Hartree Type Nonlinearity Research Articles

Existence of least energy nodal solution for Kirchhoff-type system with Hartree-type nonlinearity

This paper deals with following Kirchhoff-type system with critical growth $ \begin{cases} -(a+ b\int _{\mathbb{R}^3}|\nabla u|^{2}dx)\Delta u+ V(x)u+\phi|u|^{p-2}u = |u|^{4}u+\mu f(u), ~ x\in\mathbb{R}^3, (-\Delta)^{\alpha/2}\phi = l|u|^p, ~ x\in \mathbb{R}^3, \end{cases} $ where $a, \mu \gt 0$, $b, l\geq0$, $\alpha\in(0, 3)$, $p\in[2, 3)$ and $\phi|u|^{p-2}u$ is a Hartree-type nonlinearity. By the minimization argument on the nodal Nehari manifold and the quantitative deformation lemma, we prove that the above system has a least energy nodal solution. Our result improve and generalize some interesting results which were obtained in subcritical case.

Small Data Scattering for Cubic Dirac Equation with Hartree Type Nonlinearity in $\mathbb{R}^{1+3}$

We prove that the initial value problem for the Dirac equation $(-i\gamma^\mu \partial_\mu + m) \psi= ( \frac{e^{- |x|}}{|x|} \ast ( \overline \psi \psi)) \psi \quad \text{in } \mathbb{R}^{1+3}$ ...

Finite time blow-up and global solutions for a nonlocal parabolic equation with Hartree type nonlinearity

This paper is concerned with the initial boundary value problem of a nonlocal parabolic equation. By establishing the comparison principle and studying the long-time behavior of its flow, we find the criteria for finite time blow-up and global existence of solutions respectively, which in particular includes the results of arbitrarily high energy initial data. We also characterize the asymptotic profile to both solutions vanishing at infinity and blowing up in finite time.

Ground state solutions for a class of Schrödinger-Poisson systems with Hartree-type nonlinearity

In this paper, we consider the following Schrödinger-Poisson system with Hartree-type nonlinearity where , , is a Riesz potential and . By using the Pohozaev type identity and the filtration of Nehari manifold, we show the existence of positive ground state solutions for the above system.

Asymptotics of The Spectrum of a Two-Dimensional Hartree-Type Operator with a Coulomb Self-Action Potential Near the Lower Boundaries of Spectral Clusters

We consider the eigenvalue problem for a perturbed two-dimensional oscillator where the perturbation is an integral Hartree-type nonlinearity with a Coulomb self-action potential. We obtain asymptotic eigenvalues and asymptotic eigenfunctions near the lower boundaries of spectral clusters formed in a neighborhood of the eigenvalues of the unperturbed operator and construct an asymptotic expansion near a circle where the solution is localized.

The Concentration Behavior of Ground States for a Class of Kirchhoff-type Problems with Hartree-type Nonlinearity

Abstract In this paper, we consider the Kirchhoff equation with Hartree-type nonlinearity { - ( ε 2 ⁢ a + ε ⁢ b ⁢ ∫ ℝ 3 | ∇ ⁡ u | 2 ⁢ d ⁢ x ) ⁢ Δ ⁢ u + V ⁢ ( x ) ⁢ u = ε μ - 3 ⁢ ( ∫ ℝ 3 K ⁢ ( y ) ⁢ F ⁢ ( u ⁢ ( y ) ) | x - y | μ ⁢ d ⁢ y ) ⁢ K ⁢ ( x ) ⁢ f ⁢ ( u ) , u ∈ H 1 ⁢ ( ℝ 3 ) , \left\{\begin{aligned} \displaystyle-&\displaystyle\biggl{(}\varepsilon^{2}a+% \varepsilon b\int_{\mathbb{R}^{3}}\lvert\nabla u\rvert^{2}\mathop{}\!dx\biggr{% )}\Delta u+V(x)u=\varepsilon^{\mu-3}\biggl{(}\int_{\mathbb{R}^{3}}\frac{K(y)F(% u(y))}{\lvert x-y\rvert^{\mu}}\mathop{}\!dy\biggr{)}K(x)f(u),\\ &\displaystyle u\in H^{1}(\mathbb{R}^{3}),\end{aligned}\right. where ε > 0 {\varepsilon>0} is a small parameter, a , b > 0 {a,b>0} , μ ∈ ( 0 , 3 ) {\mu\in(0,3)} , V , K {V,K} are two positive continuous function and F is the primitive function of f which is superlinear but subcritical at infinity in the sense of the Hardy–Littlewood–Sobolev inequality. We show that the equation admits a positive ground state solution for ε > 0 {\varepsilon>0} sufficiently small. Furthermore, we prove that these ground state solutions concentrate around such points which are both the minima points of the potential V and the maximum points of the potential K as ε → 0 {\varepsilon\to 0} .

