## On the equation $$div(| \nabla u| ^{p-2}\nabla u)+\lambda | u| ^{p-2}u=0$$.(English)Zbl 0714.35029

The paper deals with the multiplicity of solutions of the equation $div(| \nabla u|^{p-2}\nabla u)+\lambda | u|^{p- 2}u=0\text{ in } W_ 0^{1,p}(\Omega),$ where $$\Omega$$ is any bounded domain, $$1<p<+\infty$$, and $$\lambda$$ is the minimum of the Rayleigh quotient $$\int_{\Omega}| \nabla u|^ p dx/\int_{\Omega}| u|^ p dx$$ as u varies in $$W_ 0^{1,p}(\Omega)$$. It is shown that any two solutions are proportional. This result was known in the linear case $$(p=2)$$ and, for general p, under special assumptions on $$\partial \Omega$$. By improving the method of A. Anane [C. R. Acad. Sci., Paris, Sér. I 305, 725-728 (1987; Zbl 0633.35061)] the author obtains the optimal result.
Reviewer: L.Ambrosio

### MSC:

 35J60 Nonlinear elliptic equations 35J70 Degenerate elliptic equations 35P30 Nonlinear eigenvalue problems and nonlinear spectral theory for PDEs 35J20 Variational methods for second-order elliptic equations

### Keywords:

p-Laplacian; multiplicity of solutions

Zbl 0633.35061
Full Text: