Existence of groundstates for a class of nonlinear Choquard equations

By Vitaly Moroz, Jean Van Schaftingen. 2012

463 citations recorded by the source corpus.

We prove the existence of a nontrivial solution 𝑢 ∈ H¹ (ℝ^N) to the nonlinear Choquard equation -Δ 𝑢 + 𝑢 = (I_α * 𝐹 (𝑢)) 𝐹' (𝑢) in ℝ^N, where I_α is a Riesz potential, under almost necessary conditions on the nonlinearity 𝐹 in the spirit of Berestycki and Lions. This solution is a groundstate; if moreover 𝐹 is even and monotone on (0, ∞), then 𝑢 is of constant sign and radially symmetric.

Subfields: Applied Mathematics, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/824ac36244e718e5891dc31d761a1ce6ab70f10e

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In Applied Mathematics

1. Xianhua Tang
2. Daniele Cassani
3. Fashun Gao
4. Xinfu Li
5. Minbo Yang
6. Sitong Chen
7. Jianjun Zhang
8. Binlin Zhang

In Mathematical Physics

1. Xinfu Li
2. Minbo Yang
3. Fashun Gao
4. Sitong Chen
5. Binlin Zhang
6. Xianhua Tang
7. Hongyu Ye
8. Jean Van Schaftingen

Citations in the corpus

• The existence of least energy nodal solutions for some class of Kirchhoff equations and Choquard equations (2015, background): “(Iα ∗|ψ|p)|ψ|p−2ψ in R+ × RN, 5 see [13, 26]. In the past years, the existence of positive solutions, ground states and semiclassical states for (1.15) have been studied by many researchers, see e.g. [10, 20, 24, 27,28], etc. However, there arefew works considering the existence ofsignchanging solut”
• Standing waves to upper critical Choquard equation with a local perturbation: Multiplicity, qualitative properties and stability (2021, background): “We follow the arguments of [29] which rely on polarization.”
• Choquard-type equations with Hardy–Littlewood–Sobolev upper-critical growth (2018, methodology): “Here we borrow some ideas from [4, 45] to establish boundedness, decay, positivity and symmetry of ground state solutions to (1.”
• Existence and non-existence results for Kirchhoff-type problems with convolution nonlinearity (2018, methodology): “Inspired by [24, 27, 28], we introduce the following basic assumption on f : (F1) f ∈ C(R,R), f(t) = o(|t|α/3) as |t|→ 0 and f(t) = o(|t|2+α) as |t|→∞.”
• Semi-classical states for the Choquard equation (2013, background): “The necessity of the restrictions on p follows from the adapted Pohožaev identity [19, lemma 2.1; 36; 42, theorem 2; 43, proposition 3.5].”