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.