Existence of infinitely many small solutions for sublinear fractional Kirchhoff-Schrodinger-Poisson systems

Abstract

We study the Kirchhoff-Schrödinger-Poisson system m([u]2α) (-Δ)α u + V(x)u + k(x)φu = ƒ(x, u), x ∈ ℝ3, (-Δ)β φ = k(x)u2, x ∈ ℝ3, where [∙]α denotes the Gagliardo semi-norm, (-Δ)α denotes the fractional Laplacian operator with α, β ∈ (0, 1], 4α + 2β ≥ 3 and m : [0, +∞) → [0, +∞) is a Kirchhoff function satisfying suitable assumptions. The functions V(x) and k(x) are nonnegative and the nonlinear term ƒ(x, s) satisfies certain local conditions. By using a variational approach, we use a Kajikiya's version of the symmetric mountain pass lemma and Moser iteration method to prove the existence of infinitely many small solutions.