Sign-changing solution for logarithmic elliptic equations with critical exponent

TL;DR

This paper establishes the existence of sign-changing solutions with specific nodal properties for a class of logarithmic elliptic equations involving critical Sobolev exponents, applicable to arbitrary bounded domains and specifically to balls.

Contribution

It proves the existence of sign-changing solutions with exactly two nodal domains for general domains and constructs infinitely many radial solutions with prescribed nodal patterns in balls.

Findings

Existence of sign-changing solutions with two nodal domains in arbitrary domains.

Construction of infinitely many radial sign-changing solutions in balls.

Solutions involve critical Sobolev exponents and logarithmic nonlinearities.

Abstract

In this paper, we consider the logarithmic elliptic equations with critical exponent \begin{equation} \begin{cases} -\Delta u=\lambda u+ |u|^{2^*-2}u+\theta u\log u^2, \\ u \in H_0^1(\Omega), \quad \Omega \subset \R^N. \end{cases} \end{equation} Here, the parameters $N\geq 6$, $\lambda\in \R$, $\theta>0$ and $ 2^*=\frac{2N}{N-2} $ is the Sobolev critical exponent. We prove the existence of sign-changing solution with exactly two nodal domain for an arbitrary smooth bounded domain $\Omega\subset \mathbb{R}^{N}$. When $\Omega=B_R(0)$ is a ball, we also construct infinitely many radial sign-changing solutions with alternating signs and prescribed nodal characteristic.