In this paper we deal with large solutions to

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.

在本文中，我们针对

其中\（\ Omega \ subset {\ mathbb {R}} ^ N \）和\（N \ ge 1 \）是一个光滑，开放，连通且有界的域\（p \ ge 2 \），\（\ beta> 0 \），\（p-1 <q \ le p \）和\（f \ in C（\ Omega）\ cap L ^ {\ infty}（\ Omega} \）。我们对研究p的行为有兴趣。我们的主要结果表明，在某种意义上，如果域\（\ Omega \）足够大，则此类解在本地均匀收敛到一个极限函数，而该极限函数原来是一个合适的极限方程的大解（涉及到\（\ infty \）-拉普拉斯语。否则，如果\（\ Omega \）很小，那么我们将完全崩溃。