High energy solutions of the Choquard equation

Cao, Daomin; Li, Hang

doi:10.3934/dcds.2018129

Cited by 12 publications

(4 citation statements)

References 18 publications

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“…The qualitative properties of solutions to fractional order or higher order Hartree or Choquard type equations have been extensively studied, for instance, see [1,12,14,15,29,31,32,34,35,36,57,59,62,66,67,68] and the references therein. In [35], Dai, Liu and Qin proved the super poly-harmonic properties of nonnegative classical solutions by using the outer-spherical average associated with (−∆)…”

Section: 2mentioning

confidence: 99%

Classification of solutions to conformally invariant systems with mixed order and exponentially increasing or nonlocal nonlinearity

Dai¹,

Qin²

2021

Preprint

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In this paper, under extremely mild assumption u(x) = O(|x| K ) at ∞ for some K ≥ 1 arbitrarily large, we prove classification of solutions to the following conformally invariant system with mixed order and exponentially increasing nonlinearity in R 2 :where p ∈ (0, +∞), u ≥ 0 and satisfies the finite total curvature condition R 2 u 4 (x)dx < +∞.In order to show integral representation formula and crucial asymptotic property for v, we derive and use an exp L +L ln L inequality, which is itself of independent interest. When p = 3 2 , the system is closely related to single conformally invariant equations (−∆)1 2 u = u 3 and (−∆)v = e 2v on R 2 , which have been quite extensively studied. We also derive classification results for nonnegative solutions to conformally invariant system with mixed order and Hartree type nonlocal nonlinearity in R 3 .

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Section: 2mentioning

confidence: 99%

Classification of solutions to conformally invariant systems with mixed order and exponentially increasing or nonlocal nonlinearity

Dai¹,

Qin²

2021

Preprint

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“…where N ≥ 3 and α ∈ (0, N ), We can refer to [5,15,17,20]. In [20], Moroz and Van Schaftingen obtained that problem (1.2) has a nontrivial solution when…”

Section: Introductionmentioning

confidence: 99%

Normalized solutions for Choquard equations with general nonlinearities

Yuan¹,

Chen²,

Tang³

2020

era

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In this paper, we prove the existence of positive solutions with prescribed L 2-norm to the following Choquard equation: −∆u − λu = (Iα * F (u))f (u), x ∈ R 3 , where λ ∈ R, α ∈ (0, 3) and Iα : R 3 → R is the Riesz potential. Under the weaker conditions, by using a minimax procedure and some new analytical techniques, we show that for any c > 0, the above equation possesses at least a couple of weak solution (ūc,λc) ∈ Sc × R − such that ūc 2 2 = c.

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“…In the cases γ > 0 and b < 0, (1.2) represents a Hartree model for crystals (see [37]). For more related results in three and higher dimensions, please refer to [4,3,7,8,15,16,17,19,22,35,36,33,43] and the references therein.…”

Section: Introductionmentioning

confidence: 99%

Existence and symmetry of solutions to 2-D Schrödinger-Newton equations

Cao,

Dai,

Zhang

2020

Preprint

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In this paper, we consider the following 2-D Schrödinger-Newton equationsBy using ideas from [13,21,42], under mild assumptions, we obtain existence of ground state solutions and mountain pass solutions to the above equations for p ≥ 2 and q ≥ 2p − 2 via variational methods. The auxiliary functional J 1 plays a key role in the cases p ≥ 3. We also prove the radial symmetry of positive solutions (up to translations) for p ≥ 2 and q ≥ 2. The corresponding results for planar Schrödinger-Poisson systems will also be obtained. Our theorems extend the results in [13,21] from p = 2 and b = 1 to general p ≥ 2 and b ≥ 0.

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High energy solutions of the Choquard equation

Cited by 12 publications

References 18 publications

Classification of solutions to conformally invariant systems with mixed order and exponentially increasing or nonlocal nonlinearity

Classification of solutions to conformally invariant systems with mixed order and exponentially increasing or nonlocal nonlinearity

Normalized solutions for Choquard equations with general nonlinearities

Existence and symmetry of solutions to 2-D Schrödinger-Newton equations

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