A dichotomy result for a modified Schrödinger equations on unbounded domains

Abstract

This article aims to investigate the existence of bounded positive solutions of problem \[ (P)\qquad \left\{ \begin{array}{ll} - {\rm div} (a(x,u,\nabla u)) + A_t(x,u,\nabla u) = g(x,u) &\hbox{in $\hat{I}\text{c◯}$,}\\ u\ = \ 0 & \hbox{on $\partial \hat{I}\text{c◯}$,} \end{array}\right.\] with $A_t(x,t,\hat{I}¾)=\frac{\partial A}{\partial t}(x,t,\hat{I}¾)$, $a(x,t,\hat{I}¾)={\nabla }_{\hat{I}}¾A(x,t,\hat{I}¾)$ for a given $A(x,t,\hat{I}¾)$ which grows as $|\hat{I}¾{|}^p+|t{|}^p$ , $p>1$, where $\hat{I}\text{c◯}\subseteq {\mathbb{R}}^N$, $N\ge 2$, is an open connected domain with Lipschitz boundary and infinite Lebesgue measure, eventually $\hat{I}\text{c◯}={\mathbb{R}}^N$, which generalizes the modified Schrödinger equation \[ - {\rm div} ((A^*_1(x) + A^*_2(x)|u|^{s}) \nabla u) + \frac{s}2 A^*_2(x)\ |u|^{s - 2} u\ |\nabla u|^2 + u\ =\ |u|^{μ-2}u \quad\hbox{in ${\mathbb{R}}^3$.} \] Under suitable assumptions on $A(x,t,\hat{I}¾)$ and $g(x,t)$, problem $(P)$ has a variational structure. Then, even in lack of radial symmetry hypotheses, one bounded positive solution of $(P)$ can be found by passing to the limit on a sequence $(u_k{)}_k$ of bounded solutions on bounded domains. Furthermore, if stronger hypotheses are satisfied, either such a solution is nontrivial or a constant $\bar{\hat{I}}»>0$ and a sequence of points $(y_k{)}_k\subset {\mathbb{R}}^N$ exist such that \[ |y_k| \to +\infty\qquad \hbox{and}\qquad \int_{B_1(y_k)} |u_k|^p dx \ge \barλ\quad \hbox{for all $k\ge 1$.} \]