Boundary blow-up solutions to a class of degenerate elliptic equations

Abstract

Let $\Omega$ be a bounded domain in $\mathbb {R}^N=\mathbb {R}^{N_1} \times \mathbb {R}^{N_2}$ with $N_1, N_2 \ge 1$, and $N(s) = N_1 + (1+s)N_2$ be the homogeneous dimension of $\mathbb {R}^N$ for $s \ge 0$. In this paper, we prove the existence and uniqueness of boundary blow-up solutions to the following semilinear degenerate elliptic equation

$$\begin{aligned} {\left\{ \begin{array}{ll} G_s u = {|x|^{2s}} u^p_+ \; &{}\text { in } \Omega ,\\ u(z)\rightarrow +\infty \; &{}\text { as } {d}(z) \rightarrow 0, \end{array}\right. } \end{aligned}$$

where $u_+ = \max \{u,0\}$, $1<p<{{N(s)+2s} \over {N(s)-2}}$, and d(z) denotes the Grushin distance from z to the boundary of $\Omega$. Here $G_s$ is the Grushin operator of the form

$$\begin{aligned} G_s u= \Delta _x u + |x|^{2s}\Delta _y u, \; s\ge 0. \end{aligned}$$

It is worth noticing that our results do not require any assumption on the smoothness of the domain $\Omega$, and when $s=0$, we cover the previous results for the Laplace operator $\Delta$.