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

Anane, Aomar

C. R. Acad. Sci., Paris, Sér. I 305, 725-728 (1987).

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)

Keywords:

first eigenvalue; Dirichlet problem; simple; isolated; positive eigenvalue; negative eigenfunction