Electronic Journal of Differential Equations,
Vol. 2006(2006), No. 02, pp. 1-8.
Title: Existence of large solutions for a semilinear elliptic
problem via explosive sub- supersolutions
Author: Zhijun Zhang (Yantai Univ., Shandong, China)
Abstract:
We consider the boundary blow-up nonlinear elliptic problems
$\Delta u\pm\lambda |\nabla u|^q=k(x)g(u)$ in a bounded
domain with boundary condition $u|_{\partial \Omega}=+\infty$,
where $q\in [0, 2]$ and $\lambda\geq0$.
Under suitable growth assumptions on $k$ near the boundary
and on $g$ both at zero and at infinity, we show the existence
of at least one solution in $C^2(\Omega)$.
Our proof is based on the method of explosive sub-supersolutions,
which permits positive weights $k(x)$ which are unbounded
and/or oscillatory near the boundary. Also, we show
the global optimal asymptotic behaviour of the solution
in some special cases.
Submitted May 21, 2005. Published January 6, 2006.
Math Subject Classifications: 35J60, 35B25, 35B50, 35R05.
Key Words: Semilinear elliptic equations; explosive subsolutions;
explosive superbsolutions; existence;
global optimal asymptotic behaviour.