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

2018
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Discrete and Continuous Dynamical Systems. Series S
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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