Infinitely many solutions for the stationary fractional ${\varvec{p}}$-Kirchhoff problems in $\pmb {\mathbb {R}}^{{\varvec{N}}}$

Abstract

In the present paper, we investigate the existence of multiple solutions for the nonhomogeneous fractional p-Kirchhoff equation

$$\begin{aligned}&M\left( \iint \limits _{ {\mathbb {R}} ^{2N}}\frac{\left| u(x)-u(y)\right| ^{p}}{\left| x-y\right| ^{N+ps}}\mathrm{d}x\mathrm{d}y+\int \limits _{ {\mathbb {R}} ^{N}}V(x)\left| u\right| ^{p}\mathrm{d}x\right) \\&\quad \times (( -\Delta )_{p}^{s}u+V(x)\left| u\right| ^{p-2}u) =f(x,u)\text { in } {\mathbb {R}} ^{N}, \end{aligned}$$

where $\left( -\Delta \right) _{p}^{s}$ is the fractional p-Laplacian operator, $0<s<1<p<\infty$ with $sp<N$, $M: {\mathbb {R}} _{0}^{+}\rightarrow {\mathbb {R}} _{0}^{+}$ is a nonnegative, continuous and increasing Kirchhoff function, the nonlinearity $f: {\mathbb {R}} ^{N}\times {\mathbb {R}} \rightarrow {\mathbb {R}}$ is a Carathéodory function that obeys some conditions which will be stated later and $V\in C( {\mathbb {R}} ^{N}, {\mathbb {R}} ^{+})$ is a non-negative potential function. We first establish the Bartsch–Pankov–Wang type compact embedding theorem for the fractional Sobolev spaces. Then multiplicity results are obtained by using the variational method, $(S_{+})$ mapping theory and Krasnoselskii’s genus theory.