. Buoso, L. Provenzano, On the eigenvalues of a biharmonic Steklov problem, Integral Methods in Science and Engineering: Theoretical and Computational Advances, Springer, 2015. [FGW] A. Ferrero, F. Gazzola, T. Weth, On a fourth order Steklov eigenvalue problem, Analysis (Munich) 25 (2005), no. 4, 315-332. • Beomjun Choi, Multi-Dimensional Fast Diffusion Equation via Diffusive Scaling of Generalized Carleman Kinetic Equation — Slides In this talk, we investigate certain generalized Carleman kinetic equations for $$n\ge2$$ and prove convergence towards the solution of Fast Diffusion Equation(FDE) or Porous Medium Equation $$u_t=\Delta u^m$$ ($$0\le m\le2$$) in its diffusive hydrodynamic limit. Using comparison principle of system combined with fixed speed propagation property of transport equation, we create a new barrier argument for this hyperbolic system. It is crucial to construct explicit local sub and solution of system and this is done by employing an ansatz from the second order asymptotic expansion. This allow us prove diffusive limit toward subcritical FDE, which is thought to be difficult with previous method due to the lack of mass conservation. Moreover, we can also prove convergence with growing initial data in slow diffusion range (including $$n=1$$), which was also unknown before. This is a joint work by Beomjun Choi and Ki-Ahm Lee. • Tomasz Cieślak, Chemorepulsion- the role of the sign In my talk I will review both, known results, as well as recent computations related to the fully parabolic problem of chemorepulsion. Che­mo­re­pul­sion is described by a system of two parabolic equations similar to the Keller-Segel system, however the sign in the upper equation differs. Hence one expects that cells and a chemical substance being produced by them repel each other and a unique solution is bounded and can be prolonged for an arbitrary time. This result holds in 2d (a common result with Ph. Laurençot and C. Morales-Rodrigo), moreover global-in-time weak solutions can be constructed in dimensions $$3$$ and $$4$$ (again common results with PhL and CMR). The question is whether solutions in higher dimensions are regular. • Isabel Coelho, Positive solutions of a singular Minkowski-curvature equation — Slides We discuss the existence of positive solutions for the quasilinear equation $$\label{rad} - \bigg( \frac{u'}{ \sqrt{1-|u'|^2}} \bigg)' - \frac{N-1}{r} \frac{u'}{ \sqrt{1-|u'|^2}} + a(r)u = b(r)g(u)$$ satisfying the initial condition $$u^\prime(0) = 0$$ in bounded intervals of the type $$[0,R]$$. We consider $$N\ge2$$ and that the functions $$a$$, $$b$$ and $$g$$ are smooth and $$g$$ is superlinear at infinity. Solutions of this problem correspond to the radially symmetric solutions of the $$N$$-dimensional Minkowski-curvature equation $$\label{PDE} -\div\biggl( \frac{\nabla u} {\sqrt{1 - |\nabla u|^2}}\biggr) + a(|x|) u = b(|x|)g(u)$$ in the open ball $$B_R$$ of radius $$R$$ centered at $$0$$ in $$\mathbb{R}^N$$. We prove the existence of positive decreasing solutions of \eqref{rad} satisfying $$u(R) = 0$$ or $$u'(R) = 0$$ , that is, solutions of \eqref{PDE} satisfying homogeneous Dirichlet or Neumann boundary conditions, respectively. Then we briefly discuss how to infer the existence of a positive decreasing solution of \eqref{rad} defined in $$[0,+\infty[$$ and satisfying $$u(+\infty)=0$$. Our approach combines variational and topological methods. #### References [D] Coelho, Corsato, Rivetti, Positive radial solutions of the Dirichlet problem for the Minkowski-curvature equation in a ball, Topol. Methods Nonlinear Anal. 44 (2014), no. 1, 23–39. [N] Bonheure, Coelho, De Coster, in preparation. • Matteo Cozzi, Plane-like minimizers for a non-local Ginzburg-Landau-type energy in a periodic medium — Slides We consider a non-local energy of the form \[ \mathcal{E}(u) := \frac{1}{2} \int_{\mathbb{R}^n}\!\int_{\mathbb{R}^n} \bigl| u(x) - u(y) \bigr|^2 K(x, y) \intd x \intd y + \int_{\mathbb{R}^n} W\bigl(x, u(x)\bigr) \intd x,$ where $$K: \mathbb{R}^n \times \mathbb{R}^n \to [0, +\infty]$$ is a measurable function comparable to the kernel of the fractional Laplacian of order $$2 s$$, with $$s \in (0, 1)$$, and $$W: \mathbb{R}^n \times \mathbb{R} \to [0, +\infty)$$ is a smooth double-well potential, having zeroes in $$\pm 1$$. Both $$K$$ and $$W$$ are assumed to be $$\mathbb{Z}^n$$-periodic.

