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On the equation $div(\vert \nabla u\vert \sp{p-2}\nabla u)+\lambda \vert u\vert \sp{p-2}u=0$. (English) Zbl 0714.35029
The paper deals with the multiplicity of solutions of the equation $$ div(\vert \nabla u\vert\sp{p-2}\nabla u)+\lambda \vert u\vert\sp{p- 2}u=0\text{ in } W\sb 0\sp{1,p}(\Omega), $$ where $\Omega$ is any bounded domain, $1<p<+\infty$, and $\lambda$ is the minimum of the Rayleigh quotient $\int\sb{\Omega}\vert \nabla u\vert\sp p dx/\int\sb{\Omega}\vert u\vert\sp p dx$ as u varies in $W\sb 0\sp{1,p}(\Omega)$. 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 {\it 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

35J60Nonlinear elliptic equations
35J70Degenerate elliptic equations
35P30Nonlinear eigenvalue problems for PD operators; nonlinear spectral theory
35J20Second order elliptic equations, variational methods
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