For a weighted variable exponent Sobolev space, the compact and bounded embedding results are proved. For that, new boundedness and compact action properties are established for Hardy’s operator and its conjugate in weighted variable exponent Lebesgue spaces. Furthermore, the obtained results are applied to the existence of positive eigenfunctions for a concrete class of nonlinear ode with nonstandard growth condition.

中文翻译：

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关于可变指数Sobolev空间中的紧有界嵌入及其应用

对于加权变量指数Sobolev空间，证明了紧嵌入和有界嵌入的结果。为此，为Hardy算子及其在加权变量指数Lebesgue空间中的共轭建立了新的有界性和紧迫性。此外，将所得结果应用于具有非标准生长条件的具体一类非线性颂歌的正本征函数的存在。