Positive Energy Solutions Research Articles

Least energy solutions for a weakly coupled fractional Schrödinger system

In this paper, we consider the following system of two weakly coupled fractional nonlinear Schrödinger equations {(−Δ)su+u=(∣u∣2p+b(x)∣u∣p−1∣v∣p+1)ux∈RN(−Δ)sv+ω2sv=(∣v∣2p+b(x)∣v∣p−1∣u∣p+1)vx∈RN. By use of the s-harmonic extension technique, we establish the existence of a nontrivial least energy solution of the system via variational methods. Especially, in the autonomous case i.e. b(x)≡b, a positive least energy solution with both nontrivial components is obtained.

On a class of coupled Schrödinger systems with critical Sobolev exponent growth

Consider the following two critical nonlinear Schrödinger systems: urn:x-wiley:mma:media:mma3598:mma3598-math-0002 urn:x-wiley:mma:media:mma3598:mma3598-math-0003 where is a smooth bounded domain, N≥3,−λ(Ω) < λ1,λ2<0,μ1,μ2>0,α,β≥1 with α + β = 2∗,γ ≠ 0,λ(Ω) is the first eigenvalue of −Δ with the Dirichlet boundary condition and urn:x-wiley:mma:media:mma3598:mma3598-math-0004 For N = 3,λ1=λ2,γ > 0 small, we obtain the existence of positive least energy solution of and . For N≥5,γ > 0, the existence of positive least energy solution of is established. For N≥5,γ ≠ 0, we prove that possesses a positive least energy solution. The limit behavior of the positive least energy solutions when γ→−∞ and phase separation for are also considered. Copyright © 2015 John Wiley & Sons, Ltd.

A semiclassical kinetic theory of Dirac particles and Thomas precession

Kinetic theory of Dirac fermions is studied within the matrix valued differential forms method. It is based on the symplectic form derived by employing the semiclassical wave packet build of the positive energy solutions of the Dirac equation. A satisfactory definition of the distribution matrix elements imposes to work in the basis where the helicity is diagonal which is also needed to attain the massless limit. We show that the kinematic Thomas precession correction can be studied straightforwardly within this approach. It contributes on an equal footing with the Berry gauge fields. In fact in equations of motion it eliminates the terms arising from the Berry gauge fields.

Blow up of arbitrarily positive initial energy solutions for a viscoelastic wave equation

In this paper, we consider the nonlinear viscoelastic equation: utt−△u+∫0tg(t−τ)△u(τ)dτ+|ut|m−2ut=|u|p−2u,inΩ×[0,T], with initial conditions and Dirichlet boundary conditions. For nonincreasing positive functions g, we show the finite time blow up of some solutions whose initial data have arbitrarily high initial energy.

Screening of electron electric dipole moment through the Foldy-Wouthuysen representation

The existent of the intrinsic electric dipole moments (EDM) lead to CP violation in a physical system. In the non-relativistic and point like limits, the effects of them in atoms are canceled which is well-known as Schiff screening effects. It is why that the energy shift due to the EDM is proportional to the expectation value of which vanishes in non-relativistic limit. In this paper, using Foldy-Wouthuysen representation we remove the odd terms (those terms mix the positive and negative energy solutions) up to order and then study the Schiff screening effects.

Global nonexistence of positive initial energy solutions for a viscoelastic wave equation

In this paper, we consider the nonlinear viscoelastic equation: |ut|ρutt−△u+∫0tg(t−τ)△u(τ)dτ+|ut|m−2ut=|u|p−2u,in Ω×[0,T], with initial conditions and Dirichlet boundary conditions. For nonincreasing positive functions g, we prove the nonexistence of global solutions with positive initial energy.

Critical conditions and finite energy solutions of several nonlinear elliptic PDEs in [formula omitted

This paper is concerned with the critical conditions of nonlinear elliptic equations and the corresponding integral equations involving Riesz potentials and Bessel potentials. We show that the equations and some energy functionals are invariant under the scaling transformation if and only if the critical conditions hold. In addition, the Pohozaev identity shows that those critical conditions are the necessary and sufficient conditions for existence of the finite energy positive solutions. Finally, we discuss respectively the existence of the negative solutions of the k-Hessian equations in the subcritical case, critical case and supercritical case. Here the Serrin type critical exponent and the Sobolev type critical exponent play key roles.

Least Energy Solutions for Nonlinear Schrödinger Systems with Mixed Attractive and Repulsive Couplings

Abstract In this paper we study the ground state solutions for a nonlinear elliptic system of three equations which comes from models in Bose-Einstein condensates. Comparing with existing works in the literature which have been on purely attractive or purely repulsive cases, our investigation focuses on the effect of mixed interaction of attractive and repulsive couplings. We establish the existence of least energy positive solutions and study asymptotic profile of the ground state solutions, giving indication of co-existence of synchronization and segregation. In particular we show symmetry breaking for the ground state solutions.

