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boundedness of eigenfunction; weighted Sobolev space; Schauder fixed point theorem; degenerated quasilinear partial differential equations; weak solutions; eigenvalue problems; boundedness of the solution
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.
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