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

Lindqvist, Peter

Proc. Am. Math. Soc. 109, No. 1, 157-164 (1990).

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

Citations:

Zbl 0633.35061