Electron. J. Diff. Eqns., Vol. 2004(2004), No. 19, pp. 1-10.

Existence of solutions to nonlocal and singular elliptic problems via Galerkin method

Francisco Julio S. A. Correa & Silvano D. B. Menezes

Abstract:
We study the existence of solutions to the nonlocal elliptic equation
$$ -M(\|u\|^2)\Delta u = f(x,u) $$
with zero Dirichlet boundary conditions on a bounded and smooth domain of $\mathbb{R}^n$. We consider the $M$-linear case with $f\in H^{-1}(\Omega )$, and the sub-linear case $f(u)=u^{\alpha}$, $0 less than\alpha less than 1$. Our main tool is the Galerkin method for both cases when $M$ continuous and when $M$ is discontinuous.

Submitted December 15, 2003. Published February 11, 2004.
Math Subject Classifications: 35J60, 35J25.
Key Words: Nonlocal elliptic problems, Galerkin Method.

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Francisco Julio S. A. Correa
Departamento de Matematica-CCEN
Universidade Federal do Para
66.075-110 Belem Para Brazil
email: fjulio@ufpa.br
Silvano D. B. Menezes
Departamento de Matematica-CCEN
Universidade Federal do Para
66.075-110 Belem Para Brazil
email: silvano@ufpa.br

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