Nontrivial solutions for perturbations of the \(p\)-Laplacian on unbounded domains (Q1900386)

The authors are concerned with existence of nontrivial solutions for \[ - \Delta_p u\equiv -\text{div}(|\nabla u|^{p- 2} \nabla u)= f(u)\quad \text{in }\Omega,\quad u= 0\quad \text{on } \partial\Omega,\tag{P} \] where \(\Omega\subset \mathbb{R}^N\) is of the form \(\Omega= \Omega'\times \mathbb{R}^{N- K}\) with \(\Omega'\subset \mathbb{R}^K\) a bounded smooth domain, \(N\geq 1\), \(K\geq 0\), and where \(1< p< N\). \(f: \mathbb{R}\to \mathbb{R}\) is a continuous function with subcritical growth: \[ |f(s)|\leq a_0|s|^{p- 1}+ b_0|s|^{p^*-1}\quad \forall s\in \mathbb{R},\tag{f} \] for some constants \(a_0, b_0> 0\), where \(p^*= Np/(N- p)\). Set \(F(s)= \int^s_0 f(t)dt\). The assumptions on \(f\) and \(F\) are the following: \[ \limsup_{s\to \infty} {F(s)\over |s|^q}\leq b< + \infty,\quad \text{for } p< q< p^*;\leqno{(\text{F}_1)_q} \] \[ f(s) s- pF(s)\geq a|s|^\mu> 0\quad \forall s\in \mathbb{R}\backslash \{0\}\quad \text{with } 0< \mu< p^*;\leqno{(\text{F}_2)_\mu} \] \[ \limsup_{|s|\to 0} {f(s)\over s|s|^{p- 2}}\leq M< + \infty,\leqno{(\text{f}_1)} \] \[ \lim_{|s|\to \infty} {f(s)\over |s|^{p^*-1}}= 0.\leqno{(\text{f}_2)} \] \[ \limsup_{s\to 0} {pF(s)\over |s|^p}\leq \alpha< \lambda_{1p}< \beta\leq \liminf_{|s|\to \infty} {pF(s)\over |s|^p},\tag{F\(_3\)} \] where \[ \lambda_{1p}= \inf_{0\neq u\in W^{1, p}_0(\Omega)} {\int_\Omega |\nabla u|^p dx\over \int_\Omega |u|^p dx}. \] Let \(k\) be a given number depending on \(N\): for a covering of \(\mathbb{R}^N\) by balls \((B_i)_{i\in I}\), each point of \(\mathbb{R}^N\) belongs to at most \(k\) balls. For \(N< 8\), \(k= N\). Theorem 1.1. Assume that \(F\) satisfies \((\text{f})\), \((\text{f}_1)\), \((\text{f}_2)\) and \((\text{F}_3)\), \((\text{F}_1)_q\), \((\text{F}_2)_\mu\) with \(\mu> (N/ p)(q- p)\). Then, problem (P) has a nontrivial solution \(u\in W^{1, p}_0(\Omega)\), provided that the following condition holds: \(M< \lambda_{1p}/k\). The proof uses Ekeland's variational principle; more precisely, an improvement by Schechter of the mountain-pass theorem of Ambrosetti and Rabinowitz, combined with the concentration-compactness principle of Lions. \((\text{F}_3)\) is a condition on ``crossing'' the first eigenvalue \(\lambda_{1p}\). The other hypotheses are done in order to apply the concentration-compactness principle. This kind of problem was studied earlier by several authors (e.g. Berestycki and Lions, Burton, Esteban, Lions and Strauss when \(p= 2\); Jianfu and Xiping, Schindler, Yu when \(p>2\)).