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Electronic Journal of Differential Equations Electron. J. Diff. Equ., Vol. 2009(2009), No. 160, pp. 1-13.

Existence of weak solutions for degenerate semilinear elliptic equations in unbounded domains

Venkataramanarao Raghavendra, Rasmita Kar

Abstract:
In this study, we prove the existence of a weak solution for the degenerate semilinear elliptic Dirichlet boundary-value problem
$$\displaylines{ Lu-\mu u g_{1} + h(u) g_{2}= f\quad \hbox{in }\Omega,\cr u = 0\quad \hbox{on }\partial\Omega }$$
in a suitable weighted Sobolev space. Here the domain $\Omega\subset\mathbb{R}^{n}$, $n\geq 3$, is not necessarily bounded, and $h$ is a continuous bounded nonlinearity. The theory is also extended for $h$ continuous and unbounded.

Submitted October 25, 2009. Published December 15, 2009.
Math Subject Classifications: 35J70, 35D30.
Key Words: Degenerate equations; weighted Sobolev space; unbounded domain.

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Venkataramanarao Raghavendra
Department of Mathematics and Statistics
Indian Institute of Technology, Kanpur, India 208016
email: vrag@iitk.ac.in
Rasmita Kar
Department of Mathematics and Statistics
Indian Institute of Technology, Kanpur, India 208016
email: rasmita@iitk.ac.in

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