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Solutions for a class of quasilinear Choquard equations with Hardy–Littlewood–Sobolev critical nonlinearity
Nonlinear Analysis ( IF 1.3 ) Pub Date : 2020-04-04 , DOI: 10.1016/j.na.2020.111888
Sihua Liang , Lixi Wen , Binlin Zhang

In this article, we consider the quasilinear Choquard equation with critical nonlinearity in RN: ε2Δu+V(x)uε2uΔu2=RN|u|22μ|xy|μdy|u|22μ2u+h(x,u)u,where 0<μ<N, N3, 2μ=(2Nμ)(N2) is the critical exponent in the sense of Hardy–Littlewood–Sobolev inequality, ε is a real parameter. By using a change of variables, the quasilinear equations are reduced to a semilinear one, whose associated functionals are well defined in the usual Sobolev space and satisfy the geometric conditions of the mountain pass theorem. Under suitable assumptions on V and h, we investigate the existence and multiplicity of solutions for the above problem by using the mountain pass theorem and index theory. In order to overcome the lack of compactness, we apply the concentration-compactness principle.



中文翻译:

具有Hardy–Littlewood–Sobolev临界非线性的一类拟线性Choquard方程的解

在本文中,我们考虑了具有临界非线性的拟线性Choquard方程 [Rñ-ε2Δü+VXü-ε2üΔü2=[Rñ|ü|22μ|X-ÿ|μdÿ|ü|22μ-2ü+HXüü哪里 0<μ<ññ32μ=2ñ-μñ-2 是Hardy–Littlewood–Sobolev不等式意义上的关键指数, ε是一个实参。通过使用变量的变化,将拟线性方程简化为一个半线性方程,其相关函数在通常的Sobolev空间中得到了很好的定义,并满足了山口定理的几何条件。在适当的假设下VH,我们使用山路定理和指标理论研究上述问题的解的存在性和多重性。为了克服紧凑性的不足,我们应用了浓度-紧凑性原理。

更新日期:2020-04-04
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