Exhaustive existence and non-existence results for some prototype polyharmonic equations in the whole space

JOURNAL OF DIFFERENTIAL EQUATIONS

卷:269

Article

Quoc Anh Ngo^1,2  Van Hoang Nguyen^3,4  Quoc Hung Phan^5,6  Ye, Dong^7,8,9 

[1] Univ Tokyo, Grad Sch Math Sci, Megum Ku, 3-8-1 Komaba, Tokyo 1538914, Japan

[2] Vietnam Natl Univ, Univ Sci, Hanoi, Vietnam

[3] Vietnam Acad Sci & Technol, Inst Math, Hanoi, Vietnam

[4] FPT Univ, Dept Math, Hanoi, Vietnam

[5] Duy Tan Univ, Inst Res & Dev, Da Nang, Vietnam

[6] Duy Tan Univ, Fac Nat Sci, Da Nang, Vietnam

[7] East China Normal Univ, Ctr Partial Differential Equat, Sch Math Sci, Shanghai 200062, Peoples R China

[8] East China Normal Univ, Shanghai Key Lab PMMP, Shanghai 200062, Peoples R China

[9] Univ Lorraine, IECL, UMR 7502, F-57073 Metz, France

关键词: Polyharmonic equation;    Existence and non-existence;    Liouville theorem;    Maximum principle type result;   

DOI  :  10.1016/j.jde.2020.07.041

来源: Elsevier

【 摘 要 】

In this paper, we are interested in entire, non-trivial, non-negative solutions and/or entire positive solutions to the simplest models of polyharmonic equations with power-type nonlinearity with n >= 1, m >= 1, and alpha is an element of R. We aim to study the existence and non-existence of such classical solutions to the above equations in the full range of the constants n, m and alpha. Remarkably, we are able to provide necessary and sufficient conditions on the exponent a to guarantee the existence of such solutions in R-n. Finally, we identify all the situations where any entire non-trivial, non-negative classical solution must be positive everywhere. (C) 2020 Elsevier Inc. All rights reserved.

【 授权许可】

Free