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

Advances in Nonlinear Analysis

article

Mingqi Xiang¹  Binlin Zhang²  Vicenţiu D. Rădulescu³ 

[1] College of Science, Civil Aviation University of China;College of Mathematics and System Science, Shandong University of Science and Technology;Faculty of Applied Mathematics, AGH University of Science and Technology, Poland and Department of Mathematics, University of Craiova

关键词: Schrödinger–Kirchhoff problem;    Fractional p–Laplacian;    Multiple solutions;    Critical exponent;    Principle of concentration compactness;   

DOI  :  10.1515/anona-2020-0021

学科分类：社会科学、人文和艺术（综合）

来源: De Gruyter

【 摘 要 】

This paper concerns the existence and multiplicity of solutions for the Schrődinger–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(−Δ)psu+V(x)| u |p−2u ]=| u |ps⋆−2u+f(x,u) in ℝN,‖ u ‖λ=(λ∫ℝ∫2N| u(x)−u(y) |p| x−y |N+psdxdy+∫ℝNV(x)| u |pdx)1/p $$\begin{align}& \left\| u \right\|_{\lambda }^{\left( \theta -1 \right)p}\left[ \lambda \left( -\Delta \right)_{p}^{s}u+V\left( x \right){{\left| u \right|}^{p-2}}u \right]={{\left| u \right|}^{p_{s}^{\star }-2}}u+f\left( x,u \right)\,in\,{{\mathbb{R}}^{N}}, \\ & {{\left\| u \right\|}_{\lambda }}={{\left( \lambda \int\limits_{\mathbb{R}}{\int\limits_{2N}{\frac{{{\left| u\left( x \right)-u\left( y \right) \right|}^{p}}}{{{\left| x-y \right|}^{N+ps}}}}dxdy+\int\limits_{{{\mathbb{R}}^{N}}}{V\left( x \right){{\left| u \right|}^{p}}dx}} \right)}^{{1}/{p}\;}} \\ \end{align}$$ where (−Δ)ps $\left( -\Delta \right)_{p}^{s}$is the fractional p –Laplacian with 0 0 is a real parameter, 1 0 such that the above problem has m pairs of solutions for all λ ∈ (0, Λ m ]. For θ=ps⋆/p, $\theta ={p_{s}^{\star }}/{p}\;,$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 inf x ∈ℝ 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.

【 授权许可】

CC BY