Sign-changing solution for logarithmic elliptic equations with critical exponent

Abstract

In this paper, we consider the logarithmic elliptic equations with critical exponent \begin{equation} \begin{cases} -Δu=λu+ |u|^{2^*-2}u+θu\log u^2, \\ u \in H_0^1(Ω), \quad Ω\subset \R^N. \end{cases} \end{equation} Here, the parameters $N\ge 6$, $\lambda \in \mathbb{R}$, $\theta >0$ and $2^{\ast }=\frac{2N}{N-2}$ 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.