Full paper in PDF format:
$%D. Cao and J. Chabrowski, Critical Neumann problem with competing Hardy potentials, Rev. Mat. Complut. 20 (2007), no. 2, 309–338.%$

Critical Neumann Problem with Competing Hardy Potentials
Daomin CAO and Jan CHABROWSKI
Institute of Applied Mathematics
Academy of Mathematics and Systems Science
Chinese Academy of Sciences
Beijing 100080 — PR China
dmcao@amt.ac.cn
Department of Mathematics
University of Queensland
St. Lucia 4072, Qld — Australia
jhc@maths.uq.edu.au

Received: September 9, 2006
Accepted: March 1, 2007

ABSTRACT

In this paper we investigate the solvability of the nonlinear Neumann problem

{-Δu + λ--u-2 - γ-u2 = Q(x)|u|2*-2u in Ω, ∂u |x-a| |x| ∂ν = 0 on ∂Ω, u> 0 on Ω,

involving a critical Sobolev nonlinearity and two competing Hardy potentials in a bounded domain. We examine the common effect of the shape of the graph of the weight function, the mean curvature of the boundary and Hardy potentials on the existence of solutions of this problem. We are mainly interested in the existence of positive solutions. We also obtain the existence of sign-changing solutions.

Key words: critical Sobolev exponent, Hardy potential, concentration-compactness principle, Neumann problem.
2000 Mathematics Subject Classification: 35B33, 35J65, 35Q55.