Positive high energy solution for Kirchhoff equation in $\mathbb{R}^{3}$ with superlinear nonlinearities via Nehari-Pohožaev manifold

In this paper, we study the following nonlinear problem of Kirchhoff type: \begin{equation}\label{(0.1)} \left\{% \begin{array}{ll} -\left(a+b\int\limits_{\mathbb{R}^3}|\nabla u|^2\right)\Delta u+V(x)u=f(u), & \hbox{$x\in \mathbb{R}^3$}, \\ u>0, & \hbox{$x\in \mathbb{R}^3$}, (0.1) \end{array}% \right.\end{equation} where $a,$ $b>0$ are constants, $V:\mathbb{R}^3\rightarrow\mathbb{R}$ and $f(t)$ is subcritical and superlinear at infinity. Under certain assumptions on non-constant potential $V$, we prove the existence of positive high energy solutions by using a linking argument with a barycenter map restricted on a Nehari-Pohožaev type manifold. Our main result has solved Kirchhoff equation (0.1) with superlinear nonlinearities, which has not been studied, and can be viewed as a partial extension of a recent result of He and Zou in [9] concerning Kirchhoff equations with 4-superlinear nonlinearities.

Nondegeneracy of positive solutions of semilinear elliptic problems in the hyperbolic space

In this article, we will study the nondegeneracy properties of positive finite energy solutions of the equation -Δ𝔹u - λu = |u|p-1u in the hyperbolic space. We will show that the degeneracy occurs only in an N-dimensional subspace. We will prove that the positive solutions are nondegenerate in the case of geodesic balls.

Multiplicity of nontrivial solutions for system of nonhomogenous Kirchhoff-type equations inRN

In this paper, we are interested in looking for multiple solutions for the following system of nonhomogenous Kirchhoff-type equations: (1.1) where constants a,c > 0;b,d,λ≥0, N = 1,2 or 3, f,g∈L2(RN) and f,g≢0, F∈C1(RN×R2,R), , V∈C(RN,R) satisfy some appropriate conditions. Under more relaxed assumptions on the nonlinear term F, the existence of one negative energy solution and one positive energy solution for the nonhomogenous system 2.1 is obtained by Ekeland's variational principle and Mountain Pass Theorem, respectively. Copyright © 2014 John Wiley & Sons, Ltd.

Positive least energy solutions for a coupled Schrödinger system with critical exponent

In this paper, we consider the following coupled Schrödinger system with doubly critical exponents, which can be seen as a counterpart of the Brezis–Nirenberg problem:{−Δu+λ1u=μ1u5+βu2v3,x∈Ω,−Δv+λ2v=μ2v5+βv2u3,x∈Ω,u>0,v>0,x∈Ω,u=v=0,x∈∂Ω, where Ω⊂R3 is a smooth bounded domain, λ1,λ2<0, μ1,μ2>0 and β>0. Under certain conditions on λ1,λ2 and β, we show that this problem has at least one positive least energy solution.

Positive least energy solutions and phase separation for coupled Schrödinger equations with critical exponent: higher dimensional case

We study the following nonlinear Schrodinger system which is related to Bose–Einstein condensate: $$\begin{aligned} \left\{ {\begin{array}{lll} -\Delta u +\lambda _1 u = \mu _1 u^{2^*-1}+\beta u^{\frac{2^*}{2}-1}v^{\frac{2^*}{2}}, \quad x\in \Omega -\Delta v +\lambda _2 v =\mu _2 v^{2^*-1}+\beta v^{\frac{2^*}{2}-1} u^{\frac{2^*}{2}}, \quad x\in \Omega u\ge 0, v\ge 0 \,\,\hbox {in } \Omega ,\quad u=v=0 \,\,\hbox {on } \partial \Omega . \end{array}}\right. \end{aligned}$$ Here $$\Omega \subset \mathbb R^N$$ is a smooth bounded domain, $$2^*:=\frac{2N}{N-2}$$ is the Sobolev critical exponent, $$-\lambda _1(\Omega )<\lambda _1,\lambda _2<0$$ , $$\mu _1,\mu _2>0$$ and $$\beta \ne 0$$ , where $$\lambda _1(\Omega )$$ is the first eigenvalue of $$-\Delta $$ with the Dirichlet boundary condition. When $$\beta =0$$ , this is just the well-known Brezis–Nirenberg problem. The special case $$N=4$$ was studied by the authors in (Arch Ration Mech Anal 205:515–551, 2012). In this paper we consider the higher dimensional case $$N\ge 5$$ . It is interesting that we can prove the existence of a positive least energy solution $$(u_\beta , v_\beta )$$ for any $$\beta \ne 0$$ (which can not hold in the special case $$N=4$$ ). We also study the limit behavior of $$(u_\beta , v_\beta )$$ as $$\beta \rightarrow -\infty $$ and phase separation is expected. In particular, $$u_\beta -v_\beta $$ will converge to sign-changing solutions of the Brezis–Nirenberg problem, provided $$N\ge 6$$ . In case $$\lambda _1=\lambda _2$$ , the classification of the least energy solutions is also studied. It turns out that some quite different phenomena appear comparing to the special case $$N=4$$ .

