Positive solutions to the sublinear Lane-Emden equation are isolated

TL;DR

This paper proves that positive least energy solutions to the sublinear Lane-Emden equation are isolated on smooth bounded domains, impacting the spectral properties of the Dirichlet-Laplacian and extending to Lipschitz domains.

Contribution

It establishes the isolation of positive solutions for the sublinear Lane-Emden equation and shows the first eigenvalue is not an accumulation point of the spectrum.

Findings

Positive solutions are isolated on smooth bounded domains.

First eigenvalue is not an accumulation point of the spectrum.

Results extend to Lipschitz domains.

Abstract

We prove that on a smooth bounded set, the positive least energy solution of the Lane-Emden equation with sublinear power is isolated. As a corollary, we obtain that the first $q-$eigenvalue of the Dirichlet-Laplacian is not an accumulation point of the $q-$spectrum, on a smooth bounded set. Our results extend to a suitable class of Lipschitz domains, as well.