Electron. J. Diff. Eqns., Vol. 2006(2006), No. 02, pp. 1-8.

Existence of large solutions for a semilinear elliptic problem via explosive sub- supersolutions

Zhijun Zhang

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.

Show me the PDF file (225K), TEX file, and other files for this article.

Zhijun Zhang
Department of Mathematics and Informational Science
Yantai University, Yantai, Shandong, 264005, China
email: zhangzj@ytu.edu.cn

Return to the EJDE web page