Dirichlet problem for a class of nonlinear degenerate elliptic operators with critical growth and logarithmic perturbation

In this paper, we investigate the existence of weak solutions for a class of degenerate elliptic Dirichlet problems with critical nonlinearity and a logarithmic perturbation, i.e. 0.2 \(\begin{aligned} \Big \{\begin{array}{l} -(\Delta _{x} u+(\alpha +1)^2|x|^{2 \alpha } \Delta _{y} u)=u^{\frac{Q+2}{Q-2}} + \lambda u\log u^2,\\ u=0~~ \text { on } \partial \Omega , \end{array} \end{aligned}\) $$\begin{array}{c}\hfill \{\begin{array}{c}-({\mathrm{\Delta }}_{x}u+{(\alpha +1)}^2{|x|}^{2\alpha }{\mathrm{\Delta }}_{y}u)=u^{\frac{Q+2}{Q-2}}+\lambda ulog{u}^2,\hfill \\ u=0\text{on}\partial \mathrm{\Omega },\hfill \end{array}\end{array}$$ where \((x,y)\in \Omega \subset \mathbb {R}^N = \mathbb {R}^m \times \mathbb {R}^n\) $(x,y)\in \mathrm{\Omega }\subset {\mathbb{R}}^N={\mathbb{R}}^m\times {\mathbb{R}}^n$ with \(m \ge 1\) $m\ge 1$ , \(n\ge 0\) $n\ge 0$ , \(\Omega \cap \{x=0\}\ne \emptyset \) $\mathrm{\Omega }\cap \{x=0\}\ne \mathrm{\varnothing }$ is a bounded domain, the parameter \(\alpha \ge 0\) $\alpha \ge 0$ and \( Q=m+ n(\alpha +1)\) $Q=m+n(\alpha +1)$ denotes the “homogeneous dimension” of \(\mathbb {R}^N\) ${\mathbb{R}}^N$ . When \(\lambda =0\) $\lambda =0$ , we know that from [23] the problem (0.2) has a Pohožaev-type non-existence result. Then for \(\lambda \in \mathbb {R}\backslash \{0\}\) $\lambda \in \mathbb{R}\backslash \{0\}$ , we establish the existences of non-negative ground state weak solutions and non-trivial weak solutions subject to certain conditions.