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

[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.

[发布日期] 2020-12-05

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