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

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.

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

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