The Brezis-Nirenberg type critical problem for the nonlinear Choquard equation

By Fashun Gao, Minbo Yang. 2016

327 citations recorded by the source corpus.

We establish some existence results for the Brezis-Nirenberg type problem of the nonlinear Choquard equation $$ - \Delta u = \left( {\int_\Omega {\frac{{{{\left| {u\left( y \right)} \right|}^{2_\mu ^*}}}}{{{{\left| {x - y} \right|}^\mu }}}dy} } \right){\left| u \right|^{2_\mu ^* - 2}}u + \lambda uin\Omega ,$$−Δu=(∫Ω|u(y)|2μ*|x−y|μdy)|u|2μ*−2u+λuinΩ,, where Ω is a bounded domain of RN with Lipschitz boundary, λ is a real parameter, N ≥ 3, $$2_\mu ^* = \left( {2N - \mu } \right)/\left( {N - 2} \right)$$2μ*=(2N−μ)/(N−2) is the critical exponent in the sense of the Hardy-Littlewood-Sobolev inequality.

Subfields: Applied Mathematics, Computational Theory and Mathematics, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/fba8beede8bc6a973c42c8d29c160d0a89d208fb

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In Applied Mathematics

1. Fashun Gao
2. Xinfu Li
3. Minbo Yang
4. Daniele Cassani
5. Jianjun Zhang

In Computational Theory and Mathematics

1. Xinfu Li
2. Fashun Gao
3. Minbo Yang

In Mathematical Physics

1. Fashun Gao
2. Xinfu Li
3. Minbo Yang

Citations in the corpus

• Singularly perturbed critical Choquard equations (2016, unknown): “t m1,ν max,e ≤ m1,ν max,ν max, applying Lemma 2.8 with the fact that e<νmax, this yields a contradiction, proving the lemma. The following Brezis-Lieb type lemma for the nonlocal term is proved in [21]. Lemma 3.5. Let N≥ 3 and 0 <µ<N. If (un) is a bounded sequence in L 2N N−2(RN) such that ”
• Multiple solutions for critical Choquard-Kirchhoff type equations (2020, unknown): “3 of [40] in the subcase s = 1 and p = 2, see also [16], yields ‖wn‖ 2·2μ * = ‖un‖ 2·2μ * − ‖u‖ 2·2μ * + o(1).”
• Standing waves to upper critical Choquard equation with a local perturbation: Multiplicity, qualitative properties and stability (2021, background): “While Gao and Yang [24] obtained the solution to the equation u I u u u xΔ ,α p p N¯ ¯ 2 ∣ ∣ ∣ ∣( )− = ∗ ∈− (1.6) in D N1,2 ( ).”
• On nonlocal Choquard equations with Hardy-Littlewood-Sobolev critical exponents (2016, background): “δ and let ψ∈C0∞(Ω) be such that ψ(x) = ˆ 1 if x∈Bδ, 0 if x∈RN \Ω, 0 ≤ψ(x) ≤1 ∀x∈RN, |Dψ(x)|≤C= const ∀x∈RN. We define, for ε>0, Uε(x) := ε 2−N 2 U( x ε ), uε(x) := ψ(x)Uε(x). (2.6) From [28], we know that Z Ω |∇uε|2dx= C(N,µ)) N−2 2N−µ· N 2 S2 H,L +O(εN−2) (2.7) 10 and Z Ω Z Ω |uε(x)|2 ∗ µ |u”