Relationship between variational problems with norm constraints and ground state of semilinear elliptic equations in $$\mathbb {R}^2$$

IF 2.1 2区数学 Q1 MATHEMATICS Calculus of Variations and Partial Differential Equations Pub Date : 2024-05-11 DOI:10.1007/s00526-024-02710-y

Masato Hashizume

引用次数: 0

Abstract

In this paper, we investigate variational problems in $\mathbb {R}^2$ with the Sobolev norm constraints and with the Dirichlet norm constraints. We focus on property of maximizers of the variational problems. Concerning variational problems with the Sobolev norm constraints, we prove that maximizers are ground state solutions of corresponding elliptic equations, while we exhibit an example of a ground state solution which is not a maximizer of corresponding variational problems. On the other hand, we show that maximizers of maximization problems with the Dirichlet norm constraints and ground state solutions of corresponding elliptic equations are the same functions, up to scaling, under suitable setting.

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带规范约束的变分问题与 $$\mathbb {R}^2$ 中半线性椭圆方程的基态之间的关系

在本文中，我们研究了在（\mathbb {R}^2\）中具有索波列夫规范约束和狄利克特规范约束的变分问题。我们重点研究变分问题最大化的性质。关于具有 Sobolev norm 约束的变分问题，我们证明了最大值是相应椭圆方程的基态解，同时我们举例说明了基态解不是相应变分问题的最大值。另一方面，我们证明了在适当的设置下，带狄利克特准则约束的最大化问题的最大化和相应椭圆方程的基态解是相同的函数，且不受限于缩放。

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

Calculus of Variations and Partial Differential Equations 数学-数学

CiteScore

3.30

自引率

4.80%

发文量

224

审稿时长

6 months

期刊介绍： Calculus of variations and partial differential equations are classical, very active, closely related areas of mathematics, with important ramifications in differential geometry and mathematical physics. In the last four decades this subject has enjoyed a flourishing development worldwide, which is still continuing and extending to broader perspectives. This journal will attract and collect many of the important top-quality contributions to this field of research, and stress the interactions between analysts, geometers, and physicists. The field of Calculus of Variations and Partial Differential Equations is extensive; nonetheless, the journal will be open to all interesting new developments. Topics to be covered include: - Minimization problems for variational integrals, existence and regularity theory for minimizers and critical points, geometric measure theory - Variational methods for partial differential equations, optimal mass transportation, linear and nonlinear eigenvalue problems - Variational problems in differential and complex geometry - Variational methods in global analysis and topology - Dynamical systems, symplectic geometry, periodic solutions of Hamiltonian systems - Variational methods in mathematical physics, nonlinear elasticity, asymptotic variational problems, homogenization, capillarity phenomena, free boundary problems and phase transitions - Monge-Ampère equations and other fully nonlinear partial differential equations related to problems in differential geometry, complex geometry, and physics.