### The least eigenvalues of nonhomogeneous degenerated quasilinear eigenvalue problems

Pavel Drábek (1995)

Mathematica Bohemica

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We prove the existence of the least positive eigenvalue with a corresponding nonnegative eigenfunction of the quasilinear eigenvalue problem $$\begin{array}{ccc}\hfill -div\left(a(x,u)\right|{|}^{p-2}\nabla u)=& {\lambda b(x,u)\left|u\right|}^{p-2}u\phantom{\rule{1.0em}{0ex}}\phantom{\rule{4.0pt}{0ex}}\text{in}\phantom{\rule{4.0pt}{0ex}}\Omega ,u=\hfill & \hfill 0\phantom{\rule{56.9055pt}{0ex}}\phantom{\rule{4.0pt}{0ex}}\text{on}\phantom{\rule{4.0pt}{0ex}}\partial \Omega ,\end{array}$$ 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 ${L}^{\infty}\left(\Omega \right)$. The main tool is the investigation of the associated homogeneous eigenvalue problem...