Multiple solutions for nonhomogeneous Schrödinger–Kirchhoff type equations involving the fractional p-Laplacian in $${\mathbb {R}}^N$$ R N

Abstract

In this paper we investigate the existence of multiple solutions for the nonhomogeneous fractional p-Laplacian equations of Schrodinger–Kirchhoff type $$\begin{aligned} M\left( \iint _{R^{2N}}\frac{|u(x)-u(y)|^p}{|x-y|^{N+ps}}dxdy\right) (-\varDelta )^s_pu+V(x)|u|^{p-2}u=f(x,u)+g(x) \end{aligned}$$ in $${\mathbb {R}}^N$$ , where $$(-\varDelta )^s_p$$ is the fractional p-Laplacian operator, with $$0<s<1<p<\infty $$ and $$ps<N$$ , the nonlinearity $$f:{\mathbb {R}}^N\times {\mathbb {R}}\rightarrow {\mathbb {R}}$$ is a Caratheodory function and satisfies the Ambrosetti–Rabinowitz condition, $$V:{\mathbb {R}}^N\rightarrow {\mathbb {R}}^+$$ is a potential function and $$g:{\mathbb {R}}^N\rightarrow {\mathbb {R}}$$ is a perturbation term. We first establish Batsch–Wang type compact embedding theorem for the fractional Sobolev spaces. Then multiplicity results are obtained by using the Ekeland variational principle and the Mountain Pass theorem.