Existence and Uniqueness of Multi-Bump Solutions for Nonlinear Schrödinger–Poisson Systems

In this paper, we study the following Schrödinger–Poisson equations: { - ε 2 ⁢ Δ ⁢ u + V ⁢ ( x ) ⁢ u + K ⁢ ( x ) ⁢ ϕ ⁢ u = | u | p - 2 ⁢ u , x ∈ ℝ 3 , - ε 2 ⁢ Δ ⁢ ϕ = K ⁢ ( x ) ⁢ u 2 , x ∈ ℝ 3 , \left\{\begin{aligned} &\displaystyle{-}\varepsilon^{2}\Delta u+V(x)u+K(x)\phi u% =\lvert u\rvert^{p-2}u,&\hskip 10.0ptx&\displaystyle\in\mathbb{R}^{3},\\ &\displaystyle{-}\varepsilon^{2}\Delta\phi=K(x)u^{2},&\hskip 10.0ptx&% \displaystyle\in\mathbb{R}^{3},\end{aligned}\right. where p ∈ ( 4 , 6 ) {p\in(4,6)} , ε > 0 {\varepsilon>0} is a parameter, and V and K are nonnegative potential functions which satisfy the critical frequency conditions in the sense that inf ℝ 3 ⁡ V = inf ℝ 3 ⁡ K = 0 {\inf_{\mathbb{R}^{3}}V=\inf_{\mathbb{R}^{3}}K=0} . By using a penalization method, we show the existence of multi-bump solutions for the above problem, with several local maximum points whose corresponding values are of different scales with respect to ε → 0 {\varepsilon\rightarrow 0} . Moreover, under suitable local assumptions on V and K, we prove the uniqueness of multi-bump solutions concentrating around zero points of V and K via the local Pohozaev identity.