On periodic and compactly supported least energy solutions to semilinear elliptic equations with non-Lipschitz nonlinearity

TL;DR

This paper investigates the existence and non-existence of least energy solutions to a class of semilinear elliptic equations with non-Lipschitz nonlinearities, focusing on solutions that are periodic in one variable and compactly supported in others, using variational methods.

Contribution

It introduces a novel approach combining the Nehari manifold method with Pohozaev constraints to identify critical parameters for solution existence.

Findings

Identifies the parameter range where solutions exist or do not exist.

Shows solutions are non-trivial in the periodic variable.

Establishes that solutions are not globally compactly supported.

Abstract

We discuss the existence and non-existence of periodic in one variable and compactly supported in the other variables least energy solutions for equations with non-Lipschitz nonlinearity of the form: $-\Delta u=\lambda u^p - u^q$ in $\mathbb{R}^{N+1}$, where $ 0< q < p \leq 1$, $\lambda \in \mathbb{R}$. The approach is based on the Nehari manifold method supplemented by a one-sided constraint given through the functional of the suitable Pohozaev identity. The limit value of the parameter $\lambda$, where the approach is applicable, corresponds to the existence of periodic in one variable and compactly supported in the other variables least energy solutions. This value is found through the extrem values of nonlinear generalized Rayleigh quotients and the so-called curve of the critical exponents of $p,q$. Important properties of the solutions are derived, such as that they are not trivial with respect to the periodic variable and do not coincide with compactly supported solutions on the entire space $\mathbb{R}^{N+1}$.