Groundstates of nonlinear Choquard equations: existence, qualitative properties and decay asymptotics

By Vitaly Moroz, Jean Van Schaftingen. 2012

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We consider a semilinear elliptic problem \[ - \Delta u + u = (I_\alpha \ast \abs{u}^p) \abs{u}^{p - 2} u\quad\text{in $\R^N$,} \] where $I_\alpha$ is a Riesz potential and $p>1$. This family of equations includes the Choquard or nonlinear Schr\"odinger--Newton equation. For an optimal range of parameters we prove the existence of a positive groundstate solution of the equation. We also establish regularity and positivity of the groundstates and prove that all positive groundstates are radially symmetric and monotone decaying about some point. Finally, we derive the decay asymptotics at infinity of the groundstates.

Subfields: Applied Mathematics, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/55c9cea367a0ea9ed507ceee55eb9b5b750277ae

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Citations in the corpus

Mass minimizers and concentration for nonlinear Choquard equations in $\R^N$ (2015, methodology): “ W0(( α+2 N ) 1 q+2λx) (1.12) in L2Ns N+α (RN) for N+α N ≤ s< N+α (N−2)+ as k→ +∞. Moreover, there exists xj0 ∈ {xi| λi = λ, 1 ≤ i≤ m} such that xk → xj0 as k→ +∞. Remark1.6. It has been proved in [19] that for α∈ [N−2,N) if N≥ 3 and α∈ (0,N) if N= 1,2, then each groundstate solution uof (1.11) s”
Singularly perturbed critical Choquard equations (2016, background): “In [24], Moroz and Van Schaftingen completely investigated the qualitative properties of solutions of (1.”
Semi-classical states for the Choquard equation (2013, methodology): “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].”
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”
On fractional Choquard equations (2014, background): “We refer the reader to [21] for sharp results about the exponential decay of ground state solutions in the case s = 1.”
Groundstates for a local nonlinear perturbation of the Choquard equations with lower critical exponent (2017, background): “1) The following classical Hardy–Littlewood–Sobolev inequality is a starting point of the variational approach to the problem (C∗) and implies by standard arguments [22,24] the well-definiteness, continuity and differentiability of the nonlocal term in the functional J defined by (2.”