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Existence of positive solutions to the nonlinear Choquard equation with competing potentials [PDF]
Electronic Journal of Differential Equations, 2018 This article concerns the existence of positive solutions of the nonlinear Choquard equation $$ -\Delta u+a(x)u=b(x)\Big(\frac{1}{|x|}*|u|^2\Big)u,\quad u\in H^{1}({\mathbb R}^3), $$ where the coefficients a and b are positive functions such that
Jun Wang, Mengmeng Qu, Lu Xiao
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Infinitely many solutions for nonhomogeneous Choquard equations [PDF]
Electronic Journal of Qualitative Theory of Differential Equations, 2019 In this paper, we study the following nonhomogeneous Choquard equation \begin{equation*} \begin{split} -\Delta u+V(x)u=(I_\alpha*|u|^p)|u|^{p-2}u+f(x),\qquad x\in \mathbb{R}^N, \end{split} \end{equation*} where $N\geq3,\alpha\in(0,N),p\in \big[\frac{N ...
Tao Wang, Hui Guo
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Ground state solutions fora nonlinear Choquard equation [PDF]
, 2016 We discuss the existence of ground state solutions for the Choquard equation-Δu + u-(Iα←F(u)) F'(u) inRN. We prove the existence of solutions under general hypotheses, investigating in particular the case of a homogeneous nonlinearity F(u) =|u|p/p.
Battaglia, L.
core +5 more sources
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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Advances in Mathematical Physics, Volume 2023, Issue 1, 2023., 2023 This article is concerned with the initial‐value problem of a Schrödinger–Hartree equation in the presence of anisotropic partial/whole harmonic confinement. First, we get a sharp threshold for global existence and finite time blow‐up on the ground state mass in the L2‐critical case. Then, some new cross‐invariant manifolds and variational problems are
Min Gong, Hui Jian, Igor Freire
wiley +1 more source
Existence of Two Solutions for a Critical Elliptic Problem with Nonlocal Term in ℝ4
Journal of Function Spaces, Volume 2022, Issue 1, 2022., 2022 In this paper, we prove the existence of two positive solutions for a critical elliptic problem with nonlocal term and Sobolev exponent in dimension four.
Khadidja Sabri +4 more
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Advances in Mathematical Physics, Volume 2022, Issue 1, 2022., 2022 In this paper, we consider the following fourth order elliptic Kirchhoff‐type equation involving the critical growth of the form Δ2u−a+b∫ℝN∇u2dxΔu+Vxu=Iα∗Fufu+λu2∗∗−2u,in ℝN,u∈H2ℝN, where a > 0, b ≥ 0, λ is a positive parameter, α ∈ (N − 2, N), 5 ≤ N ≤ 8, V : ℝN⟶ℝ is a potential function, and Iα is a Riesz potential of order α.
Li Zhou, Chuanxi Zhu, Sergey Shmarev
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