Bound states for a stationary nonlinear Schrodinger-Poisson system with sign-changing potential in $R^3$

We study the following Schrödinger-Poisson system (P_λ){ll} -Δu + V(x)u+λϕ(x) u =Q(x)u^{p}, x\in \mathbb{R}^3 \\ -Δϕ= u^2, \lim\limits_{|x|\to +\infty}ϕ(x)=0, u>0, where $λ\geqslant0$ is a parameter, $1 < p < +\infty$, $V(x)$ and $Q(x)$ are sign-changing or non-positive functions in $L^{\infty}(\mathbb{R}^3)$. When $V(x)\equiv Q(x)\equiv1$, D.Ruiz \cite{RuizD-JFA} proved that ($P_λ$) with $p\in(2,5)$ has always a positive radial solution, but ($P_λ$) with $p\in(1,2]$ has solution only if $λ>0$ small enough and no any nontrivial solution if $λ\geqslant{1/4}$. By using sub-supersolution method, we prove that there exists $λ_0>0$ such that ($P_λ$) with $p\in(1,+\infty)$ has always a bound state ($H^1(\mathbb{R}^3)$ solution) for $λ\in[0,λ_0)$ and certain functions $V(x)$ and $Q(x)$ in $L^{\infty}(\mathbb{R}^3)$. Moreover, for every $λ\in[0,λ_0)$, the solutions $u_λ$ of $\rm (P_λ)$ converges, along a subsequence, to a solution of ($P_0$) in $H^1$ as $λ\to 0$.

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