Pseudo-relativistic Hartree Equation Research Articles

Limit behaviors of pseudo-relativistic Hartree equation with power-type perturbations

We consider the following pseudo-relativistic Hartree equations with power-type perturbation,i∂tψ=−Δ+m2ψ−(1|x|⁎|ψ|2)ψ+ε|ψ|p−2ψ, with (t,x)∈R×R3 where 2<p<3, ε>0 and m>0, p=83 can be viewed as a Slater modification. We mainly focus on the normalized ground state solitary waves φε, where ‖φε‖22=N. Firstly, we prove the existence and nonexistence of normalized ground states under L2-subcritical, L2-critical (p=83) and L2-supercritical perturbations. Secondly, we classify perturbation limit behaviors of ground states when ε→0+, and obtain two different blow-up profiles for N=Nc and N>Nc, where Nc be regard as “Chandrasekhar limiting mass”. We prove that 〈φε,−Δφε〉∼ε−23p−4 for N=Nc and 2<p<3, while 〈φε,−Δφε〉∼ε−23p−8 for N>Nc and 83<p<3. Finally, we study the asymptotic behavior for ε→+∞, and obtain an energy limit limε→+∞⁡eε(N)=12mN and a vanishing rate ∫R3|φε|pdx≲ε−1 when N>Nc and 83<p<3.

Vortex solutions for pseudo-relativistic Hartree equations

In this paper, we study k-vortex solutions of the form of the pseudo-relativistic Hartree equation 1 iψt(x,t)=(−Δ+m2−m)ψ(x,t)−(|x|−1∗|ψ|2)ψ(x,t), (x,t)∈R3×R, under the constraint Such solutions are obtained as minimizers of the problem 2 ek(N)=inf{Ek(u):u∈Hs∖{0},∫ℝ3|u(x,0)|2dx=N>0} with the associated functional of (1). We show that there is a threshold value such that problem (2) admits a nonnegative minimizer u N if , and there exists no minimizer for if . Moreover, the stability of the vortex solution is considered, and the limiting behavior of the minimizer u N as is described.

A constrained minimization problem related to two coupled pseudo-relativistic Hartree equations

We are concerned with the following constrained minimization problem:e(a1,a2,β):=inf⁡{Ea1,a2,β(u1,u2):‖u1‖L2(R3)=‖u2‖L2(R3)=1}, where Ea1,a2,β is the energy functional associated to two coupled pseudo-relativistic Hartree equations involving three parameters a1,a2,β and two trapping potentials V1(x) and V2(x). In this paper, we obtain the existence of minimizers of e(a1,a2,β) for possible a1,a2 and β under suitable conditions on the potentials, which generalizes the results of the papers [17–19] in different senses.

Standing waves for the pseudo-relativistic Hartree equation with Berestycki-Lions nonlinearity

We study the following class of pseudo-relativistic Hartree equations − ε 2 Δ + m 2 u + V ( x ) u = ε μ − N ( | x | − μ ⁎ F ( u ) ) f ( u ) in R N , where the nonlinearity satisfies general hypotheses of Berestycki-Lions type. By using the method of penalization arguments, we prove the existence of a family of localized positive solutions that concentrate at the local minimum points of the indefinite potential V ( x ) , as ε → 0 .

Stable solitary waves for pseudo-relativistic Hartree equations with short range potential

We consider solitary waves with prescribed stellar mass for the pseudo-relativistic Hartree equation, which is known to describe the dynamics of pseudo-relativistic boson stars in the meanfield limit. This leads to the study of the following nonlocal elliptic equation −Δ+m2u−(|x|−γ∗|u|2)u=μu,x∈R3under the normalized constraint ∫R3|ψ(t,x)|2dx=N,where γ∈(1,2), m>0, N>0 denotes the stellar mass of Boson stars and the frequency μ∈R is unknown and appears as Lagrange multiplier. In contrast to the mass-subcritical case γ∈(0,1) and mass-critical case γ=1, the corresponding pseudo-relativistic Hartree energy functional is always unbounded on the L2 sphere for any N>0, we prove that the above problem admits at least one solution uN, which is a ground state if N or m is sufficiently small. Furthermore, the stability of the corresponding solitary wave for the related time-dependent pseudo-relativistic Hartree equation is given. In addition, we also give an accurate description of the limiting behavior of uN as N→0+ and m→0+, respectively. The main contribution of this paper is to extend the main results in Guo and Zeng (2017), Lenzmann (2009) and Coti Zelati and Nolasco (2013) from long range potential case γ∈(0,1] to short range potential case γ∈(1,2).