For any direction $$\omega \in \mathbb{R}^n \setminus \{ 0 \}$$, we prove the existence of a class A minimizer $$u_\omega$$ of the functional $$\mathcal{E}$$, having interface $$\{ |u_\omega| < 9/10 \}$$ contained in a strip $$\{ \omega \cdot x \in [0, M_0|\omega|] \}$$ of universal width $$M_0 > 0$$. Moreover, $$u_\omega$$ enjoys a suitable periodicity/almost-pe­ri­odi­ci­ty property, in dependence of whether $$\omega$$ is rational or not.

As a result, we obtain the existence of entire solutions to the integro-differential Euler-Lagrange equation corresponding to $$\mathcal{E}$$.

This is a joint work with Prof. E. Valdinoci (WIAS, Berlin).

• Fatma Gamze Duzgun, On existence and behaviour of solution for nonlinear diffusion type equation with third type boundary value

We consider the problem \begin{multline*} \frac{\partial u}{\partial t}-\Delta u+g(x,t,u) + e(x,t) \|u\|_{L_{2}(\Omega )}(t) = h(x,t), \\ (x,t)\in Q_{T}\equiv \Omega \times (0,T) \tag{1.1} \end{multline*} \begin{equation*} u(x,0)=u_0, \quad x\in\Omega\subset \mathbb{R}^{n}, n\geq3 \tag{1.2} \end{equation*} \begin{equation*} \left. \left( \dfrac{\partial u}{\partial \eta }+a(x^{\prime },t)u\right) \right\vert _{\Sigma _{T} }=\varphi (x^{\prime },t),\ \quad (x^{\prime },t)\in \Sigma _{T}\equiv \partial \Omega \times \lbrack 0,T] , T>0\tag{1.3} \end{equation*} Here $$\Omega \subset \mathbb{R}^{n}$$, $$n\geq 3$$, is a bounded domain with sufficiently smooth boundary $$\partial \Omega$$; $$\Delta$$ denotes the Laplace operator with $$n$$-dimension ($$\Delta =\sum_{i=1}^n \frac{\partial ^{2}}{\partial x_{i\text{ }}^{2}}$$), $$g:Q_{T} \times \mathbb{R}^{1}\to \mathbb{R}^{1}$$, $$e:Q _{T} \to \mathbb{R}^{1}$$ and $$a:\Sigma _{T} \to \mathbb{R}^{1}$$ and $$u_0$$ are given functions; $$h$$, $$\varphi$$ are given generalized functions.

For the solution of a nonlinear diffusion type equation with initial and third type boundary value conditions, the existence and uniqueness theorem is proved . Also some results about the behaviour of the solution for considered problem are given.

• Giovany Figueiredo, Ground states of elliptic problems involving non homogeneous operator — Slides

We investigate the existence of ground states for functionals with nonhomogenous principal part. Roughly speaking, we show that the Nehari manifold method requires no homogeinity on the principal part of a functional. This result is motivated by some elliptic problems involving nonhomogeneous operators. As an application, we prove the existence of a ground state and infinitely many solutions for three classes of boundary value problems.

• Habib Fourti, Harmonic functions with nonlinear Neumann boundary condition and their Morse indices

We consider solutions of a nonlinear Neumann elliptic equation: \begin{equation*} \Delta u =0 \hbox{ in } \Omega ,\quad \partial u/\partial\nu = f(x,u) \text{ on } \partial\Omega, \end{equation*} where $$\Omega$$ is a bounded open smooth domain in $$\mathbb{R}^N$$, $$N\geq2$$ and $$f$$ satisfies super-linear and subcritical growth conditions and some other assumptions. We prove that $$L^{\infty}$$-bounds on solutions are equivalent to bounds on their Morse indices. This result is similar to that of A. Bahri, P. L. Lions in [BL].

To prove the a priori estimate, we used a blow-up argument which leads to deal with the following Liouville type problem \begin{equation*} (PL) \begin{cases} \Delta u=0 &\text{in } \Pi,\\ \frac{\partial u }{\partial \nu }=|u|^{p-1}u &\text{on } \partial \Pi, \end{cases} \end{equation*} where $$\Pi$$ denotes the half space $$\{x_N>0\}$$.

We prove that problem $$(PL)$$ does not possess nontrivial bounded solution with finite Morse index provided that $$1 <p <\frac{N}{N-2}$$ if $$N\geq 3$$ and $$p>1$$ if $$N=2$$.

