Sign-changing solution for logarithmic elliptic equations with critical exponent

In this paper, we consider the logarithmic elliptic equations with critical exponent

$$\begin{aligned} \left\{ \begin{array}{ll} -\Delta u=\lambda u+ |u|^{2^*-2}u+\theta u\log u^2, \\ u \in H_0^1(\Omega ), \quad \Omega \subset {{\mathbb {R}}}^N. \end{array}\right. \end{aligned}$$

Here, the parameters \(N\ge 6\), \(\lambda \in {{\mathbb {R}}}\), \(\theta >0\) and \( 2^*=\frac{2N}{N-2} \) is the Sobolev critical exponent. We prove the existence of a 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.