On Existence of Multiple Normalized Solutions to a Class of Elliptic Problems in Whole $$\mathbb {R}^N$$ Via Penalization Method

IF 1 3区数学 Q1 MATHEMATICS

Potential Analysis Pub Date : 2023-12-13 DOI:10.1007/s11118-023-10116-2

Claudianor O. Alves, Nguyen Van Thin

引用次数: 0

Abstract

In this paper we study the existence of multiple normalized solutions to the following class of elliptic problems

$$\begin{aligned} \left\{ \begin{aligned}&-\epsilon ^2\Delta u+V(x)u=\lambda u+f(u), \quad \quad \text {in }\mathbb {R}^N,\\&\int _{\mathbb {R}^{N}}|u|^{2}dx=a^{2}\epsilon ^N, \end{aligned} \right. \end{aligned}$$

where $a,\epsilon >0$, $\lambda \in \mathbb {R}$ is an unknown parameter that appears as a Lagrange multiplier, $V:\mathbb {R}^N \rightarrow [0,\infty )$ is a continuous function, and f is a continuous function with $L^2$-subcritical growth. It is proved that the number of normalized solutions is related to the topological richness of the set where the potential V attains its minimum value. In the proof of our main result, we apply minimization techniques, Lusternik-Schnirelmann category and the penalization method due to del Pino and Felmer (Calc. Var. Partial Differential Equations 4, 121–137 1996).

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论通过惩罚法解决全 $$\mathbb {R}^N$ 中一类椭圆问题的多重归一化解的存在性

本文研究了以下一类椭圆问题的多重归一化解的存在性 $$\begin{aligned}&-\epsilon ^2\Delta u+V(x)u=\lambda u+f(u)\Left\{ \begin{aligned}&-\epsilon ^2\Delta u+V(x)u=\lambda u+f(u), \quad \quad \text {in }\mathbb {R}^N,\&\int _\{mathbb {R}^{N}}|u|^{2}dx=a^{2}\epsilon ^N, \end{aligned}.\对\end{aligned}$ 其中$a,epsilon >0$, $\lambda \in \mathbb {R}$是一个作为拉格朗日乘数出现的未知参数，$V:\mathbb {R}^N \rightarrow [0,\infty )$ 是一个连续函数，f是一个具有$L^2$-次临界增长的连续函数。证明归一化解的数量与势 V 达到最小值的集合的拓扑丰富度有关。在证明我们的主要结果时，我们应用了最小化技术、Lusternik-Schnirelmann 范畴以及 del Pino 和 Felmer 的惩罚法（Calc. Var. Partial Differential Equations 4, 121-137 1996）。

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.