In this talk, I will focus on the proof of this Liouville type theorem. Our method relies on careful estimates of the energy $$\int |u|^{p+1}$$ on balls or spherical annuli on the boundary, mixing the spectral information contained in the finiteness of the Morse index of $$u$$ and identities (Pohozaev identity and other identities) satisfied by solution of $$(PL)$$.

#### References

[BL] A. Bahri and P. L. Lions, Solutions of superlinear elliptic equations and their Morse indices, Comm. Pure. App. Math. 45 (1992), 1205–1215.

• Thomas Gallouët, Finite time Blow-up for a particle system modeling a 1D Keller-Segel equation with non linear diffusion — Slides

In this talk we discuss a dichotomy theorem for a particle gradient flow of homogenous reaction-diffusion functional in $$\mathbb R^N$$. This problem can be seen as a discrete and deterministic approximation of the gradient flow, in Wasserstein space, of the energy \begin{align*} {\cal F}[\rho]=\frac{1}{m-1} &\int_{\mathbb R^d}\rho^m(x)\intd x\\ &-\chi \frac{1}{d(m-1)} \iint_{\mathbb R^{d}\times \mathbb R^{d}} |x-y|^{d(1-m)}\rho(x)\rho(y)\intd x \intd y. \end{align*} It is an extension, with a non linear diffusion, of the classical Keller-Segel problem for which it is easy to prove a dichotomy theorem [BDP]. We focus on the super critical case and we show that, unless for very peculiar situations, the blow-up occurs in finite time. This result is the first one to describe precisely the super critical case for a degenerate Keller-Segel model. The best results, in the continuous settings and dimension higher than three, are obtained in [BCL].

#### References

[BDP] A. Blanchet, J. Dolbeault, and B. Perthame. Two-dimensional Keller-Segel model: optimal critical mass and qualitative properties of the solutions, Electron. J. Differential Equations, pages No. 44, 32 pp. (electronic), 2006.

[BCL] A. Blanchet, J. A. Carrillo, and P. Laurençot. Critical mass for a Patlak-Keller-Segel model with degenerate diffusion in higher dimensions, Calc. Var. Partial Differential Equations, 35(2): 133–168, 2009.

• Shoichi Hasegawa, A critical exponent for Hénon type equation on the hyperbolic space — Slides

We discuss a critical exponent with respect to the stability of solutions to an Hénon type equation on the hyperbolic space. In Euclidean space, there exists a critical exponent on stable solutions of an Hénon type equation ([DDG, F]). In particular, this exponent is called the Joseph-Lundgren exponent in case of the Lane-Emden equation. However, on the hyperbolic space, the Lane-Emden equation admits no such critical exponent on stability ([BFG14]). In this talk, we show the existence of a critical exponent of Joseph-Lundgren type for a weighted Lane-Emden equation. More precisely, we prove the nonexistence of non-trivial stable solutions of the Hénon type equation for the subcritical case. Moreover, we show that the Hénon type equation has stable, positive, and radial solutions for the supercritical case. We also state an existence of unstable radial solutions and give some properties of stable radial solutions.

#### References

[BFG14] E. Berchio, A. Ferrero, G. Grillo, Stability and qualitative properties of radial solutions of the Lane-Emden-Fowler equation on Riemannian models. J. Math. Pures Appl. (9) 102(1), 1–35 (2014)

[DDG] E. N. Dancer, Y. Du, Z. Guo, Finite Morse index solutions of an elliptic equation with supercritical exponent. J. Differential Equations 250(8), 3281–3310 (2011)

[F] A. Farina On the classification of solutions of the Lane-Emden equation on unbounded domains of $$\mathbb{R}^{N}$$. J. Math. Pures Appl. (9) 87(5), 537–561 (2007)

• Jesús Hernández, Positive and compact support solutions for some semilinear singular problems and the linear Schrödinger equation — Slides

We study the existence (or not) of positive and compact support solutions to some singular semilinear equations related with the linear Schrödinger equation. In the one-dimensional case we give a complete description of the solution set by using energy methods. In the general case we obtain partial results by using both asymptotic bifurcation and variational methods (Nehari manifolds) together with Pohozaev identity. This is joint work with J.I. Díaz and Y. Ilyasov.

