Electron. J. Diff. Eqns., Vol. 2007(2007), No. 54, pp. 1-13.

Existence of bounded solutions for nonlinear degenerate elliptic equations in Orlicz spaces

Ahmed Youssfi

Abstract:
We prove the existence of bounded solutions for the nonlinear elliptic problem
$$ -\hbox{\rm div}a(x,u,{\nabla}u)=f \quad\hbox{in }{\Omega}, $$
with $u\in W^1_0L_M({\Omega})\cap L^{\infty}(\Omega)$, where
$$ a(x,s,\xi)\cdot\xi\geq {\overline M}^{-1}M(h(|s|))M(|\xi|), $$
and $h:{\mathbb{R}^+}{\to }{]0,1]}$ is a continuous monotone decreasing function with unbounded primitive. As regards the $N$-function $M$, no $\Delta_2$-condition is needed.

Submitted December 11, 2006. Published April 10, 2007.
Math Subject Classifications: 46E30, 35J70, 35J60.
Key Words: Orlicz-Sobolev spaces; degenerate coercivity; L-infity-estimates; rearrangements.

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Ahmed Youssfi
Department of Mathematics and Informatics
Faculty of Sciences Dhar El Mahraz
University Sidi Mohammed Ben Abdallah
PB 1796 Fez-Atlas, Fez, Morocco
email: Ahmed.youssfi@caramail.com

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