Large solutions to quasilinear problems involving the p-Laplacian as p diverges

Abstract

In this paper we deal with large solutions to

$$\begin{aligned} {\left\{ \begin{array}{ll} u - \Delta _{p} u + \beta |\nabla u|^{q} =f&{} \text{ in } \,\Omega ,\\ u (x) = +\infty &{} \text{ on } \,\partial \Omega , \end{array}\right. } \end{aligned}$$

where $\Omega \subset {\mathbb {R}}^N$ , with $N\ge 1$, is a smooth, open, connected, and bounded domain, $p \ge 2$, $\beta >0$, $p-1<q\le p$ and $f\in C(\Omega )\cap L^{\infty }(\Omega )$. We are interested in studying their behavior as p diverges. Our main result states that, if, in some sense, the domain $\Omega$ is large enough, such solutions converge locally uniformly to a limit function that turns out to be a large solution of a suitable limit equation (that involves the $\infty$-Laplacian). Otherwise, if $\Omega$ is small, we have a complete blow-up.