• Norihisa Ikoma, Existence of positive solutions of fully nonlinear equations — Slides

This talk is based on joint work with Patricio Felmer (Universidad de Chile) and devoted to the existence and nonexistence result of positive solutions of the following fully nonlinear equations: $$\label{1} \begin{cases} - \mathcal{M}^+_{\lambda,\Lambda} (u'') + V(x) u = f(u) \quad \text{in } \mathbb{R}, \\ u > 0 \text{ in } \mathbb{R}, \quad u(x) \to 0 \text{ as } |x| \to \infty, \end{cases}$$ and $$\label{2} \begin{cases} - \mathcal{M}^-_{\lambda,\Lambda} (u'') + V(x) u = f(u) \quad \text{in } \mathbb{R}, \\ u > 0 \text{ in } \mathbb{R}, \quad u(x) \to 0 \text{ as } |x| \to \infty. \end{cases}$$ Here $$0<\lambda \leq \Lambda < \infty$$ are constants and $$\mathcal{M}^\pm_{\lambda,\Lambda}(s)$$ stand for the Pucci operators given by $$\mathcal{M}^+_{\lambda,\Lambda}(s) := \Lambda s_+ - \lambda s_-$$ and $$\mathcal{M}^-_{\lambda,\Lambda}(s) := \lambda s_+ - \Lambda s_-$$ for $$s \in \mathbb{R}$$ where $$s_+ := \max\{0,s\}$$ and $$s_- := \max\{ 0,-s \}$$. The functions $$V, f \in C^1(\mathbb{R})$$ are given and a typical model of $$f(s)$$ is a power type nonlinearity $$f(s) = |s|^{p-1}s$$ ($$1<p<\infty$$). Under some conditions for $$V(x)$$ and $$f(s)$$, we prove the existence and nonexistence of positive solutions of \eqref{1} and \eqref{2}.

• Kerime Kallı, Solvability and Long Time Behavior of Semilinear Parabolic Problem

We study some third type boundary value problems for a general semi­linear parabolic equation in divergence form: \begin{gather} \frac{\partial u}{\partial t} + Lu + g(x,t,u) = h(x,t), \quad (x,t)\in Q_{T}\equiv \Omega \times (0,T], \label{eq:d1}\\ u(x,0)=u_{0}(x), \quad x\in\Omega\subset \mathbb{R}^{n}, n\ge 3, \\ \left. \left( \frac{\partial u}{\partial \nu }+k(x,t)u\right) \right\vert _{\Gamma_{T}}=\varphi (x,t), \quad \Gamma_{T}\equiv \partial \Omega \times \lbrack 0,T],\ T>0. \label{eq:d3} \end{gather} Here $$\Omega$$ is a bounded domain with sufficiently smooth boundary $$\partial\Omega$$; $$L$$ denotes a second order linear elliptic operator: \begin{equation*} Lu := -\sum_{i,j=1}^n D_{i} \bigl(a_{ij}\left( x,t\right) D_{j}u\bigr) + \sum_{i=1}^n b_{i}(x,t) D_{i}u + c(x,t) u, \end{equation*} where $$a_{ij}$$, $$b_{i}$$ and $$c$$ are given coefficient functions ($$i,j=1,\dotsc,n$$); $$g:Q_{T} \times \mathbb{R} \to \mathbb{R}$$ and $$k:\Gamma_{T} \to \mathbb{R}$$ are given functions; $$h$$ and $$\varphi$$ are given generalized functions.

For the existence and the uniqueness of the generalized solution of problem \eqref{eq:d1}–\eqref{eq:d3} we obtain sufficient conditions for $$L,g$$ and $$k$$. Under these conditions we prove that problem \eqref{eq:d1}–\eqref{eq:d3} is uniquely solvable in corresponding spaces.

For the long time behavior of solution, we obtained the existence of the absorbing sets in two different spaces for the autonomous case of the problem.

• Anatolii Kulikov, Local bifurcations in the Kawahara-Kuramoto-Sivashinsky equation

Consider the periodic boundary value problem for the Kawahara-Kuramoto-Sivashinsky equation. Local bifurcations in the neighbourhood of the trivial solution are considered. Sufficiency conditions for the birth of spatially nonhomogeneous t-periodic solutions and invariant two-dimensional tori are obtained. In particular, a periodic boundary value problem can have a two-dimensional local attractor formed by unstable t-periodic solutions.

• Dmitrii Kulikov, Local bifurcations in the Kawahara-Kuramoto-Sivashinsky equation — Slides

Consider the periodic boundary value problem for the Kawahara-Kuramoto-Sivashinsky equation. Local bifurcations in the neighbourhood of the trivial solution are considered. Sufficiency conditions for the birth of spatially nonhomogeneous t-periodic solutions and invariant two-dimensional tori are obtained. In particular, a periodic boundary value problem can have a two-dimensional local attractor formed by unstable t-periodic solutions.

