Existence and nonexistence of least energy nodal solution for a class of elliptic problem in $\mathbb{R}^{2}$

TL;DR

This paper investigates the existence and nonexistence of least energy nodal solutions for a class of elliptic problems in two-dimensional spaces, considering critical exponential growth nonlinearities in bounded and unbounded domains.

Contribution

It establishes the existence of least energy nodal solutions in bounded and unbounded domains and proves nonexistence in the autonomous case in the entire plane.

Findings

Existence of least energy nodal solutions in bounded domains.

Existence of least energy nodal solutions in unbounded domains.

Nonexistence of such solutions in the autonomous case in .

Abstract

In this work, we prove the existence of least energy nodal solution for a class of elliptic problem in both cases, bounded and unbounded domain, when the nonlinearity has exponential critical growth in $\mathbb{R}^2$. Moreover, we also prove a nonexistence result of least energy nodal solution for the autonomous case in whole $\mathbb{R}^{2}$.