Groundstates for Kirchhoff-Type Equations with Hartree-Type Nonlinearities

In this paper, we consider the following nonlinear problem of Kirchhoff-type with Hartree-type nonlinearities: $$\begin{aligned} \left\{ \begin{array}{ll} -\left( a+b\int _{\mathbb {R}^N}|Du|^2\right) \Delta u+V(x)u=(I_{\alpha }*|u|^{p})|u|^{p-2}u,&{}\quad x\in \mathbb {R}^N,\\ \\ u\in H^1(\mathbb {R}^N),\quad u>0,&{}\quad x\in \mathbb {R}^N, \end{array}\right. \end{aligned}$$ where $$N\ge 3$$ , $$\max \{0,N-4\}<\alpha <N$$ , $$2<p<\frac{N+\alpha }{N-2}$$ , $$a>0,b\ge 0$$ are constants, $$I_{\alpha }$$ is the Riesz potential and $$V{:}\,\mathbb {R}^N\rightarrow \mathbb {R}$$ is a potential function. Under certain assumptions on V, we prove that the problem has a positive ground state solution by using global compactness lemma, monotonicity technique and some new tricks recently given in the literature.

Ground states for Kirchhoff equation with Hartree-type nonlinearities

In this paper we study the Kirchhoff equation with Hartree-type nonlinearities − ( a + b ∫ R 3 | ∇ u | 2 ) Δ u + V ( x ) u = ( I α ⁎ | u | p ) | u | p − 2 u in R 3 , where a , b > 0 , V : R 3 → R is a potential function and I α is a Riesz potential of order α ∈ ( 0 , 3 ) . Under certain assumptions on V ( x ) , we prove the equation has a ground state for all ( 3 + α ) / 3 < p < 3 + α by variational methods.

Scattering results for Dirac Hartree-type equations with small initial data

We consider the Dirac equations with cubic Hartree-type nonlinearity which are derived by uncoupling the Dirac-Klein-Gordon systems. We prove small data global well-posedness and scattering results in the full scaling subcritical regularity regime. The strategy of the proof relies on the localized Strichartz estimates and bilinear estimates in \begin{document}$ V^2 $\end{document} spaces, together with the use of the null structure that the nonlinear term exhibits. This result is shown to be almost optimal in the sense that the iteration method based on Duhamel's formula fails over the supercritical range.

Classification of nonnegative solutions to a bi-harmonic equation with Hartree type nonlinearity

AbstractIn this paper, we are concerned with the following bi-harmonic equation with Hartree type nonlinearity$$\Delta ^2u = \left( {\displaystyle{1 \over { \vert x \vert ^8}}* \vert u \vert ^2} \right)u^\gamma ,\quad x\in {\open R}^d,$$where 0 &amp;lt; γ ⩽ 1 andd⩾ 9. By applying the method of moving planes, we prove that nonnegative classical solutionsuto (𝒫γ) are radially symmetric about some pointx0∈ ℝdand derive the explicit form foruin theḢ2critical case γ = 1. We also prove the non-existence of nontrivial nonnegative classical solutions in the subcritical cases 0 &amp;lt; γ &amp;lt; 1. As a consequence, we also derive the best constants and extremal functions in the corresponding Hardy-Littlewood-Sobolev inequalities.