• Annunziata Loiudice, Semilinear critical problems with singular nonlinearities on Carnot groups — Slides

$$\newcommand{\GG}{\mathbb G}$$

We present regularity, existence and non-existence results for the semilinear subelliptic problem \begin{equation*} \begin{cases} -\Delta_{\GG} u = \psi ^{\alpha}(\xi)\dfrac{|u|^{p_\alpha -2}u}{d(\xi)^\alpha} + \lambda u & \text{in }\Omega ,\\ u = 0 & \text{on } \partial\Omega, \end{cases} \label{e:probl} \end{equation*} where $$\Delta_{\GG}$$ is a sub-Laplacian on a Carnot group $$\GG$$ of homogeneous dimension $$Q$$, $$\Omega$$ is an open subset of $$\GG$$, $$0\in \Omega$$, $$0<\alpha<2$$, $$p_\alpha= 2(Q-\alpha)/(Q-2)$$ is the related critical exponent, $$d=d(\xi)$$ is the natural homogeneous norm associated to the fundamental solution of $$\Delta_\GG$$ on $$\GG$$, $$\psi:=|\nabla_\GG d|$$, where $$\nabla_\GG$$ is the subelliptic gradient associated to $$\Delta_{\GG}$$, $$\lambda \in \mathbb{R}$$. These results are contained in [L2].

The variational formulation of the problem is based on suitable Har­dy-Sobolev type inequalities for subelliptic gradients.

We prove non-existence results, for starshaped domains with respect to the natural anisotropic dilations of $$\GG$$, by means of suitable Pohozaev-type identities modelled on the stratified geometry. To implement such identities in the present singular case a deep analysis of regularity of solutions is performed, which has no analogue in the Euclidean canonical setting, by suitably adapting methods in [LU]. In the existence results, a crucial tool is the knowledge of the asymptotic behavior of the appropriate Hardy-Sobolev extremals, proved in [L1].

#### References

[L2] A. Loiudice, “Critical growth problems with singular nonlinearities on Carnot groups'', preprint 2015, submitted.

[L1] A. Loiudice, “$$L^p$$-weak regularity and asymptotic behavior of solutions for critical equations with singular potentials on Carnot groups'', NoDEA Nonlinear Differential Equations and Applications 17 (2010), 575-589.

[LU] E. Lanconelli, F. Uguzzoni, “Non-existence results for semilinear Kohn-Laplace equations in unbounded domains'', Comm. Partial Differential Equations 25 (2000), 1703-1739.

• Liliane Maia, A non periodic and asymptotically linear indefinite variational problem in $$\mathbb{R}^N$$ — Slides

A nonlinear Schrödinger equation which models a light beam propagating in a saturable medium may present a sign changing potential in the linear term and lead to a semilinear elliptic equation in $$\mathbb{R}^N$$ with a potential that has a negative part, see [CS1]. We will present some recent results on the existence of nontrivial solution for \begin{equation*} -\Delta u + V(x) u = f(u) \qquad\text{in } \mathbb{R}^N, \tag{$$P$$} \end{equation*} $$N\geq 3$$, with a continuous potential $$V$$ which may change sign and non-periodic, with an asymptotic limit $$V_\infty$$ at infinity and a function $$f$$ asymptotically linear at infinity.

We do not assume any monotonicity at $$s\mapsto f(s)/s$$. For this reason we do not use projections on the Nehari manifold either apply the generalized Nehari method as in [PA1]. We apply the classical linking theorem with Cerami condition. This is possible by using the positive ground state solution $$u_0$$ of limit problem \begin{equation*} -\Delta u + V_\infty u = f(u) \qquad\text{in } \mathbb{R}^N, \tag{$$P_\infty$$} \end{equation*} projected on a infinite dimensional subspace of $${H^1(\mathbb{R}^N)}$$ with finite codimension. Moreover, it is crucial to estimate the interactions of the translated of $$u_0$$ in order to obtain the linking geometry. Furthermore, the lack of compactness due to working with a problem in the unbounded domain $$\mathbb{R}^N$$ is circumvent by an assumption of a spectral gap of the operator $$-\Delta + V$$ following the ideas of [DR].

This is a work in collaboration with José Carlos de Oliveira Jr. and Ricardo Ruviaro (UnB, Brazil).

#### References

[DR] Y. Ding and B. Ruf, Solutions of a nonlinear Dirac equation with external fields, Arch. Rational Mech. Anal., 190 (2008), 57–82.

[PA1] A. A. Pankov, Periodic nonlinear Schrödinger equation with application to photonic crystals, Milan J. Math., 73 (2005), 259–287.

