Improved Sobolev inequalities involving weighted Morrey norms and the existence of nontrivial solutions to doubly critical elliptic systems involving fractional Laplacian and Hardy terms

Gongbao Li, ,Hubei Key Laboratory of Mathematical Sciences & School of Mathematics and Statistics, Central China Normal University, Wuhan, 430079, China, Tao Yang
2018 Discrete and Continuous Dynamical Systems. Series S  
In this paper, we prove two new improved Sobolev inequalities involving weighted Morrey norms inḢ s (R n )×Ḣ s (R n ) and D 1,p (R n )×D 1,p (R n ). For instance, the corresponding inequality inḢ s (R n ) ×Ḣ s (R n ) states that: there exists C = C(n, s, α, η 1 , η 2 ) > 0 such that for each (u, v) ∈Ḣ s (R n ) × H s (R n ), p ∈ [2, 2 * s (α)) and θ ∈ (θ, 2η 1 2 * s (α) ), it holds that 2010 Mathematics Subject Classification. Primary: 35J50; Secondary: 35A23, 35B33.
doi:10.3934/dcdss.2020469 fatcat:bzzzh6yehzevnkekakcqikemiu