On the first eigenvalue of some quasilinear elliptic equations (Q1113381)

Let \(\Omega\) be a bounded domain in \({\mathbb{R}}^ N\) with smooth boundary \(\partial \Omega\). For given \(p\in (1,+\infty)\), \[ a\in L^{\infty}_+(\Omega)=\{f\in L^{\infty}(\Omega);\quad f(x)\geq 0\quad a.e.\quad x\in \Omega \} \] and \[ b\in L^{\infty}_ 0(\Omega)=\{f\in L^{\infty}(\Omega);\quad f^+(.)=\max (f(.),0)\not\equiv 0\}, \] we consider the following eigenvalue problem \[ (E)_{\lambda}\quad - \Delta_ pu(x)+a(x)| u|^{p-2} u(x)=\lambda b(x)| u|^{p-2} u(x),\quad x\in \Omega,\quad \lambda >0,\quad u(x)=0,\quad x\in \partial \Omega, \] where \(\Delta_ pu(x)=div(| \nabla u|^{p-2} \nabla u(x)).\) The main purpose of this paper is to show that there exists a positive number \(\lambda_ 1\), the first eigenvalue, such that \((E)_{\lambda}\) admits a positive solution if and only if \(\lambda =\lambda_ 1\) and that \(\lambda_ 1\) is simple, i.e., solutions of \((E)_{\lambda_ 1}\) forms a one dimensional subspace of \(W_ 0^{1,p}(\Omega)\).