LAUSR

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.