The existence and boundary behavior of large solutions to semilinear elliptic equations with nonlinear gradient terms. (English) Zbl 1296.35067
Summary: In this paper, for more general \(f\), \(g\) and \(a\), \(b\), we obtain conditions about the existence and boundary behavior of solutions to boundary blow-up elliptic problems
\[
\Delta u=a(x)g(u)+b(x)f(u)|\nabla u|^q,\quad x\in \Omega, u|{\partial\Omega}=+\infty
\]
and improve and generalize most of the previously available results in the literature, where \(\Omega\) is a bounded domain with smooth boundary in \(\mathbb{R}^N, q\in(0,2], a,b\in C^\nu(\overline{\Omega})\) which are positive in \(\Omega\), may be vanishing on the boundary, and \(f,g\in C[0,\infty)\cap C^1(0,\infty)\) or \(f,g\in C^1(\mathbb{R}\)), which are increasing.
MSC:
35J61 | Semilinear elliptic equations |
35J15 | Second-order elliptic equations |
35J25 | Boundary value problems for second-order elliptic equations |