Existence results for a fractional Schrödinger–Poisson equation with concave–convex nonlinearity in ℝ3

Abstract

In this paper, we consider the following nonhomogeneous fractional Schrödinger–Poisson equations: false(−normalΔfalse)su+Vfalse(xfalse)u+ϕu=λffalse(x,ufalse)+gfalse(x,ufalse)0.1em0.1em0.3emin0.5emℝ3,false(−normalΔfalse)tϕ=u20.1em0.1em0.3emin0.5emℝ3, where λ ≥ 0, s ∈ (3/4, 1], t ∈ (0, 1], (− Δ)s denotes the fractional Laplacian. g(x, u) is of general 3‐superlinear growth at infinity, f(x, u) satisfies sublinear growth conditions. By means of some critical point theorems, we obtain the following results: (1) the existence of at least one solution for λ=0, (2) the existence of at least two nontrivial solutions for λ > 0 small enough, (3) the infinitely many solutions when f(x, u) is odd in u and λ > 0.