A Neumann problem with critical exponent in nonconvex domains and Lin-Ni’s conjecture
Liping Wang, Juncheng Wei, Shusen Yan · Transactions of the American Mathematical Society · 2010
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