A nonlinear general Neumann problem involving two critical exponents

Article type: Research Article

Authors: Hadiji, Rejeb^; | Yazidi, Habib

Affiliations: Université Paris-Est, LAMA, Laboratoire d'Analyse et de Mathématiques Appliquées, UMR 8050, UPEC, F-94010, Créteil, France. E-mails: {hadiji, habib.yazidi}@u-pec.fr

Note: [] Corresponding author. E-mail: hadiji@u-pec.fr

Abstract: We discuss the existence of solutions to the following nonlinear problem involving two critical Sobolev exponents −div (p(x)∇u)=β|u|2*−2u+f(x,u)   in Ω, u≢0   in Ω, ∂u/∂ν=Q(x)|u|2*−2u   on ∂Ω, where β≥0, Q is continuous on ∂Ω, p∈H1(Ω) is continuous and positive in Ω¯ and f is a lower-order perturbation of |u|2*−1 with f(x,0)=0.

Keywords: Sobolev critical exponent, the trace embedding, variational problem, critical nonlinearity in the boundary, Palais–Smale condition, the mean curvature

DOI: 10.3233/ASY-141242

Journal: Asymptotic Analysis, vol. 89, no. 1-2, pp. 123-150, 2014