On the Schrödinger-Poisson system with steep potential well and indefinite potential

In this paper, we study the following Schrödinger-Poisson system: $$ \left\{\aligned&-Δu+V_λ(x)u+K(x)ϕu=f(x,u)&\quad\text{in }\bbr^3,\\ &-Δϕ=K(x)u^2&\quad\text{in }\bbr^3,\\ &(u,ϕ)\in\h\times\D,\endaligned\right.\eqno{(\mathcal{SP}_λ)} $$ where $V_λ(x)=λa(x)+b(x)$ with a positive parameter $λ$, $K(x)\geq0$ and $f(x,t)$ is continuous including the power-type nonlinearity $|u|^{p-2}u$. By applying the method of penalized functions, the existence of one nontrivial solution for such system in the less-studied case $3<p\leq4$ is obtained for $λ$ sufficiently large. The concentration behavior of this nontrivial solution for $λ\to+\infty$ are also observed. It is worth to point out that some new conditions on the potentials are introduced to obtain this nontrivial solution and the Schrödinger operator $-Δ+V_λ(x)$ may be strong indefinite in this paper.

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