Nodal solutions for pseudo-relativistic Hartree equations

In this paper, we investigate the existence of nodal solutions for the following problem [Display omitted] where 12<α<1,0<μ≤2 and 2N−μN−1<p<2N−μN−2α. We show that for any positive integer k, problem (P0) possesses a radially symmetrical solution changing sign exactly k-times.

Asymptotic behavior of ground states of generalized pseudo-relativistic Hartree equation

With appropriate hypotheses on the nonlinearity f , we prove the existence of a ground state solution u for the problem − Δ + m 2 u + V u = ( W ∗ F ( u ) ) f ( u ) in R N , where V is a bounded potential, not necessarily continuous, and F the primit

Blow-Up Profile of Pseudo-relativistic Hartree Equations with Singular Potentials

We consider the minimizers for the nonlinear Schrodinger functional $$\begin{aligned} \mathcal {E}_a(u)=\int _{{\mathbb {R}}^3}\left( u\sqrt{-\Delta +m^2}u+V(x)u^2\right) \hbox {d}x -\frac{a}{2}\int _{{\mathbb {R}}^3}\left( |x|^{-1}*u^2\right) u^2\hbox {d}x \end{aligned}$$ with the mass constraint $$\Vert u\Vert _{L^2({\mathbb {R}}^3)}=1$$ , for $$m\in {\mathbb {R}}$$ , $$a>0$$ and V(x) is a nonpositive singular potential. Our main results are the existence and blow-up behavior of minimizers when a tends to a critical value $$a^*=\Vert Q\Vert ^2_{L^2({\mathbb {R}}^3)}$$ , where Q is a positive ground state of the following problem $$\begin{aligned} \sqrt{-\Delta }u+u-\left( |x|^{-1}*u^2\right) u=0\quad \text{ in }\, {\mathbb {R}}^3,\quad u\in H^{\frac{1}{2}}({\mathbb {R}}^3). \end{aligned}$$

Vortex-type solutions for magnetic pseudo-relativistic Hartree equation

We deal with a class of magnetic pseudo-relativistic Hartree type equation where , m>0, is a continuous vector potential, is an external continuous scalar potential and is a convolution kernel, is a positive constant, , . Under the action of some subgroup of linear isometries on potential A and W, and some assumptions on the decay of A and W at infinity, we prove the existence of vortex-type solutions to this problem by using variational methods and asymptotic estimates as p = 2.

On critical pseudo-relativistic Hartree equation with potential well

The aim of this paper is to investigate the existence and asymptotic behavior of the solutions for the critical pseudo-relativistic Hartree equation $$ \sqrt{-\Delta+m^{2}}u+(\beta V(x)-\lambda)u =\bigg(\int_{\mathbb{R}^{N}}\frac{|u(z)|^{2_{\mu}^{\ast}}}{|x-z|^{\mu}}dz\bigg)|u| ^{2_{\mu}^{\ast}-2}u $$% for $\mathbb{R}^{N}$, where $m, \lambda, \beta\in\mathbb{R}^+$, $0&lt; \mu&lt; N$, $N\geq3$, $2_{\mu}^{\ast}=({2N-\mu})/({N-1})$ plays the role of critical exponent due to the Hardy-Littlewood-Sobolev inequality. By transforming the nonlocal problem into a local one via the Dirichlet-to-Neumann map, we are able to obtain the existence of the solutions by variational methods. Suppose that $0&lt; \lambda&lt; \lambda_{1}(\Omega)$ with $\lambda_{1}(\Omega)$ the first eigenvalue and the parameter $\beta$ is large enough, we can prove the existence of ground state solutions. Furthermore, for any sequences $\beta_{n}\rightarrow\infty$, we can show that the ground state solutions $\{u_{n}\}$ converges to a solution of $$ \sqrt{-\Delta+m^{2}}u-\lambda u= \bigg(\int_{\Omega}\frac{|u(z)|^{2_{\mu}^{\ast}}}{|x-z|^{\mu}}dz\bigg) |u|^{2_{\mu}^{\ast}-2}u \quad \mbox{in } \Omega, $$ where $\Omega :=\mbox{int} V^{-1}(0)$ is a nonempty bounded set with smooth boundary. By the way we also establish the existence and nonexistence results for the ground state solutions of the problems set on bounded domain.

Remarks about a generalized pseudo-relativistic Hartree equation

With appropriate hypotheses on the nonlinearity f, we prove the existence of a ground state solution u for the problem(−Δ+m2)σu+Vu=(W⁎F(u))f(u)in RN, where 0<σ<1, V is a bounded continuous potential and F the primitive of f. We also show results about the regularity of any solution of this problem.

