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

By Liping Wang, Jun-cheng Wei, Shusen Yan. 2010

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We consider the following nonlinear Neumann problem: {―Au + μu = u N+2 N―2 , u > 0 in Ω, ∂u ∂n = 0 on ∂Ω, where Ω C 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.

Subfields: Applied Mathematics, Computational Theory and Mathematics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/d091cf57b60af75cff0b8ea4de028384d18196d5