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

Abstract:

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