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Ground state solutions for a nonlinear Choquard equation [PDF]
, 2016 We discuss the existence of ground state solutions for the Choquard equation $$-Δu=(I_α*F(u))F'(u)\quad\quad\quad\text{in }\mathbb R^N.$$ We prove the existence of solutions under general hypotheses, investigating in particular the case of a homogeneous nonlinearity $F(u)=\frac{|u|^p}p$. The cases $N=2$ and $N\ge3$ are treated differently in some steps.
Battaglia, L.
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Dynamics of blow-up solutions for the Schrödinger–Choquard equation
Boundary Value Problems, 2018 In this paper, we study the dynamics of blow-up solutions for the nonlinear Schrödinger–Choquard equation iψt+Δψ=λ1|ψ|p1ψ+λ2(Iα∗|ψ|p2)|ψ|p2−2ψ. $$i\psi_{t}+\Delta \psi =\lambda_{1}\vert \psi \vert ^{p_{1}}\psi +\lambda_{2}\bigl(I _{\alpha }\ast \vert ...
Cunqin Shi, Kun Liu
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Choquard equations with mixed potential
Partial Differential Equations and Applications, 2023 In this paper, we study the following class of nonlinear Choquard equation, $$-Δu+a(z)u=K(u)f(u)\quad \text{in}\quad \R^N,$$ where $\R^N=\R^L\times\R^M$, $L\geq2$, $K(u)=|.|^{-γ}*F(u)$, $γ\in(0,N)$, $a$ is a continuous real function and $F$ is the primitive function of $f$. Under some suitable assumptions mixed on the potential $a$.
Romildo N. de Lima, Marco A. S. Souto
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Boundary Value Problems for Choquard Equations
Nonlinear Analysis, 2023 We prove existence of a positive radial solution to the Choquard equation $$-Δu +V u=(I_α\ast |u|^p)|u|^{p-2}u\qquad\text{in}\,\,\,Ω$$ with Neumann or Dirichlet boundary conditions, when $Ω$ is an annulus, or an exterior domain of the form $\mathbb{R}^N\setminus \bar{B}_a(0)$.
Bernardini C., Cesaroni A.
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On fractional Choquard equations [PDF]
Mathematical Models and Methods in Applied Sciences, 2015 We investigate a class of nonlinear Schrödinger equations with a generalized Choquard nonlinearity and fractional diffusion. We obtain regularity, existence, nonexistence, symmetry as well as decays properties.
D'AVENIA, Pietro +2 more
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Linear Barycentric Rational Method for Solving Schrodinger Equation
Journal of Mathematics, Volume 2021, Issue 1, 2021., 2021 A linear barycentric rational collocation method (LBRCM) for solving Schrodinger equation (SDE) is proposed. According to the barycentric interpolation method (BIM) of rational polynomial and Chebyshev polynomial, the matrix form of the collocation method (CM) that is easy to program is obtained.
Peichen Zhao, Yongling Cheng, Ram Jiwari
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Fractional Choquard equation with critical nonlinearities [PDF]
Nonlinear Differential Equations and Applications NoDEA, 2017 32 pages.
Mukherjee, T., Sreenadh, K.
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Ground State Solutions of Fractional Choquard Problems with Critical Growth
Fractal and Fractional, 2023 In this article, we investigate a class of fractional Choquard equation with critical Sobolev exponent. By exploiting a monotonicity technique and global compactness lemma, the existence of ground state solutions for this equation is obtained.
Jie Yang, Hongxia Shi
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Saddle solutions for the fractional Choquard equation [PDF]
Zeitschrift für angewandte Mathematik und Physik, 2022 We study the saddle solutions for the fractional Choquard equation \begin{align*} (-Δ)^{s}u+ u=(K_α\ast|u|^{p})|u|^{p-2}u, \quad x\in \mathbb{R}^N \end{align*} where $s\in(0,1)$, $N\geq 3$ and $K_α$ is the Riesz potential with order $α\in (0,N)$. For every Coxeter group $G$ with rank $1\leq k\leq N$ and $p\in[2,\frac{N+α}{N-2s})$, we construct a $G ...
Ying-Xin Cui, Jiankang Xia
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Advances in Nonlinear Analysis, 2020 In this paper, we study the singularly perturbed fractional Choquard ...
Yang Zhipeng, Zhao Fukun
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