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

• 作者: Gongbao LI Tao YANG
• 来源期刊: 数学物理学报（英文版）
• 年，卷（期）: 2020,（6）
• 页码范围: 1808-1830
• 页数: 23页
• 语种: 英文

摘要

In this paper, we consider the existence of nontrivial weak solutions to a double critical problem involving a fractional Laplacian with a Hardy term:(-△)su-γu/|x|zs=|u |2*s(β)-2/|x|β+ [Iμ*Fα(·,u)](x)fα(x,u),u∈Hs(Rn),(0.1) where s ∈ (0,1), 0 ≤ α,β ＜ 2s ＜ n, μ∈(0, n), γ ＜γH, Iμ(x) = |x|-μ，Fα(x,u) =|u(x)|2//μ(α)/|x|δμ(α) fα(x,u) =|u(x)|2//μ(α)-2u(x)/|x|δμ(α),2#μ(α) = (1-μ/2n) ?2*s(α), δμ(α) = (1-μ/2n)α, 2*s(α) =2(n-α)/n-2s and γH＝4sr2(n+2s/4)/r2(n-2s/ 4).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 s∈(0,1), 0 ＜ α ＜ 2s ＜ n, p∈[1, 2*s(α)) and r =2(α/2*s(α)). We also establish an improved Sobolev inequality,(∫RN|u(y)|2*s(α)/|y|αdy) 1/2*s(α)≤C‖u‖θHs(Rn)‖u‖1-θLp,n-2s/2P+Pr(Rn,|y|-pr), ?uHs(Rn), (0.3) where s ∈ (0,1), 0 ＜ α ＜ 2s ＜ n, p E [1,2*s(α)), r = α/2*s(α),0 ＜ max{α/2*s(α),2*s-1/2*s(α)}＜θ ＜ 1, 2S = 2n/n-2s and C = C(n,s,α) ＞ 0 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].