A Neumann problem with critical exponent in nonconvex domains and Lin-Ni’s conjecture

Abstract

We consider the following nonlinear Neumann problem: \[ { − Δ u + μ u = u N + 2 N − 2 , u > 0 a m p ; in Ω , ∂ u ∂ n = 0 a m p ; on ∂ Ω , \left \{\begin {array}{lll} -\Delta u + \mu u = u^{\frac {N+2}{N-2}},\quad u>0 \quad & \mbox {in} \ \Omega , \\ \frac {\partial u}{\partial n}=0 & \mbox {on} \ \partial \Omega , \end {array}\right . \] where Ω ⊂ R N \Omega \subset \mathbb {R}^N is a smooth and bounded domain, μ > 0 \mu > 0 and n n denotes the outward unit normal vector of ∂ Ω \partial \Omega . Lin and Ni (1986) conjectured that for μ \mu small, all solutions are constants. We show that this conjecture is false for all dimensions in some (partially symmetric) nonconvex domains Ω \Omega . Furthermore, we prove that for any fixed μ \mu , there are infinitely many positive solutions, whose energy can be made arbitrarily large. This seems to be a new phenomenon for elliptic problems in bounded domains.