On a conjecture of Lin-Ni for a semilinear Neumann problem

Abstract

Let Ω \Omega be a bounded domain in R n {\mathbb {R}^n} ( n ≥ 3 ) (n \geq 3) and λ > 0 \lambda > 0 . We consider \[ − Δ u + λ u = u ( n + 2 ) / ( n − 2 ) a m p ; in Ω , u > 0 a m p ; in Ω , ∂ u ∂ ν = 0 a m p ; on ∂ Ω , \begin {array}{*{20}{c}} { - \Delta u + \lambda u = {u^{(n + 2)/(n - 2)}}} & {{\text {in}}} \; {\Omega ,} \\ {u > 0} & {{\text {in}}} \; {\Omega ,} \\ {\frac {{\partial u}} {{\partial \nu }} = 0} & {{\text {on}}} \; {\partial \Omega ,} \\ \end {array} \] and show that for λ \lambda sufficiently small, the minimal energy solutions are only constants.