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

We study the compact embedding from 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, 00 such that q(x0)=2N/(N−2) and q(x)⩽2N/(N−2)−C0/(logl(1/|x−x0|)) for |x−x0|⩽η. As an application, we show an existence of a positive solution to the nonlinear elliptic boundary value problem −Δu=uq(x)−1 in Ω, u(x)=0 on ∂Ω.