Existence and concentration of solutions for a fractional Schrödinger–Poisson system with discontinuous nonlinearity

Abstract

In this paper, we study the following fractional Schrödinger–Poisson system with discontinuous nonlinearity:ε2s(−Δ)su+V(x)u+ϕu=H(u−β)f(u),inR3,ε2s(−Δ)sϕ=u2,inR3,u>0,inR3, $$\begin{cases}^{2s}{\left(-{\Delta}\right)}^{s}u+V\left(x\right)u+\phi u=H\left(u-\beta \right)f\left(u\right),\quad \hfill & \text{in} {\mathbb{R}}^{3},\hfill \\ {\varepsilon }^{2s}{\left(-{\Delta}\right)}^{s}\phi ={u}^{2},\quad \hfill & \text{in} {\mathbb{R}}^{3},\hfill \\ u{ >}0,\quad \hfill & \text{in} {\mathbb{R}}^{3},\hfill \end{cases}$$ where ɛ > 0 is a small parameter, s∈(34,1) $s\in \left(\frac{3}{4},1\right)$ , β > 0, H is the Heaviside function, (−Δ)s u is the fractional Laplacian operator, V:R3→R $V :{\mathbb{R}}^{3}\to \mathbb{R}$ is a continuous potential and f:R→R $f :\mathbb{R}\to \mathbb{R}$ is superlinear continuous nonlinearity with subcritical growth at infinity. By using nonsmooth analysis, we investigate the existence and concentration of solutions for the above problem. Moreover, we obtain some properties of these solutions, such as convergence and decay estimate.

Keywords

• fractional schrödinger–poisson system
• discontinuous nonlinearity
• nonsmooth analysis
• 35r11
• 35a15
• 35b40
• 47g20