具有某种对称性的Sobolev紧嵌入定理及其应用
摘要
在这篇博士学位论文中,我们主要考虑有界光滑的柱对称区域Ω上变指数Sobolev空间W_0~(1,p(x))(Ω)中特殊的紧嵌入定理,即W_0~(1,p(x))(Ω)空间中具有与Ω相同对称性的函数可以嵌入到加权的L~(q(x))(Ω)空间中,其中q(x)可以超越临界Sobolev指数p~*(x)=(?).从而,应用此嵌入定理,变分方法,以及椭圆方程的正则性理论,我们分别得到了具有超临界非线性项的如下三类椭圆问题和解的存在性,多重性和正则性.
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 W_0~(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.
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 W_0~(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.
引文
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