Infinitely many solutions for a fractional Kirchhoff type problem via Fountain Theorem

Abstract

In this paper, we use the Fountain Theorem and the Dual Fountain Theorem to study the existence of infinitely many solutions for Kirchhoff type equations involving nonlocal integro-differential operators with homogeneous Dirichlet boundary conditions. A model for these operators is given by the fractional Laplacian of Kirchhoff type:{M(∬R2N|u(x)−u(y)|2|x−y|N+2sdxdy)(−Δ)su(x)−λu=f(x,u)in Ωu=0in RN∖Ω, where Ω is a smooth bounded domain of RN, (−Δ)s is the fractional Laplacian operator with 0<s<1 and 2s<N, λ is a real parameter, M is a continuous and positive function and f is a Carathéodory function satisfying the Ambrosetti–Rabinowitz type condition.