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

TL;DR

This paper thoroughly investigates the existence and non-existence of classical solutions to polyharmonic equations with power nonlinearities in the whole space, providing precise conditions on parameters for solution existence.

Contribution

It offers necessary and sufficient conditions on the exponent for solutions to exist and characterizes when solutions must be positive, filling gaps in the understanding of these equations.

Findings

Derived exact conditions for solution existence based on parameters.

Identified parameter regimes where solutions are necessarily positive.

Provided comprehensive classification of solutions in the entire space.

Abstract

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 \[ \Delta^m u = \pm u^{\alpha} \quad \text{ in } \mathbb R^n \] with $n \geqslant 1$, $m \geqslant 1$, and $\alpha \in \mathbb R$. 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 $\alpha$ to guarantee the existence of such solutions in $\mathbb R^n$. Finally, we identify all the situations where any entire non-trivial, non-negative classical solution must be positive.