具有临界增长的分数阶Choquard-Kirchhoff型问题解的存在性

Existence of solutions for fractional Choquard-Kirchhoff type problems with critical growth

本文研究一类无界区域上分数阶Choquard-Kirchhoff型问题,此类问题源于横振动中对弦长的非局部测量引起的张力,也可用于刻画量子机械波函数的自引力坍缩.该方程的非线性项包含临界项μ(I_(a)^(*)|u|^(2_(a,s)^(*)-2)u|)和扰动项λf(x)uq-1,其中μ,λ均为正参数,2^(*)_(a,s)为分数阶Hardy-Littlewood-Sobolev临界指数,f(x)为连续函数.本文首先利用Nehari流形及Ekeland变分原理证得问题对应的能量泛函具有Palais-Smale序列,然后对参数μ的上界进行估计,在参数λ和次数q选取适当范围时借助Vitali定理及山路引理获得了问题正解的存在性及多重性.最后,当参数λ充分大时,本文利用强极大值原理及临界点定理建立了问题具有正解及无穷多对不同解的存在性定理.

In this paper,the fractional Choquard-Kirchhoff type problems on unbounded domains are consid‐ered.These problems stem from the tension arising from nonlocal measurements of length of a string during transverse vibration and can also be used to describe the self-gravitational collapse of a quantum mechanical wave function.In the problem,the critical termμ(I_(a)^(*)|u|^(2_(a,s)^(*)-2)u|)and perturbation termλf(x)uq-1 are contained in the nonlinear terms,whereμ,λare positive parameters,2^(*)_(α,s)is the fractional Hardy-LittlewoodSobolev critical exponent,and f(x)is a continuous function.First,the Palais-Smale sequences of energy functional corresponding to the problem are obtained by using the Nehari manifold and Ekeland’s variational principle.Second,the upper bound of parameterμis estimated.When appropriate ranges on the parameterλand power q are chosen,the existence and multiplicity of positive solutions of the problem are further ob‐tained by adopting the Vitali theorem and mountain pass lemma.Finally,when the parameterλis sufficiently large,by using the strong maximum principle and critical point theorem,existence theorems for the positive solutions and infinitely many pairs of different solutions of the problem are established.