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

Abstract:

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