无界域中的索波列夫紧凑嵌入及其在椭圆方程中的应用

IF 1.2 3区数学 Q1 MATHEMATICS

Journal of Mathematical Analysis and Applications Pub Date : 2024-10-26 DOI:10.1016/j.jmaa.2024.129001

Ryuji Kajikiya

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引用次数: 0

摘要

我们给出了无界域 Ω 的 Sobolev 空间 W0m,p(Ω) 的紧凑嵌入的必要条件和充分条件。利用这个条件，我们就能判断紧凑嵌入是否成立。我们举了几个满足紧凑嵌入的无界域 Ω 的例子。利用我们的条件，我们研究了无界域中的一个半线性椭圆方程，并证明了一个正解和无穷多个解的存在。

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Sobolev compact embeddings in unbounded domains and its applications to elliptic equations

We give a necessary and sufficient condition for the compact embedding of the Sobolev space

W_{0}^{m, p} (Ω)

for unbounded domains Ω. Applying this condition, we can decide whether the compact embedding holds or not. We give several examples of unbounded domains Ω satisfying the compact embedding. Using our condition, we study a semilinear elliptic equation in unbounded domains and prove the existence of a positive solution and infinitely many solutions.

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来源期刊

Journal of Mathematical Analysis and Applications 数学-数学

CiteScore

2.50

自引率

7.70%

发文量

790

审稿时长

6 months

期刊介绍： The Journal of Mathematical Analysis and Applications presents papers that treat mathematical analysis and its numerous applications. The journal emphasizes articles devoted to the mathematical treatment of questions arising in physics, chemistry, biology, and engineering, particularly those that stress analytical aspects and novel problems and their solutions. Papers are sought which employ one or more of the following areas of classical analysis: • Analytic number theory • Functional analysis and operator theory • Real and harmonic analysis • Complex analysis • Numerical analysis • Applied mathematics • Partial differential equations • Dynamical systems • Control and Optimization • Probability • Mathematical biology • Combinatorics • Mathematical physics.