Existence of groundstates for a class of nonlinear Choquard equations
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In this article, the authors prove the existence of a nontrivial solution to the nonlinear Choquard equation in ℝ^N, where I_α is a Riesz potential.Abstract:
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.read more
Citations
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Groundstates of nonlinear Choquard equations: existence, qualitative properties and decay asymptotics
TL;DR: In this paper, the authors considered a semilinear elliptic problem and proved the existence of a positive groundstate solution of the Choquard or nonlinear Schr\"odinger--Newton equation for an optimal range of parameters.
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A guide to the Choquard equation
TL;DR: A survey of the existence and properties of solutions to the Choquard type equations can be found in this paper, where some variants and extensions of its variants can also be found.
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The Brezis-Nirenberg type critical problem for the nonlinear Choquard equation
Fashun Gao,Minbo Yang +1 more
TL;DR: In this paper, the Brezis-Nirenberg type problem of the nonlinear Choquard equation was studied and existence results for the problem were established for the case where Ω is a bounded domain of R with Lipschitz boundary.
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Nodal solutions for the Choquard equation
TL;DR: In this paper, the general Choquard equation is considered and minimal action odd solutions for p ∈ (N + α N, N + α n − 2 ) and minimal-action nodal solutions for n ∈ 2, N+α N − 2, N+ α N−2 ) are presented.
Journal ArticleDOI
A guide to the Choquard equation
TL;DR: A survey of the existence and properties of solutions to the Choquard type equations can be found in this article, where some variants and extensions of its variants can also be found.
References
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