Results 51 to 60 of about 1,353 (146)
Existence of groundstates for a class of nonlinear Choquard equations in the plane
, 2017 We prove the existence of a nontrivial groundstate solution for the class of nonlinear Choquard equation $$ -\Delta u+u=(I_\alpha*F(u))F'(u)\qquad\text{in }\mathbb{R}^2, $$ where $I_\alpha$ is the Riesz potential of order $\alpha$ on the plane $\mathbb{R}
Battaglia, Luca, Van Schaftingen, Jean
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Groundstates of the Choquard equations with a sign-changing self-interaction potential
, 2018 We consider a nonlinear Choquard equation $$ -\Delta u+u= (V * |u|^p )|u|^{p-2}u \qquad \text{in }\mathbb{R}^N, $$ when the self-interaction potential $V$ is unbounded from below.
Battaglia, Luca, Van Schaftingen, Jean
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Nonlinear Choquard equations: Doubly critical case
Applied Mathematics Letters, 2018 Consider nonlinear Choquard equations \begin{equation*} \left\{\begin{array}{rcl} - u +u & = &(I_ *F(u))F'(u) \quad \text{in } \mathbb{R}^N, \\ \lim_{x \to \infty}u(x) & = &0, \end{array}\right. \end{equation*} where $I_ $ denotes Riesz potential and $ \in (0, N)$. In this paper, we show that when $F$ is doubly critical, i.e. $F(u) =
openaire +3 more sources
Solitary waves for Schrödinger–Choquard equation [PDF]
AIP Conference Proceedings, 2018 In this paper, by variational methods and the profile decomposition of bounded sequences in H1, we study the existence of solitary waves for the Schrodinger–Choquard equation with an L2-critical nonlinearity.
Boyan Kostadinov +2 more
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Existence of stable standing waves for the Schrödinger–Choquard equation
Boundary Value Problems, 2018 In this paper, by variational methods and the profile decomposition of bounded sequences in H1 $H^{1}$ we study the existence of stable standing waves for the Schrödinger–Choquard equation with an L2 $L^{2}$-critical nonlinearity. Our results extend some
Kun Liu, Cunqin Shi
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, 2017 We prove the existence of ground state solutions by variational methods to the nonlinear Choquard equations with a nonlinear perturbation \[ -{\Delta}u+ u=\big(I_\alpha*|u|^{\frac{\alpha}{N}+1}\big)|u|^{\frac{\alpha}{N}-1}u+f(x,u)\qquad \text{ in ...
Van Schaftingen, Jean, Xia, Jiankang
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Concentration phenomena for a fractional Choquard equation with magnetic field
, 2018 We consider the following nonlinear fractional Choquard equation $$ \varepsilon^{2s}(-\Delta)^{s}_{A/\varepsilon} u + V(x)u = \varepsilon^{\mu-N}\left(\frac{1}{|x|^{\mu}}*F(|u|^{2})\right)f(|u|^{2})u \mbox{ in } \mathbb{R}^{N}, $$ where $\varepsilon>0 ...
Ambrosio, Vincenzo
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Ground states of a non‐local variational problem and Thomas–Fermi limit for the Choquard equation
Journal of the London Mathematical Society, Volume 111, Issue 3, March 2025.Abstract We study non‐negative optimisers of a Gagliardo–Nirenberg‐type inequality ∫∫RN×RN|u(x)|p|u(y)|p|x−y|N−αdxdy⩽C∫RN|u|2dxpθ∫RN|u|qdx2p(1−θ)/q,$$\begin{align*} & \iint\nolimits _{\mathbb {R}^N \times \mathbb {R}^N} \frac{|u(x)|^p\,|u(y)|^p}{|x - y|^{N-\alpha }} dx\, dy\\ &\quad \leqslant C{\left(\int _{{\mathbb {R}}^N}|u|^2 dx\right)}^{p\theta } {\
Damiano Greco +3 more
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On a critical Choquard-Kirchhoff p-sub-Laplacian equation in ℍn
Analysis and Geometry in Metric SpacesThis article is devoted to the study of a critical Choquard-Kirchhoff pp-sub-Laplacian equation on the entire Heisenberg group Hn{{\mathbb{H}}}^{n}, where the Kirchhoff function KK can be zero at zero, i.e., the equation can be degenerate, and involving ...
Liang Sihua +3 more
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Abstract and Applied Analysis, Volume 2025, Issue 1, 2025.This paper is concerned with the positive solutions to a fractional‐order system with Hartree‐type nonlinearity and its equivalent integral system. We firstly use the regularity lifting lemma to obtain the integrability and smoothness of the solutions.
Yu-Cheng An +2 more
wiley +1 more source