Abstract
We give a sucient condition for the compact embedding from W k;p(¢) 0 (›) to L q(¢) (›) in case essinfx2›(Np(x)=(N i kp(x)) i q(x)) = 0, where › is a bounded open set in R N. As an application, we find a nontrivial nonnegative weak solution of the nonlinear elliptic equation idiv ‡ jru(x)j p(x)i2 ru(x) · = ju(x)j q(x)i2 u(x) in ›, u(x) = 0 on @›. We also consider the existence of a weak solution to the problem above even if the embedding is not compact.
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