Existence and mass concentration of pseudo-relativistic Hartree equation

In this paper, we investigate the constrained minimization problem e(a):=inf{u∈H,∥u∥22=1}Ea(u), where the energy functional Ea(u)=∫R3(u−Δ+m2 u + Vu2) dx − a2∫R3(|x|−1 * u2)u2 dx with m∈R, a&amp;gt;0, is defined on a Sobolev space H. We show that there exists a threshold a*&amp;gt;0 so that e(a) is achieved if 0&amp;lt;a&amp;lt;a* and has no minimizers if a≥a*. We also investigate the asymptotic behavior of non-negative minimizers of e(a) as a→a*.

The fractional Hartree equation without the Ambrosetti–Rabinowitz condition

We consider a class of pseudo-relativistic Hartree equations in presence of general nonlinearities not satisfying the Ambrosetti–Rabinowitz condition. Using variational methods based on critical point theory, we show the existence of two non trivial signed solutions, one positive and one negative.

Semiclassical analysis for pseudo-relativistic Hartree equations

In this paper we study the semiclassical limit for the pseudo-relativistic Hartree equation−ε2Δ+m2u+Vu=(Iα⁎|u|p)|u|p−2u,in RN, where m>0, 2≤p<2NN−1, V:RN→R is an external scalar potential, Iα(x)=cN,α|x|N−α is a convolution kernel, cN,α is a positive constant and (N−1)p−N<α<N. For N=3, α=p=2, our equation becomes the pseudo-relativistic Hartree equation with Coulomb kernel.

Ground states for the pseudo-relativistic Hartree equation with external potential

We prove the existence of positive ground state solutions to the pseudo-relativistic Schrödinger equationwhere N ≥ 3, m &gt; 0, V is a bounded external scalar potential and W is a radially symmetric convolution potential satisfying suitable assumptions. We also provide some asymptotic decay estimates of the found solutions.

Ground states for pseudo-relativistic Hartree equations of critical type

We study the existence of ground state solutions for a class of nonlinear pseudo-relativistic Schrödinger equations with critical two-body interactions. Such equations are characterized by a nonlocal pseudo-differential operator closely related to the square root of the Laplacian. We investigate this problem using variational methods after transforming the problem to an elliptic equation with a nonlinear Neumann boundary conditions.

Pseudorelativistic Hartree Equation with General Nonlinearity: Existence, Non-existence and Variational Identities

Abstract We prove several existence and non existence results of solitary waves for a class of nonlinear pseudo-relativistic Hartree equations with general nonlinearities. We use variational methods and some new variational identities involving the half Laplacian.

Uniqueness of ground states for pseudorelativistic Hartree equations

We prove uniqueness of ground states $Q$ in $H^{1/2}$ for pseudo-relativistic Hartree equations in three dimensions, provided that $Q$ has sufficiently small $L^2$-mass. This result shows that a uniqueness conjecture by Lieb and Yau in [CMP 112 (1987),147--174] holds true at least under a smallness condition. Our proof combines variational arguments with a nonrelativistic limit, which leads to a certain Hartree-type equation (also known as the Choquard-Pekard or Schroedinger-Newton equation). Uniqueness of ground states for this limiting Hartree equation is well-known. Here, as a key ingredient, we prove the so-called nondegeneracy of its linearization. This nondegeneracy result is also of independent interest, for it proves a key spectral assumption in a series of papers on effective solitary wave motion and classical limits for nonrelativistic Hartree equations.

Stellar Collapse in the Time Dependent Hartree-Fock Approximation

We prove blow-up in finite time for radially symmetric solutions to the pseudo-relativistic Hartree-Fock equation with negative energy. The non-linear Hartree-Fock equation is commonly used in the physics literature to describe the dynamics of white dwarfs. We extend thereby recent results by Fr\ohlich and Lenzmann, who established in \cite{FL1,FL2} blow-up for solutions to the pseudo-relativistic Hartree equation. As key ingredient for handling the exchange term we use the conservation of the expectation of the square of the angular momentum operator.

Effective dynamics for boson stars

We study solutions close to solitary waves of the pseudo-relativistic Hartree equation describing boson stars under the influence of an external gravitational field. In particular, we analyse the long-time effective dynamics of such solutions. In essence, we establish a (long-time) stability result for solutions describing boson stars that move under the influence of an external gravitational field. The proof of our main result tackles difficulties that are absent when deriving similar results on effective solitary wave motions for nonlinear Schrödinger equations or nonlinear wave equations. This is due to the fact that the pseudo-relativisitic Hartree equation does not exhibit Galilean or Lorentz covariance.