Electron. J. Diff. Eqns., Vol. 2003(2003), No. 16, pp. 1-16.

Existence of positive solutions for two nonlinear eigenvalue problems

Nedra Belhaj Rhouma & Lamia Maatoug

Abstract:
We study the existence of positive solutions for the following two nonlinear eigenvalue problems
$$\displaylines{ \Delta u-g(.,u)u+\lambda f(.,u)u=0, \cr \Delta u-g(.,u)u+\lambda f(.,u)=0, }$$
in a bounded regular domain in $\mathbb{R}^{2}$ with $u=0$ on the boundary. We assume that $f$ and $g$ are in Kato class of functions.

Submitted October 14, 2002. Published February 15, 2003.
Math Subject Classifications: 31A25, 31A35, 34B15, 34B27, 35J65
Key Words: Eigenvalue, Kato class, Green's function, superharmonic function, Shauder fixed point theorem, maximum principle.

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Nedra Belhaj Rhouma
Institut Preparatoire aux Etudes d'ingenieurs de Tunis,
2 Rue Jawaherlal Nehru,
1008 Montfleury, Tunis, Tunisia.
e-mail: Nedra.Belhajrhouma@ipeit.rnu.tn
Lamia Maatoug
Departement de Mathematiques
Faculte des Sciences de Tunis
Campus universitaire 1060 Tunis, Tunisia
e-mail: Lamia.Maatoug@ipeit.rnu.tn

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