We study the following elliptic equation * \(\begin{aligned} \left\{ \begin{array}{ll} - \Delta u= \mu |u|^{p-2} u+\lambda |u|^{q-2}u \ln |u|, & \quad x \in \Omega , \\ \qquad u=0, & \quad x \in \partial \Omega , \end{array}\right. \end{aligned}\) where \(\Omega \subset \mathbb {R}^{N}\) is a bounded smooth domain, \(N\ge 3\) , \(\mu \) , \(\lambda \in \mathbb {R}\) are parameters, \(1<q< 2^{*}\) , \(1<p<\infty \) and \(2^{*}:=2N/(N-2)\) is the critical Sobolev exponent. For Eq. (*), we establish the existence, nonexistence and multiplicity of positive solutions, and show the existence of infinitely many solutions. The uncertainty of the sign of the logarithmic perturbation and the challenges in estimation represent the key difficulties and innovations of this paper, as well as our primary focus. Furthermore, this work can also be viewed as an extension of the study with general logarithmic perturbation \(u\ln |u|\) .
