具有临界或亚临界指数的 $$\mathbb {R}^N$$ 中 (p, q) - 拉普拉斯方程的多重解

IF 2.1 2区数学 Q1 MATHEMATICS

Calculus of Variations and Partial Differential Equations Pub Date : 2024-08-14 DOI:10.1007/s00526-024-02811-8

Shibo Liu, Kanishka Perera

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引用次数: 0

摘要

本文研究了以下具有临界指数的拉普拉斯方程 $$\begin{aligned} -\Delta _{p}u-\Delta _{q}u=\lambda h(x)|u|^{r-2}u+g(x)|u|^{p^{*}-2}u \quad \text {in }\mathbb {R}^{N} , \end{aligned}$$where$1<q\le p<r<p^{*}$.在利用 Lions 的集中紧凑性原理为某个常数 $c^*$建立了 $c\in (0,c^*)$ 的 $(PS)_c$ 条件之后，通过应用佩雷拉（Perera）的临界点定理，得到了 $\lambda \gg 1$ 的多解 (J Anal Math, 2023. arxiv:2308.07901)。还考虑了亚临界指数的类似问题。

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Multiple solutions for (p, q)-Laplacian equations in $$\mathbb {R}^N$$ with critical or subcritical exponents

In this paper we study the following $\left( p,q\right) $-Laplacian equation with critical exponent

$$\begin{aligned} -\Delta _{p}u-\Delta _{q}u=\lambda h(x)|u|^{r-2}u+g(x)|u|^{p^{*} -2}u \quad \text {in }\mathbb {R}^{N} , \end{aligned}$$

where $1<q\le p<r<p^{*}$. After establishing $(PS)_c$ condition for $c\in (0,c^*)$ for a certain constant $c^*$ by employing the concentration compactness principle of Lions, multiple solutions for $\lambda \gg 1$ are obtained by applying a critical point theorem due to Perera (J Anal Math, 2023. arxiv:2308.07901). A similar problem with subcritical exponents is also considered.

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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.