Calculus of Variations and Partial Differential Equations ( IF 2.1 ) Pub Date : 2021-01-18 , DOI: 10.1007/s00526-020-01883-6 Stefano Buccheri , Tommaso Leonori
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.
中文翻译:
包含p -Laplacian且p发散的拟线性问题的大解
在本文中,我们针对
$$ \ begin {aligned} {\ left \ {\ begin {array} {ll} u-\ Delta _ {p} u + \ beta | \ nabla u | ^ {q} = f&{} \ text {in} \,\ Omega,\\ u(x)= + \ infty&{} \ text {on} \,\ partial \ Omega,\ end {array} \ right。} \ end {aligned} $$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。