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 \(\Delta \).