This article aims to investigate the existence of bounded positive solutions of problem \[ (P)\qquad \left\{ \begin{ array}{ ll} - { \rm div} (a(x,u,\nabla u)) + A_t(x,u,\nabla u) = g(x,u) &\hbox{ in $\Omega$,}\\ u\ = \ 0 & \hbox{ on $\partial\Omega$,} \end{ array}\right.\] with $A_t(x,t,\xi) = \frac{ \partial A}{ \partial t}(x,t,\xi)$, $a(x,t,\xi) = \nabla_\xi A(x,t,\xi)$ for a given $A(x,t,\xi)$ which grows as $|\xi|^p + |t|^p$ , $p > 1$, where $\Omega \subseteq \mathbb{ R}^N$, $N \ge 2$, is an open connected domain with Lipschitz boundary and infinite Lebesgue measure, eventually $\Omega = \mathbb{ R}^N$, which generalizes the modified Schr\"odinger equation \[ - { \rm div} ((A^*_1(x) + A^*_2(x)|u|^{ s}) \nabla u) + \frac{ s}2 A^*_2(x)\ |u|^{ s - 2} u\ |\nabla u|^2 + u\ =\ |u|^{ \mu-2}u \quad\hbox{ in $\mathbb{ R}^3$.} \] Under suitable assumptions on $A(x,t,\xi)$ 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 $\bar{ \lambda} > 0$ and a sequence of points $(y_k)_k \subset \mathbb{ R}^N$ exist such that \[ |y_k| \to +\infty\qquad \hbox{ and}\qquad \int_{ B_1(y_k)} |u_k|^p dx \ge \bar{ \lambda}\quad \hbox{ for all $k \ge 1$.} \]
Categories: math.AP