Compact Sobolev Embeddings Involving Symmetry And Its Applications

In this doctoral dissertation, we consider the special compact embedding theorems in the domainΩwhich is bounded and regular with cylindrical symmetry.We prove that functions having the same symmetry asΩin the variableexponent Sobolev space W₀^1,p(x)(Ω) can be embedded compactly into some weighted L^q(x)(Ω) spaces, with q(x) superior to the critical Sobolev exponent. Furthermore, we apply the above compact embedding theorems to the followingelliptic equations with supercritical nonlinearitiesandBy variational arguments and regularity theory of elliptic equations, we show the existence, multiplicity and regularity of the solutions of the above problems.