On the equation 𝑑𝑖𝑣(|∇𝑢|^{𝑝-2}∇𝑢)+𝜆|𝑢|^{𝑝-2}𝑢=0

Abstract

The first eigenvalue λ = λ 1 \lambda = {\lambda _1} for the equation div ⁡ ( | ∇ u | p − 2 ∇ u ) +  λ | u | p − 2 u = 0 \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 λ 1 {\lambda _1} is related to the Poincaré inequality.)