Compact embedding from W 1,2 0 (Ω) to Lq(x)(Ω) and its application to nonlinear elliptic boundary value problem with variable critical exponent

We study the compact embedding from W 1,2 0 (Ω) to Lq(x)(Ω) with a variable critical exponent 1 q(x) 2N/(N −2), N 3 if there exist a point x0 ∈ Ω, a small η >0, 0 < l <1 and C0 > 0 such that q(x0) = 2N/(N − 2) and q(x) 2N/(N − 2) − C0/ (log(1/|x−x0|))l for |x−x0| η. As an application, we show an existence of a positive solution to the nonlinear elliptic boundary value problem − u = uq(x)−1 in Ω, u(x) = 0 on ∂Ω. © 2007 Elsevier Inc. All rights reserved