In this paper, we consider the following problem: where is the p-Laplacian operator, is the critical Sobolev exponent, are the Hardy–Littlewood–Sobolev critical upper exponents, the parameters satisfy some assumptions. First, we establish the refined Sobolev inequality with Coulomb norm, and show the corresponding best constant is achieved in by a nonnegative function. Second, by using the refined Sobolev inequality with Coulomb norm, the refined Sobolev inequality with Morrey norm and variational methods, we establish the existence of nonnegative ground state solution for the above problem.

中文翻译：

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具有有限多个临界非线性的 p-Laplacian 方程的基态解

在本文中，我们考虑以下问题：其中 是 p-Laplacian 算子，是临界 Sobolev 指数，是 Hardy-Littlewood-Sobolev 临界上指数，参数满足一些假设。首先，我们建立了具有库仑范数的细化 Sobolev 不等式，并证明了相应的最佳常数是通过非负函数实现的。其次，利用库仑范数的精化Sobolev不等式、Morrey范数的精化Sobolev不等式和变分方法，我们建立了上述问题的非负基态解的存在性。