The least eigenvalues of nonhomogeneous degenerated quasilinear eigenvalue problems

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

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