Existence and concentration of nontrivial nonnegative ground state solutions to Kirchhoff-type system with Hartree-type nonlinearity

A Kirchhoff-type fractional elliptic system with Hartree-type nonlinearity is proposed to provide a unified framework for well-known nonlinear Schrodinger equations, Kirchhoff equations and Schrodinger–Poisson systems. The existence of nontrivial nonnegative ground state solutions to the system is proved when the coefficient of the potential function is larger than a threshold value, and a precise estimate of the threshold value is given for a prototypical example. It is also shown that the ground state solution concentrates on the zero set of the potential function when the coefficient tends to infinity.

Long-time Behavior of Solutions to Cubic Dirac Equation with Hartree Type Nonlinearity in ℝ1+2

Abstract In this paper we study the long-time behavior of solutions to the Dirac equation $$\begin{equation*} \big ( -i\gamma^\mu \partial_\mu + m \big) \psi= \left(V \ast ( \overline \psi \psi)\right) \psi \ \ \textrm{in } \ {\mathbb{R}}^{1+2},\end{equation*}$$where $V$ is the Yukawa potential in ${\mathbb{R}}^{2}$. It is proved that if $m&amp;gt;0$ and the initial data is small in $H^s({\mathbb{R}}^2)$ for $s&amp;gt;0$, the corresponding initial value problem is globally well posed and the solution scatters to free waves asymptotically as $t \rightarrow \pm \infty $. The main ingredients in the proof are Strichartz estimates and space-time $L^2$-bilinear null-form estimates for free waves.

Trilinear Lp estimates with applications to the Cauchy problem for the Hartree-type equation

In this paper, Lp estimates for a trilinear operator associated with the Hartree type nonlinearity are proved. Moreover, as application of these estimates, it is proved that after a linear transformation, the Cauchy problem for the Hartree-type equation becomes locally well posed in the Bessel potential and homogeneous Besov spaces under certain regularity assumptions on the initial data. This notion of well-posedness and the functional framework to solve the equation were firstly proposed by Y. Zhou.

Critical well-posedness and scattering results for fractional Hartree-type equations

Scattering for the mass-critical fractional Schrödinger equation with a cubic Hartree-type nonlinearity for initial data in a small ball in the scale-invariant space of three-dimensional radial and square-integrable initial data is established. For this, we prove a bilinear estimate for free solutions and extend it to perturbations of bounded quadratic variation. This result is shown to be sharp by proving the discontinuity of the flow map in the super-critical range.

Small data scattering of semirelativistic Hartree equation

In this paper we study the small data scattering of Hartree type semirelativistic equation in space dimension 3. The Hartree type nonlinearity is [V∗|u|2]u and the potential V which generalizes the Yukawa has some growth condition. We show that the solution scatters to linear solution if an initial data given in Hs,1 is sufficiently small and s>14. Here, Hs,1 is Sobolev type space taking in angular regularity with norm defined by ‖φ‖Hs,1=‖φ‖Hs+‖∇Sφ‖Hs. To establish the results we employ the recently developed Strichartz estimate which is Lθ2-averaged on the unit sphere S2 and construct the resolution space based on Up-Vp space.

Ground state solutions for asymptotically periodic Schr\xf6dinger\u2013Poisson systems involving Hartree-type nonlinearities

We use the non-Nehari manifold method to deal with the system \t\t\t{−Δu+V(x)u+ϕu=(∫R3Q(y)F(u(y))|x−y|μdy)Q(x)f(u(x)),x∈R3,−Δϕ=u2,u∈H1(R3),\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document}$$ \\textstyle\\begin{cases} -\\Delta u+V(x)u+\\phi u= (\\int_{\\mathbb{R}^{3}}\\frac {Q(y)F(u(y))}{|x-y|^{\\mu}}\\,dy )Q(x)f(u(x)),\\quad x\\in\\mathbb{R}^{3}, \\\\ -\\Delta\\phi=u^{2}, \\quad u \\in H^{1}(\\mathbb{R}^{3}), \\end{cases} $$\\end{document} where V(x) and Q(x) are periodic and asymptotically periodic in x. Under some mild conditions on f, we establish the existence of the Nehari type ground state solutions in two cases: the periodic one and the asymptotically periodic case.

