Non-homogeneous Schrodinger systems with sign-changing and general nonlinearities: Infinitely many solutions

Abstract

In this paper, we study the non-homogeneous nonlinear Schr\"{o}dinger system $$\left\{ \begin{array}{ll} -\triangle u_j+V_j(x) u_j=g_j(x,u_1,\cdots,u_m)+h_j(x),& x\in \Omega,\\ \\ u_j:=u_j(x)=0,& x\in \partial\Omega,\\ \\ j=1,2,\cdots,m, \end{array}\right. $$ where $\Omega\subset\mathbb{R}^{N}$ ($N\ge2$) is a bounded smooth domain, $(g_1,\cdots,g_m)$ is the gradient of $G(x,U)\in C^1(\Omega\times\mathbb{R}^m,\mathbb{R})$, $G(x,U)$ may be sign-changing, and it is super-quadratic or asymptotically-quadratic as $|U|\to\infty$. We obtain infinitely many solutions by using variational methods and perturbation methods, and we provide several typical examples to illustrate the main results. The {\bf main novelties} are as follows. (1) The nonlinearity $G$ may be sign-changing. (2) The nonlinearity $G$ is not only general, but also super-quadratic or asymptotically-quadratic at infinity and zero. (3) The nonlinearity $G$ is power-type or non-power-type. (4) We not only construct some new conditions, but also apply some conditions used in homogeneous problems to the study of non-homogeneous systems for the first time. The {\bf main difficulties} come from the following three aspects. (1) The proof of boundedness for $(PS)$ sequence of approximate functionals. (2) The detailed analysis of the asymptotic behaviors of approximate functionals. (3) The estimate of the upper and lower bounds for the minimax value sequence $\{c_k\}$ of the even function.