[CS1] C. A. Stuart, Guidance properties of nonlinear planar waveguides, Arch. Rational Mech. Anal., 125 (1993), 145–200.

• Chaieb Majda, Combined effects in a fractional semilinear Dirichlet problem in a bounded domain

We establish an existence, uniqueness and asymptotic behavior of a positive continuous solution $$u$$ for the following nonlinear fractional boundary value problem: \begin{equation*} D^{\alpha }u(x) = a_{1}(x)u^{\sigma _{1}}(x)+a_{2}(x)u^{\sigma _{2}}, \quad x\in (0,1] \text{ and } \displaystyle\lim_{x\to 0^{+}}x^{1-\alpha } u(x) = 0, \end{equation*}where $$0<\alpha <1$$, $$\sigma _{1},\sigma _{2}\in (-1,1)$$ and $$a_{1},a_{2}$$ are positive measurable functions on $$(0,1]$$ satisfying appropriate assumptions related to Karamata regular variation theory.

• Türker Özsarı, Qualitative properties of solutions for nonlinear Schrödinger equations with nonlinear boundary conditions on the half-line — Slides

In this talk, we consider the interaction between a nonlinear focusing Neumann type boundary source, a nonlinear defocusing interior source, and a weak damping term for nonlinear Schrödinger equations posed on the infinite half line. We construct solutions with negative initial energy satisfying a certain set of conditions which blow-up in finite time in $$H^1$$-sense. We obtain a sufficient condition relating the powers of nonlinearities present in the model which allows construction of blow-up solutions. In addition to the blow-up property, we also discuss the stabilization property and the critical exponent for this model. Local and global well-posedness together with a blow-up alternative is obtained by adapting the method of Bona-Sun-Zhang [BonaSunZhang2015] to the case of inhomogeneous Neumann boundary conditions. Our results generalize the previously known theory obtained by Ackleh-Deng [AASKD] where the main equation considered was only linear.

#### References

[AASKD] A.S. Ackleh and K. Deng, On the critical exponent for the Schrödinger equation with a nonlinear boundary condition, Differential Integral Equations 17 (2004), no. 11–12, 1293–1307.

[BonaSunZhang2015] J.L. Bona, S.M. Sun, B.Y. Zhang, Nonhomogeneous boundary-value problems for one-dimensional nonlinear Schrödinger equations, preprint (2015).

• Tiago Picon, $$L^1$$ Sobolev estimates for (pseudo)-differential operators — Slides

In this lecture we show that if $$A(x,D)$$ is a linear differential operator of order $$\nu$$ with smooth complex coefficients in $$\Omega\subset\mathbb{R}^N$$ from a complex vector space $$E$$ to a complex vector space $$F$$, the Sobolev a priori estimate $\|u\|_{W^{\nu-1,N/(N-1)}}\le C \,\|A(x,D)u\|_{L^{1}}$ holds locally at any point $$x_0\in\Omega$$ if and only if $$A(x,D)$$ is elliptic and the constant coefficient homogeneous operator $$A_\nu(x_0,D)$$ is canceling in the sense of Van Schaftingen [VS4] for every $$x_0\in \Omega$$ which means that $\bigcap_{\xi\in\mathbb{R}^N\setminus\{0\}}a_\nu(x_0,\xi)[E]=\{0\}.$ Here $$A_\nu(x,D)$$ is the homogeneous part of order $$\nu$$ of $$A(x,D)$$ and $$a_\nu(x,\xi)$$ is the principal symbol of $$A(x,D)$$. This result implies and unifies the proofs of several estimates for complexes and pseudo-complexes of operators of order one or higher proved recently by other methods as well as it extends —in the local setup— the characterization of Van Schaftingen to operators with variable coefficients.

This is joint work with Jorge Hounie (Universidade Federal de São Carlos).

#### References

[VS4] J. Van Schaftingen, Limiting Sobolev inequalities for vector fields and canceling linear differential operators, J. Eur. Math. Soc. 5, no. 3, 877–921 (2013).

• Marcos Tadeu de Oliveira Pimenta, Nodal solutions of a NLS equation concentrating on lower dimensional spheres — Slides

In this work we deal with a nonlinear Schrödinger equation in dimension $$N \geq 3$$, with a subcritical power-type nonlinearity and a positive potential satisfying a local condition. We prove the existence and concentration of nodal solutions which concentrate around a $$k$$-dimensional sphere in $$\mathbb{R}^N$$, where $$1 \leq k \leq N-1$$,as a positive parameter goes to zero. The radius of such sphere is related with the local minimum of a function which takes into account the potential $$V$$. Variational methods are used together with the penalization technique in order to overcome the lack of compactness.

