A dichotomy result for a modified Schrödinger equations on unbounded domains

This article aims to investigate the existence of bounded positive solutions of problem $$(P)\{\begin{array}{ll}-\mathrm{d}\mathrm{i}\mathrm{v}(a(x,u,\mathrm{\nabla }u))+A_t(x,u,\mathrm{\nabla }u)=g(x,u)& \text{in }\mathrm{\Omega }\text{,}\\ u=0& \text{on }\mathrm{\partial }\mathrm{\Omega }\text{,}\end{array}$$ with $A_t(x,t,\xi )=\frac{\mathrm{\partial }A}{\mathrm{\partial }t}(x,t,\xi )$, $a(x,t,\xi )={\mathrm{\nabla }}_{\xi }A(x,t,\xi )$ for a given $A(x,t,\xi )$ which grows as $\mathrm{\mid }\xi {\mathrm{\mid }}^p+\mathrm{\mid }t{\mathrm{\mid }}^p$ , $p>1$, where $\mathrm{\Omega }\subseteq {\mathbb{R}}^N$, $N\ge 2$, is an open connected domain with Lipschitz boundary and infinite Lebesgue measure, eventually $\mathrm{\Omega }={\mathbb{R}}^N$, which generalizes the modified Schrödinger equation $$-\mathrm{d}\mathrm{i}\mathrm{v}((A_1^{\ast }(x)+A_2^{\ast }(x)\mathrm{\mid }u{\mathrm{\mid }}^s)\mathrm{\nabla }u)+\frac{s}{2}{A}_2^{\ast }(x)\mathrm{\mid }u{\mathrm{\mid }}^{s-2}u\mathrm{\mid }\mathrm{\nabla }u{\mathrm{\mid }}^2+u=\mathrm{\mid }u{\mathrm{\mid }}^{\mu -2}u\text{in }{\mathbb{R}}^3\text{.}$$ 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 $$\mathrm{\mid }{y}_k\mathrm{\mid }\to +\mathrm{\infty }\text{and}{\int }_{B_1(y_k)}\mathrm{\mid }{u}_k{\mathrm{\mid }}^{p}dx\ge \bar{\lambda }\text{for all }k\ge 1\text{.}$$