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Simplicité et isolation de la première valeur propre du p-Laplacien avec poids. (Simplicity and isolation of first eigenvalue of the p- Laplacian with weight). (French) Zbl 0633.35061
We show that the first eigenvalue \(\lambda_ 1\) of the Dirichlet problem: \[ -div(| \nabla u|^{p-2} \nabla u)=\lambda m(x)| u|^{p-2}u\quad in\quad \Omega,\quad u=0\quad on\quad \partial \Omega, \] is simple and isolated: we also prove that it is the unique positive eigenvalue having a non negative eigenfunction.

MSC:
35P30 Nonlinear eigenvalue problems and nonlinear spectral theory for PDEs
35J65 Nonlinear boundary value problems for linear elliptic equations
35D05 Existence of generalized solutions of PDE (MSC2000)
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