# zbMATH — the first resource for mathematics

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)