Infinitely many solutions for the Hénon equation with critical exponent in non-convex domains

Article type: Research Article

Authors: Wang, Liping

Affiliations: Department of Mathematics, East China Normal University, Shanghai, P. R. China. Tel: +86 021 5434 2609; Fax: +86 021 5434 2609; E-mail: [email protected]

Abstract: We consider the Neumann problem for the Hénon equation  −Δu+u=|x|2αu(N+2)/(N−2), u>0, in Ω,  ∂u/∂n=0   on ∂Ω,  (0.1) where Ω⊂RN,N≥3 is a smooth and bounded domain, α>0 and n denotes the outward unit normal vector of ∂Ω. We show that problem (0.1) has infinitely many positive solutions, whose energy can be made arbitrarily large in some (partially symmetric) non-convex domains Ω. This seems to be a new phenomenon for the Hénon equation in bounded domains.

Keywords: Hénon equation, critical exponent, infinitely many positive solutions, energy arbitrarily large

DOI: 10.3233/ASY-2011-1037

Journal: Asymptotic Analysis, vol. 73, no. 4, pp. 203-223, 2011