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