Hostname: page-component-586b7cd67f-t8hqh Total loading time: 0 Render date: 2024-11-27T02:52:13.437Z Has data issue: false hasContentIssue false
  • English
  • Français

Compact embedding results of Sobolev spaces and positive solutions to an elliptic equation

Part of: Classical measure theory Linear function spaces and their duals Elliptic equations and systems

Published online by Cambridge University Press:  11 April 2016

Qi Han
Qi Han*
Affiliation:
Department of Mathematical Sciences, Worcester Polytechnic Institute, Worcester, MA 01609-2280, USA ([email protected])
Get access Rights & Permissions [Opens in a new window]

Extract

Using a regular Borel measure μ ⩾ 0 we derive a proper subspace of the commonly used Sobolev space D1(ℝN) when N ⩾ 3. The space resembles the standard Sobolev space H1(Ω) when Ω is a bounded region with a compact Lipschitz boundary ∂Ω. An equivalence characterization and an example are provided that guarantee that is compactly embedded into L1(RN). In addition, as an application we prove an existence result of positive solutions to an elliptic equation in ℝN that involves the Laplace operator with the critical Sobolev nonlinearity, or with a general nonlinear term that has a subcritical and superlinear growth. We also briefly discuss the compact embedding of to Lp(ℝN) when N ⩾ 2 and 2 ⩽ p ⩽ N.

Keywords

Sobolev spacescapacityelliptic equationspositive solutions

MSC classification

Secondary: 46E35: Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems 35J20: Variational methods for second-order elliptic equations 35J60: Nonlinear elliptic equations 28A12: Contents, measures, outer measures, capacities
Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 146 , Issue 4 , August 2016 , pp. 693 - 721
Copyright
Copyright © Royal Society of Edinburgh 2016 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)