Infinitely many solutions for Kirchhoff equations with sign-changing potential and Hartree nonlinearity

This paper is concerned with the following Kirchhoff-type equations: $$\begin{aligned} \left\{ \begin{array}{ll} \displaystyle -\big (a+b\int _{\mathbb {R}^{3}}|\nabla u|^{2}\mathrm {d}x\big )\Delta u+ V(x)u+\mu \phi |u|^{p-2}u=f(x, u)+g(x,u), &{} \text{ in } \mathbb {R}^{3},\\ (-\Delta )^{\frac{\alpha }{2}} \phi = \mu |u|^{p}, &{} \text{ in } \mathbb {R}^{3},\\ \end{array} \right. \end{aligned}$$ where $$a>0,~b,~\mu \ge 0$$ are constants, $$\alpha \in (0,3)$$ , $$p\in [2,3+2\alpha )$$ , the potential V(x) may be unbounded from below and $$\phi |u|^{p-2}u$$ is a Hartree-type nonlinearity. Under some mild conditions on V(x), f(x, u) and g(x, u), we prove that the above system has infinitely many nontrivial solutions. Specially, our results cover the general Schrodinger equations, the Kirchhoff equations and the Schrodinger–Poisson system.

Existence and concentration result for a class of fractional Kirchhoff equations with Hartree-type nonlinearities and steep potential well

In this paper, we study the following fractional Kirchhoff equations{(a+b∫RN|(−△)α2u|2dx)(−△)αu+λV(x)u=(|x|−μ⁎G(u))g(u),u∈Hα(RN),N≥3, where a,b>0 are constants, and (−△)α is the fractional Laplacian operator with α∈(0,1),2<2α,μ⁎=2N−μN−2α≤2α⁎=2NN−2α, 0<μ<2α, λ>0, is real parameter. 2α⁎ is the critical Sobolev exponent. g satisfies the Berestycki–Lions-type condition (see [2]). By using Pohožaev identity and concentration-compact theory, we show that the above problem has at least one nontrivial solution. Furthermore, the phenomenon of concentration of solutions is also explored. Our result supplements the results of Lü (see [8]) concerning the Hartree-type nonlinearity g(u)=|u|p−1u with p∈(2,6−α).

On the focusing energy-critical fractional nonlinear Schrödinger equations

We consider the fractional nonlinear Schrödinger equation (FNLS) with non-local dispersion $|\nabla|^{\alpha}$ and focusing energy-critical Hartree type nonlinearity $[-(|x|^{-2{\alpha}}*|u|^2)u]$. We first establish a global well-posedness of radial case in energy space by adopting Kenig-Merle arguments [20] when the initial energy and initial kinetic energy are less than those of ground state, respectively. We revisit and highlight long time perturbation, profile decomposition and localized virial inequality. As an application of the localized virial inequality, we provide a proof for finite time blowup for energy critical Hartree equations via commutator technique introduced in [2].

Multiple solutions for the Schrödinger equations with sign-changing potential and Hartree nonlinearity

In this paper, we study the following Schrödinger equation − Δ u + V λ ( x ) u + μ ϕ | u | p − 2 u = f ( x , u ) + β ( x ) | u | ν − 2 u , in R 3 , ( − Δ ) α 2 ϕ = μ | u | p , in R 3 , where μ ≥ 0 is a parameter, α ∈ ( 0 , 3 ) , ν ∈ ( 1 , 2 ) and p ∈ [ 2 , 3 + 2 α ) . V λ is allowed to be sign-changing and ϕ | u | p − 2 u is a Hartree-type nonlinearity. We require that V λ = λ V + − V − with V + having a bounded potential well Ω whose depth is controlled by λ . Under some mild conditions on V λ ( x ) and f ( x , u ) , we prove that the above system has at least two nontrivial solutions. Specially, our results cover the general Schrödinger equations and the Schrödinger–Poisson equations.