A Neumann problem with critical exponent in nonconvex domains and Lin-Ni'S conjecture

We consider the following nonlinear Neumann problem: where Ω ⊂ ℝ^N is a smooth and bounded domain, μ < 0 and n denotes the outward unit normal vector of ∂Ω. Lin and Ni (1986) conjectured that for μ small, all solutions are constants. We show that this conjecture is false for all dimensions in some (partially symmetric) nonconvex domains Ω. Furthermore, we prove that for any fixed μ 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.