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

Adimurthi; Yadava, S. L.

doi:10.1090/S0002-9947-1993-1156299-0

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

Authors: Adimurthi and S. L. Yadava
Journal: Trans. Amer. Math. Soc. 336 (1993), 631-637
MSC: Primary 35J65; Secondary 35P30
DOI: https://doi.org/10.1090/S0002-9947-1993-1156299-0
MathSciNet review: 1156299
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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.

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