A Berestycki-Lions type result for a class of degenerate elliptic problems involving the Grushin Operator

TL;DR

This paper establishes the existence of nontrivial solutions for a class of degenerate elliptic equations involving the Grushin operator, extending Berestycki-Lions type results to this setting with new compactness techniques.

Contribution

It introduces new compactness results and proves a Berestycki-Lions type theorem for degenerate elliptic equations with the Grushin operator.

Findings

Existence of nontrivial solutions for the equations studied.

Development of compactness results crucial for the analysis.

Extension of classical results to degenerate elliptic operators.

Abstract

In this work we study the existence of nontrivial solution for the following class of semilinear degenerate elliptic equations $$ -\Delta_{\gamma} u + a(z)u = f(u) ~~ \mbox{in} ~~ \mathbb{R}^{N}, $$ where $\Delta_{\gamma}$ is known as the {\it Grushin operator}, $z:=(x,y)\in\mathbb{R}^{m}\times\mathbb{R}^{k}$ and $m+k=N\geq 3$, $f$ and $a$ are continuous function satisfying some technical conditions. In order to overcome some difficulties involving this type of operator, we have proved some compactness results that are crucial in the proof of our main results. For the case $a=1$, we have showed a Berestycki-Lions type result.