• Alessio Pomponio, On Chern-Simons-Schrödinger equations including a vortex point — Slides

The study of radial stationary states with a vortex point for a gauged Schrödinger equation in dimension 2 including the so-called Chern-Si­mons term leads to the nonlocal problem \begin{multline*} - \Delta u(x) + \left( \omega + \frac{(h_u(|x|)-N)^2}{|x|^2} + \int_{|x|}^{+\infty} \frac{h_u(s)-N}{s} u^2(s) \, \mathrm{d}s \right) u(x) \\ = |u(x)|^{p-1}u(x), \end{multline*} where \begin{equation*} h_u(r)= \frac{1}{2}\int_0^{r} s u^2(s) \, \mathrm{d}s, \end{equation*} and $$N\in \mathbb{N} \cup \{0\}$$ is the order of the vortex at the origin ($$N=0$$ corresponds to the regular case).

This problem is the Euler-Lagrange equation of a certain energy functional. In this talk we present the study of the global behavior of such functional. We show that for $$p\in(1,3)$$, the functional may be bounded from below or not, depending on $$\omega$$. Quite surprisingly, the threshold value for $$\omega$$ is explicit. From this study we prove existence and non-existence of positive solutions.

The results of this talk are part of recent joint works with Yongsheng Jiang and David Ruiz.

• Carlos Román, Large conformal metrics with prescribed sign-changing Gauss curvature — Slides

Let $$(M,g)$$ be a two dimensional compact Riemannian manifold of genus $$g(M)>1$$. Let $$f$$ be a smooth function on $$M$$ such that $f \ge 0, \quad f\not\equiv 0, \quad \min_M f = 0.$ Let $$p_1,\ldots,p_n$$ be any set of points at which $$f(p_i)=0$$ and $$D^2f(p_i)$$ is non-singular. We prove that for all sufficiently small $$\lambda>0$$ there exists a family of “bubbling” conformal metrics $$g_\lambda = \e^{u_\lambda}g$$ such that their Gauss curvature is given by the sign-changing function $K_{g_\lambda}=-f+\lambda^2.$ Moreover, the family $$u_\lambda$$ satisfies $u_\lambda(p_j) = -4\log\lambda -2\log \left (\frac 1{\sqrt{2}} \log \frac 1\lambda \right ) + O(1)$ and $\lambda^2e^{u_\lambda}\rightharpoonup8\pi\sum_{i=1}^{n} \delta_{p_i},\quad as \lambda \to 0,$ where $$\delta_{p}$$ designates Dirac mass at the point $$p$$. This is joint work with Manuel del Pino.

• Dora Cecilia Salazar Lozano, Vortex–type solutions to a magnetic nonlinear Choquard equation — Slides

We consider the stationary nonlinear magnetic Choquard equation \begin{equation*} (-\I\nabla+A(x))^2 u+W(x)u = \left( \frac{1}{|x|^{\alpha }}\ast |u|^{p}\right) |u|^{p-2}u, \quad x \in \mathbb{R}^N, \end{equation*}where $$N\geq 3,$$ $$\alpha \in (0,N)$$, $$p\in \bigl[2,\frac{2N-\alpha }{N-2}\bigr),$$ $$A:\mathbb{R}^N\rightarrow \mathbb{R}^N$$ is a magnetic potential and $$W:\mathbb{R}^N\rightarrow\mathbb{R}$$ is a bounded electric potential. We assume that both $$A$$ and $$W$$ are compatible with the action of some group $$G$$ of linear isometries of $$\mathbb{R}^N$$.

We shall present some recent results on the existence of vortex-type solutions to this equation which satisfy the symmetry condition $u(g x) = \phi(g)u(x)\quad\text{for all }g\in G,\ x \in \mathbb{R}^N,$ where $$\phi:G\rightarrow \mathbb{S}^1$$ is a given continuous group homomorphism into the unit complex numbers.

• Gaetano Siciliano, A multiplicity result for a fractional Schrödinger equation in presence of a positive potential — Slides

In this talk we consider the following class of problems in $$\mathbb R^{N}$$, with $$N>2s$$, \begin{equation*} \varepsilon^{2s} (-\Delta)^{s}u + V(z)u=f(u), \quad u(z) > 0 \end{equation*} where $$0<s<1$$, $$(-\Delta)^{s}$$ is the fractional Laplacian, $$\varepsilon$$ is a positive parameter, and the potential $$V:\mathbb{R}^N \to\mathbb{R}$$ and the nonlinearity $$f:\mathbb R \to \mathbb R$$ satisfy suitable assumptions; in particular it is assumed that $$V$$ achieves its positive minimum on some set $$M.$$ By using variational methods we prove existence and multiplicity results of solutions for sufficiently small $$\varepsilon$$. In particular the multiplicity result is obtained by means of the Ljusternick-Schnirelmann and Morse theory, and depends on the “topological complexity'' of the set $$M$$.

