Boundedness of Stable Solutions to Semilinear Elliptic Equations: A Survey-Reference-Cited by-同舟云学术

Boundedness of Stable Solutions to Semilinear Elliptic Equations: A Survey

Published:2017-04-11 Issue:2 Volume:17 Page:355-368
ISSN:1536-1365
Container-title:Advanced Nonlinear Studies
language:en
Short-container-title:

Author:

Cabré Xavier¹

Affiliation:

1. Departament de Matemàtiques, Universitat Politècnica de Catalunya, Diagonal 647, 08028 Barcelona; and ICREA, Pg. Lluis Companys 23, 08010 Barcelona, Spain

Abstract

Abstract This article is a survey on boundedness results for stable solutions to semilinear elliptic problems. For these solutions, we present the currently known L ∞

${L^{\infty}}$

estimates that hold for all nonlinearities. Such estimates are known to hold up to dimension 4. They are expected to be true also in dimensions 5 to 9, but this is still an open problem which has only been established in the radial case.

Publisher

Walter de Gruyter GmbH

Subject

General Mathematics,Statistical and Nonlinear Physics

Link

https://www.degruyter.com/document/doi/10.1515/ans-2017-0008/pdf

Reference20 articles.

1. Brezis H., Is there failure of the inverse function theorem?, Morse Theory, Minimax Theory and Their Applications to Nonlinear Differential Equations (Beijing 1999), New Stud. Adv. Math. 1, International Press, Somerville (2003), 23–33.

2. Brezis H., Cazenave T., Martel Y. and Ramiandrisoa A., Blow up for ut-Δ⁢u=g⁢(u)${u_{t}-\Delta u=g(u)}$ revisited, Adv. Differential Equations 1 (1996), 73–90.

3. Brezis H. and Vázquez J. L., Blow-up solutions of some nonlinear elliptic problems, Rev. Mat. Univ. Complutense Madrid 10 (1997), 443–469.

4. Cabré X., Regularity of minimizers of semilinear elliptic problems up to dimension 4, Comm. Pure Appl. Math. 63 (2010), 1362–1380.

5. Cabré X. and Capella A., Regularity of radial minimizers and extremal solutions of semilinear elliptic equations, J. Funct. Anal. 238 (2006), 709–733.

Cited by 18 articles. 订阅此论文施引文献订阅此论文施引文献，注册后可以免费订阅5篇论文的施引文献，订阅后可以查看论文全部施引文献

1. Classification of bifurcation diagrams for semilinear elliptic equations in the critical dimension;Journal of Differential Equations;2024-07

2. A finding of the maximal saddle-node bifurcation for systems of differential equations;Journal of Differential Equations;2024-01

3. Uniform boundedness for finite Morse index solutions to supercritical semilinear elliptic equations;Communications on Pure and Applied Mathematics;2023-07-21

4. The Boundedness of Stable Solutions to Semilinear Elliptic Equations With Linear Lower Bound on Nonlinearities;International Mathematics Research Notices;2023-07-14

5. A Finding of the Maximal Saddle-Node Bifurcation for Systems of Differential Equations;2023