Non-Degeneracy and Infinitely Many Solutions for Critical SchrÖDinger-Maxwell Type Problem

IF 1 3区数学 Q1 MATHEMATICS

Potential Analysis Pub Date : 2024-01-25 DOI:10.1007/s11118-024-10123-x

Yuxia Guo, Yichen Hu, Shaolong Peng

引用次数: 0

Abstract

In this paper, we consider the following Schrödinger-Maxwell type equation with critical exponent \(-\Delta u=K(y)\Big (\frac{1}{|x|^{n-2}}*K(x)|u|^{\frac{n+2}{n-2}}\Big )u^{\frac{4}{n-2}},\quad {in}\,\, \mathbb {R}^n, \qquad \text {(0.1)}\) where the function K satisfies the assumption \(\mathcal {F}\), and \(*\) stands for the standard convolution. We first derived the non-degeneracy result for the critical Schrödinger-Maxwell equation. Then, as an application, we proved that problem Eq. (0.1) admits infinitely many non-radial positive solutions with arbitrary large energy. We believe that the various new ideas and technique computations that we used in this paper would be useful to deal with other related elliptic problems involving convolution nonlinear terms.

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临界薛定谔-麦克斯韦式问题的非退行性和无限多解

在本文中，我们考虑以下具有临界指数的薛定谔-麦克斯韦式方程（-/Delta u=K(y)\Big (\frac{1}{|x|^{n-2}}*K(x)|u|^{frac{n+2}{n-2}}\Big )u^{frac{4}{n-2}}、\quad {in}\,\mathbb {R}^n, \qquad \text {(0.1)}），其中函数 K 满足假设（\(\mathcal {F}\），而\(*\)代表标准卷积。我们首先得出了临界薛定谔-麦克斯韦方程的非退化结果。然后，作为一个应用，我们证明了问题式（0.1）中存在无限多具有任意大能量的非径向正解。我们相信，我们在本文中使用的各种新思路和计算技术将有助于处理其他涉及卷积非线性项的相关椭圆问题。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

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来源期刊

Potential Analysis 数学-数学

CiteScore

2.20

自引率

9.10%

发文量

审稿时长

>12 weeks

期刊介绍： The journal publishes original papers dealing with potential theory and its applications, probability theory, geometry and functional analysis and in particular estimations of the solutions of elliptic and parabolic equations; analysis of semi-groups, resolvent kernels, harmonic spaces and Dirichlet forms; Markov processes, Markov kernels, stochastic differential equations, diffusion processes and Levy processes; analysis of diffusions, heat kernels and resolvent kernels on fractals; infinite dimensional analysis, Gaussian analysis, analysis of infinite particle systems, of interacting particle systems, of Gibbs measures, of path and loop spaces; connections with global geometry, linear and non-linear analysis on Riemannian manifolds, Lie groups, graphs, and other geometric structures; non-linear or semilinear generalizations of elliptic or parabolic equations and operators; harmonic analysis, ergodic theory, dynamical systems; boundary value problems, Martin boundaries, Poisson boundaries, etc.