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.
Categories: math.AP