Annales Academię Scientiarum Fennicę
Mathematica
Volumen 35, 2010, 115-130
Hiroshima University, Department of Mathematics,
Graduate School of Science
Higashi-Hiroshima 739-8521, Japan;
mizuta 'at' mis.hiroshima-u.ac.jp
Hiroshima National College of Maritime Technology, General Arts
Higashino Oosakikamijima Toyotagun 725-0231, Japan;
ohno 'at' hiroshima-cmt.ac.jp
Hiroshima University, Department of Mathematics,
Graduate School of Education
Higashi-Hiroshima 739-8524, Japan;
tshimo 'at' hiroshima-u.ac.jp
Yokohama National University, Department of Mathematics
Tokiwadai, Hodogaya-ku,
Yokohama 240-8501, Japan;
shioji 'at' math.sci.ynu.ac.jp
Abstract. We give a sufficient condition for the compact embedding from W0k,p(.)(\Omega) to Lq(.)(\Omega) in case essinfx \in \Omega(Np(x)/(N-kp(x))-q(x)) = 0, where \Omega is a bounded open set in RN. As an application, we find a nontrivial nonnegative weak solution of the nonlinear elliptic equation
-div(|\nabla u(x)|p(x)-2 \nabla u(x)) = |u(x)|q(x)-2u(x) in \Omega, u(x) = 0 on \partial\Omega.
We also consider the existence of a weak solution to the problem above even if the embedding is not compact.
2000 Mathematics Subject Classification: Primary 35J20, 46B50, 46E30.
Key words: Sobolev spaces of variable exponents, compact embeddings, nonlinear elliptic problems.
Reference to this article: Y. Mizuta, T. Ohno, T. Shimomura and N. Shioji: Compact embeddings for Sobolev spaces of variable exponents and existence of solutions for nonlinear elliptic problems involving the p(x)-Laplacian and its critical exponent. Ann. Acad. Sci. Fenn. Math. 35 (2010), 115-130.
doi:10.5186/aasfm.2010.3507
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