THE EXISTENCE OF A NONTRIVIAL WEAK SOLUTION TO A DOUBLE CRITICAL PROBLEM INVOLVING A FRACTIONAL LAPLACIAN IN ■N WITH A HARDY TERM

作者：李工宝^*; 杨涛

来源：Acta Mathematica Scientia, 2020, 40(06): 1808-1830.

• existence of a weak solution
• fractional Laplacian
• double critical exponents
• Hardy term
• weighted Morrey space
• improved Sobolev inequality

In this paper,we consider the existence of nontrivial weak solutions to a double critical problem involving a fractional Laplacian with a Hardy term:■where ■,■ and ■.We show that problem(0.1) admits at least a weak solution under some conditions.To prove the main result,we develop some useful tools based on a weighted Morrey space.To be precise,we discover the embeddings ■ where ■ and ■.We also establish an improved Sobolev inequality,■ where ■,■ and ■ is a constant.Inequality(0.3) is a more general form of Theorem 1 in Palatucci,Pisante [1]. By using the mountain pass lemma along with(0.2) and(0.3), we obtain a nontrivial weak solution to problem(0.1) in a direct way. It is worth pointing out that(0.2) and(0.3)could be applied to simplify the proof of the existence results in [2] and [3].

• 单位
  华中师范大学