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

Abstract:

We consider the following nonlinear Neumann problem: \[ \left \{\begin {array}{lll} -\Delta u + \mu u = u^{\frac {N+2}{N-2}},\quad u>0 \quad & \mbox {in} \ \Omega , \\ \frac {\partial u}{\partial n}=0 & \mbox {on} \ \partial \Omega , \end {array}\right .\] where $\Omega \subset \mathbb {R}^N$ is a smooth and bounded domain, $\mu > 0$ and $n$ denotes the outward unit normal vector of $\partial \Omega$. Lin and Ni (1986) conjectured that for $\mu$ small, all solutions are constants. We show that this conjecture is false for all dimensions in some (partially symmetric) nonconvex domains $\Omega$. Furthermore, we prove that for any fixed $\mu$, 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.