The least eigenvalues of nonhomogeneous degenerated quasilinear eigenvalue problems

Drábek, Pavel

Mathematica Bohemica, Tome 120 (1995) no. 2, pp. 169-195.

Résumé

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.

Détail

DOI : 10.21136/MB.1995.126227

Classification : 35B35, 35B45, 35J20, 35J65, 35J70, 35P30, 47H12, 47N20
Keywords: boundedness of eigenfunction; weighted Sobolev space; Schauder fixed point theorem; degenerated quasilinear partial differential equations; weak solutions; eigenvalue problems; boundedness of the solution