On the equation ${\rm div}\,(\vert \nabla u\vert \sp {p-2}\nabla u)+\lambda\vert u\vert \sp {p-2}u=0$

The first eigenvalue $\lambda = {\lambda _1}$ for the equation $\operatorname{div} ({\text{\vert}}\nabla u{{\text{\vert}}^{p - 2}}\nabla u{\text{) + }}\lambda {\text{\vert}}u{{\text{\vert}}^{p - 2}}u = 0$ is simple in any bounded domain. (Through the nonlinear counterpart to the Rayleigh quotient ${\lambda _1}$ is related to the Poincaré inequality.)