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.

$$\begin{aligned} \Big \{\begin{array}{l} -(\Delta _{x} u+(\alpha +1)^2|x|^{2 \alpha } \Delta _{y} u)=u^{\frac{Q+2}{Q-2}} + \lambda u\log u^2,\\ u=0~~ \text { on } \partial \Omega , \end{array} \end{aligned}$$(0.2)

where \((x,y)\in \Omega \subset \mathbb {R}^N = \mathbb {R}^m \times \mathbb {R}^n\) with \(m \ge 1\), \(n\ge 0\), \(\Omega \cap \{x=0\}\ne \emptyset \) is a bounded domain, the parameter \(\alpha \ge 0\) and \( Q=m+ n(\alpha +1)\) denotes the “homogeneous dimension” of \(\mathbb {R}^N\). When \(\lambda =0\), we know that from [23] the problem (0.2) has a Pohožaev-type non-existence result. Then for \(\lambda \in \mathbb {R}\backslash \{0\}\), we establish the existences of non-negative ground state weak solutions and non-trivial weak solutions subject to certain conditions.