Superlinear Schrödinger-Kirchhoff type problems involving the fractional p-Laplacian and critical exponent

Abstract

This paper concerns the existence and multiplicity of solutions for the Schrodinger-Kirchhoff type problems involving the fractional p-Laplacian and critical exponent. As a particular case, we study the following degenerate Kirchhoff-type nonlocal problem: ||u||λ(θ-1)p[λ(-Delta;)spu+v(x)|u|p-2u]=|u|p∗s-2u+f(x,u)in ℝ ||u||λ=( λff∫ ∫ ℝN|u(x)-u(y)|p/|x-y|N+ps dxdy + ∫ ℝN V(x)jujpdx)1/p where is the fractional p-Laplacian with 0 < s < 1 < p 0 is a real parameter, 1 0 such that the above problem has m pairs of solutions for all λ ϵ (0, Λm]. For θ=ps∗/p, by using Krasnoselskii's genus theory, we get the existence of infinitely many solutions for the above problem for λ large enough. The main features, as well as the main difficulties, of this paper are the facts that the Kirchhoff function is zero at zero and the potential function satisfies the critical frequency infxϵℝ V(x) = 0. In particular, we also consider that the Kirchhoff term satisfies the critical assumption and the nonlinear term satisfies critical and superlinear growth conditions. To the best of our knowledge, our results are new even in p-Laplacian case.