**
MATHEMATICA BOHEMICA, Vol. 120, No. 2, pp. 169-195, 1995
**

#
The least eigenvalues of nonhomogeneous degenerated quasilinear eigenvalue problems

##
Pavel Drabek

Department of Mathematics, University of West Bohemia, P.O. Box 314, 306 14 Plzen, Czech Republic, e-mail: pdrabek@kma.zcu.cz

**Abstract:** We prove the existence of the least positive eigenvalue with a corresponding nonnegative eigenfunction of the quasilinear eigenvalue problem

\align-\operatorname{div}(a(x,u)|\nablau|^{p-2}\nabla u) = &\lambda b(x,u)|u|^{p-2}u \quad\text{ in } \Omega,

u = &0 \hskip2cm\text{ on } \partial\Omega, \endalign

where $\Omega$ is a bounded domain, $p>1$ is a real number and $a(x,u)$, $b(x,u)$ satisfy appropriate growth conditions. Moreover, the coefficient $a(x,u)$ contains a degeneration or a singularity. We work in a suitable weighted Sobolev space and prove the boundedness of the eigenfunction in $L^\infty(\Omega)$. The main tool is the investigation of the associated homogeneous eigenvalue problem and an application of the Schauder fixed point theorem.

**Keywords:** weighted Sobolev space, degenerated quasilinear partial differential equations, weak solutions, eigenvalue problems, Schauder fixed point theorem, boundedness of the solution

**Classification (MSC91):** 35J20, 35J70, 35B35, 35B45

**Full text of the article:**

[Previous Article] [Next Article] [Contents of this Number] [Journals Homepage]