Chongqing Wei, Anran Li, Multiple solutions for a class of Kirchhoff type equations with zero mass and Hardy-Littlewood-Sobolev critical nonlinearity

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DOI: 10.23952/jnva.8.2024.1.02

Volume 8, Issue 1, 1 February 2024, Pages 23-39

Abstract. In this paper, we study the multiplicity of solutions to the following Kirchhoff type equation with zero mass and Hardy-Littlewood-Sobolev critical nonlinearity
$-m(\int_{\mathbb{R}^N}|\nabla u|^2dx)\Delta u=\lambda K(x)f(u)+\left(\int_{\mathbb{R}^N}\frac{|u(y)|^{2_\mu^{*}}}{|x-y|^\mu}dy\right)|u|^{2_\mu^{*}-2}u,$ $x\in{\mathbb{R}^N},$
$u\in D^{1,2}(\mathbb{R}^N),$
where $N\geqslant3$ , $\lambda>0$ , $\mu\in(0,\min\{N,4\})$ , $2_\mu^{*}=\frac{2N-\mu}{N-2}$ is the critical exponent in the sense of Hardy-Littlewood-Sobolev inequality, and $m$ satisfies some local monotonicity conditions near zero. The nonlinearity $f$ is odd in $u$ and satisfies some classical superlinear and quasi-critical growth conditions. For any given $k\in \mathbb{N}$ , $k$ pairs of nontrivial solutions are obtained for $\lambda$ large enough by a version of the symmetric mountain pass theorem and a version of the second concentration compactness principle.

How to Cite this Article:
C. Wei, A. Li, Multiple solutions for a class of Kirchhoff type equations with zero mass and Hardy-Littlewood-Sobolev critical nonlinearity, J. Nonlinear Var. Anal. 8 (2024), 23-39.