具有临界增长的分数阶 Kirchhoff 型方程半经典解的存在性、集中性和多解性

Abstract

Abstract:

This paper studies the following fractional Kirchhoff-type equation with critical growth

$ \left(\epsilon^{2s}a+\epsilon^{4s-3}b\int_{\mathbb{R}^{3}}|(-\Delta)^{\frac{s}{2}}u|^{2}{\rm d}x\right) (-\Delta)^{s}u+V(x)u=K(x)f(u)+|u|^{2^{*}_{s}-2}u, \ \ u\in H^{s}(\mathbb{R}^{3}), $

where $\epsilon>0$ is a small parameter, $a,b>0$ are constants, $s\in(\frac{3}{4},1)$, $2^{*}_{s}=\frac{6}{3-2s}$ is the Sobolev critical exponent, the potential functions $V,K:\mathbb{R}^{3}\to\mathbb{R}$ are nonnegative continuous functions, and $f:\mathbb{R}\to\mathbb{R}$ is a continuous but non-differentiable subcritical nonlinear term. By using the generalized Nehari manifold method introduced by [Szulkin A, Weth T. Boston: International Press, 2010], the authors prove the existence of ground state solutions and their concentration properties. Furthermore, using Ljusternik-Schnirelmann category theory, they establish a relationship between the number of solutions and the topology of the sets where the potential $V$ attains its minimum and $K$ attains its maximum.

Key words: fractional Kirchhoff equation, critical growth, ground state solution, Ljusternik-Schnirelmann category theory

CLC Number:

O175.23

Lun Guo, Wentao Huang, Huifang Jia, Zheng Pan. Existence, Concentration and Multiplicity of Semiclassical Solutions for a Fractional Kirchhoff Equation with Critical Growth[J].Acta mathematica scientia,Series A, 2026, 46(4): 1634-1666.

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References 46

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Existence, Concentration and Multiplicity of Semiclassical Solutions for a Fractional Kirchhoff Equation with Critical Growth

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