Two nontrivial solutions for a nonhomogeneous fractional Schrödinger–Poisson equation in R 3 $\mathbb{R}^{3}$

Abstract

Abstract In this paper, we consider the following nonhomogeneous fractional Schrödinger–Poisson equations: { ( − Δ ) s u + V ( x ) u + ϕ u = f ( x , u ) + g ( x ) in R 3 , ( − Δ ) t ϕ = u 2 in R 3 , $$ \textstyle\begin{cases} (-\Delta )^{s}u+V(x)u+\phi u=f(x,u)+g(x)\quad \text{in }\mathbb{R}^{3}, \\ (-\Delta )^{t}\phi =u^{2}\quad \text{in }\mathbb{R}^{3}, \end{cases} $$ where s , t ∈ ( 0 , 1 ] $s,t\in (0,1]$ , 2 t + 4 s > 3 $2t+4s>3$ , ( − Δ ) s $(-\Delta )^{s}$ denotes the fractional Laplacian. By assuming more relaxed conditions on the nonlinear term f, using some new proof techniques on the verification of the boundedness of Palais–Smale sequence, existence and multiplicity of solutions are obtained.

Keywords

• Fractional Schrödinger–Poisson
• Variational methods
• Critical point theorem