The paper deals with the multiplicity of solutions of the equation
where is any bounded domain, , and is the minimum of the Rayleigh quotient as u varies in . It is shown that any two solutions are proportional. This result was known in the linear case and, for general p, under special assumptions on . 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.