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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

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
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