Schrödinger-Poisson System Involving Potential Vanishing at Infinity and Unbounded Below

This article concerns the following class of system $$\left\{ \begin{array}{lr} -\Delta u +V(x)u+\ell(x)\phi u = f(u) + \lambda|u|^{q-2}u & \text{in } \mathbb{R}^3,\\[2mm] -\Delta \phi = \ell(x)u^{2} & \text{in } \mathbb{R}^3,\\[2mm] u,\phi\in D^{1,2}(\mathbb{R}^3), \ u,\phi\geq0 & \text{in } \mathbb{R}^3, \end{array} \right.$$ where $\lambda\geq0$ and $q\geq2^*=6$ is the critical Sobolev exponent in dimension 3, the nonlinearity $f:\mathbb{R}\rightarrow \mathbb{R}$ is superlinear and has subcritical growth, $V,\ell: \mathbb{R}^3\rightarrow \mathbb{R}$ are measurable functions with $\ell\in L^2(\mathbb{R}^3)$, the potential $V$ can change sign in $\mathbb{R}^3$ and vanish at infinity, that is, $V (x) \rightarrow 0$ as $|x|\rightarrow\infty$. Our approach is based on variational method combined with Benci-Fortunato's reduction argument [ Topol. Methods Nonlinear Anal. 11 (1998) 283–293], Del Pino-Felmer's penalization technique [ Calc. Var. Partial Diff. Equations 4 (1996) 121–137] and $L^\infty$-estimate.