Superlinear Kirchhoff-type problems of the fractional p-Laplacian without the (AR) condition

Abstract

Abstract In this paper, we study the following superlinear p-Kirchhoff-type equation: {M(∫R2N|u(x)−u(y)|p|x−y|N+psdxdy)(−△)psu(x)−λ|u|p−2u=g(x,u)in Ω,u=0in RN∖Ω, $$\begin{aligned} \textstyle\begin{cases} \mathcal{M} (\int_{\mathbb{R}^{2N}}\frac { \vert u(x)-u(y) \vert ^{p}}{ \vert x-y \vert ^{N+ps}}\,dx\,dy )(-\triangle)^{s}_{p}u(x) -\lambda \vert u \vert ^{p-2}u=g(x,u)& \mbox{in }\varOmega,\\ u=0& \mbox{in }\mathbb{R}^{N}\backslash\varOmega, \end{cases}\displaystyle \end{aligned}$$ Under suitable assumptions on g(x,u) $g(x,u)$ without the (AR) condition, the existence of infinitely many solutions for the Kirchhoff equation of a fractional p-Laplacian is obtained by using the fountain theorem. Our conclusions generalize and extend some existing results.

Keywords

• Fountain theorem
• Kirchhoff-type equation
• Fractional p-Laplacian
• Ambrosetti–Rabinowitz condition