This is a joint work with Giovany M. Figueiredo (UFPA, Belém, Brazil).

• Lorenzo Tentarelli, NLS equation on metric graphs with localized nonlinearities — Slides

In this talk I will present some recent results on the NLS equation on metric graphs with localized nonlinearities. In particular, I will focus on two issues arising in the study of quantum graphs. The first one concernes existence or nonexistence of ground states, namely, minimizers of the NLS energy functional $E(u) := \frac{1}{2}\|u'\|_{L^2(\mathcal{G})}^2 - \frac{1}{p}\|u\|_{L^p(\mathcal{K})}^p\qquad(2<p<6),$ (where $$\mathcal{G}$$ is a non-compact metric graph with compact core $$\mathcal{K}$$), in the class of functions $$u\in H^1(\mathcal{G})$$ subject to the mass constraint $\|u\|_{L^2(\mathcal{G})}^2=\mu.$ The second one, concerns existence and multiplicity of nonlinear bound states for the NLS equation. These arise as critical points of the functional $$E$$ at higher levels that are located by some min-max procedure based on the notion of Krasnosel'skii genus.

• Minha Yoo, A drift approximation for non-linear parabolic PDEs with oblique boundary data — Slides

In this talk, we consider parabolic equations with singular drift, where the drift penalizes diffusion outside of a given space-time domain. Using only PDE arguments we show that the corresponding solutions converge to solutions of boundary value problems. In the case of divergence-form equations we show, by explicit formula, that the limiting boundary condition depends not only on diffusion operator but also on the space-time geometry of the confining domain. In particular if the domain is time-dependent we show that robin boundary data appears in the limit even for the most generic choice of diffusion and drift. Our result generalizes that of [AK]. This is a join work with I. C. Kim.

#### References

[AK] Alexander, Damon; Kim, Inwon, A drift approximation for parabolic PDEs with oblique boundary data. arxiv:1403.2778.

• Elvira Zappale, Equilibrium and Euler-Lagrange equations for hyperelastic materials — Slides

By means of duality we prove existence and uniqueness of equilibrium for energies described by integral functionals which fail to be convex.

This analysis is motivated by some physical models of elastic materials (cf. for istance [B,CD]) and the techniques generalize the methods first introduced in [AwG,GP]. A suitable Euler Lagrange equation characterizing the minimizers is derived.

#### References

[AwG] Awi, R. \& Gangbo, W.,A polyconvex integrand; Euler-Lagrange Equations and Uniqueness of Equilibrium ARMA 214 (2014), no. 1, 143–182.

[B] Beatty, M., Topics in finite elasticity: hyperelasticity of rubber materials, elastomers and biological tissues, with examples, Appl. Mech. Rv. Vol 40, no. 12 (1987) 1700–1734.

[CPZ] Carita G., Pisante G., \& Zappale E. In preparation.

[CD] Conti S.,\& Dolzmann G.,On the Theory of Relaxation in Nonlinear Elasticity with Constraints on the Determinant, ARMA, (2014), doi:10.1007/s00205-014-0835-9.

[GP] Gangbo, W. \& Van der Putten,Uniqueness of equilibrium configurations in solid crystals. SIAM Journal on Math. Anal. 32, (3) (2000) 465–492.

• Nikolaos Zographopoulos, Orbital Stability for the Schrödinger Operator Involving Inverse Square Potential — Slides

In this paper we prove the existence of orbitally stable standing waves for the critical Schrödinger operator, involving inverse square potential of the form: $$\label{original.problem} \I\psi_t+\Delta \psi+c_*\frac{\psi}{|x|^2}+|\psi|^{q-2}\psi = 0, \qquad x \in \mathbb{R}^N,$$ where $$c_* = \bigl(\frac{N-2}{2}\bigr)^2$$ is the best constant of the Hardy inequality. The approach, being purely variational, is based on the precompactness of any minimizing sequence with respect to the associated energy. Moreover, we discuss the presence of the Hardy energy term, in conjunction with the behavior of the standing waves.

This is a joint work with G. P. Trachanas, has been accepted for publication in J. Diff. Eq. and may be found at arXiv: 1504.06278.