Radial fast diffusion on the hyperbolic space

We consider radial solutions to the fast diffusion equation $u_t=\Delta u^m$ on the hyperbolic space $\mathbb{H}^{N}$ for $N \ge 2$, $m\in(m_s,1)$, $m_s=\frac{N-2}{N+2}$. By radial we mean solutions depending only on the geodesic distance $r$ from a given point $o \in \mathbb{H}^N$. We investigate their fine asymptotics near the extinction time $T$ in terms of a separable solution of the form ${\mathcal V}(r,t)=(1-t/T)^{1/(1-m)}V^{1/m}(r)$, where $V$ is the unique positive energy solution, radial w.r.t. $o$, to $-\Delta V=c\,V^{1/m}$ for a suitable $c>0$, a semilinear elliptic problem thoroughly studied in \cite{MS08}, \cite{BGGV}. We show that $u$ converges to ${\mathcal V}$ in relative error, in the sense that $\|{u^m(\cdot,t)}/{{\mathcal V}^m(\cdot,t)}-1\|_\infty\to0$ as $t\to T^-$. In particular the solution is bounded above and below, near the extinction time $T$, by multiples of $(1-t/T)^{1/(1-m)}e^{-(N-1)r/m}$.

Giant Faraday effect due to Pauli exclusion principle in 3D topological insulators

Experiments using ARPES, which is based on the photoelectric effect, show that the surface states in 3D topological insulators (TI) are helical. Here we consider Weyl interface fermions due to band inversion in narrow-bandgap semiconductors, such as Pb1−xSnxTe. The positive and negative energy solutions can be identified by means of opposite helicity in terms of the spin helicity operator in 3D TI as , where β is a Dirac matrix and points perpendicular to the interface. Using the 3D Dirac equation and bandstructure calculations we show that the transitions between positive and negative energy solutions, giving rise to electron–hole pairs, obey strict optical selection rules. In order to demonstrate the consequences of these selection rules, we consider the Faraday effect due to the Pauli exclusion principle in a pump–probe setup using a 3D TI double interface of a PbTe/Pb0.31Sn0.69Te/PbTe heterostructure. For that we calculate the optical conductivity tensor of this heterostructure, which we use to solve Maxwell’s equations. The Faraday rotation angle exhibits oscillations as a function of probe wavelength and thickness of the heterostructure. The maxima in the Faraday rotation angle are of the order of mrds.

Blow-up of Positive Initial Energy Solutions for A System of Nonlinear Wave Equations with Supercritical Sources

The goal of this paper is to investigate the finite time blow-up of solutions with supercritical boundary/interior sources and nonlinear boundary/interior damping. First, we prove that if the interior and boundary sources dominate their corresponding damping terms, then every weak solution blows up in finite time with positive initial energy. Second, without any restriction on the boundary source, we prove the finite time blow-up of solutions, provided that the interior sources dominate both interior and boundary damping and the initial energy is nonnegative. A similar result has been shown when the boundary source is absent. Moreover, in the absence of the interior sources, we prove that the solution grows as an exponential function.

Existence of positive solutions, negative solutions and high energy solutions for quasi-linear elliptic equations on [formula omitted

In this paper, we study the following quasi-linear elliptic equations of the form −∑i,j=1NDj(ai,j(x,u)Diu)+12∑i,j=1NDsai,j(x,u)DiuDju+V(x)u=h(x,u),x∈RN, where h∈C(RN×R,R), N≥3 and V∈C(RN,R). Under appropriate assumptions on V(x) and h(x,u), some existence results for positive solutions, negative solutions and a sequence of high energy solutions are obtained via the perturbation method.

On the interaction of the Dirac oscillator with the Aharonov–Casher system in topological defect backgrounds

In this paper, we study the influence of the Aharonov–Casher effect [Y. Aharonov, A. Casher, Phys. Rev. Lett. 53 (1984) 319] on the Dirac oscillator in three different scenarios of general relativity: the Minkowski spacetime, the cosmic string spacetime and the cosmic dislocation spacetime. In this way, we solve the Dirac equation and obtain the energy levels for bound states and the Dirac spinors for positive-energy solutions. We show that the relativistic energy levels depend on the Aharonov–Casher geometric phase. We also discuss the influence of curvature and torsion on the relativistic energy levels and the Dirac spinors due to the topology of the cosmic string and cosmic dislocation spacetimes.

Morse index and symmetry-breaking for positive solutions of one-dimensional Hénon type equations

In this paper, the Morse index and the symmetry-breaking for positive solutions of the following two-point boundary value problem{u″+h(x)f(u)=0,x∈(−1,1),u(−1)=u(1)=0, are studied, where h∈C[−1,1]∩C1([−1,1]\\{0}), h(x)>0, h(−x)=h(x) on [−1,1]\\{0}, f∈C1[0,∞), f(s)>0 for s>0, and f(0)=0. The problem for the one-dimensional Hénon equation{u″+|x|lup=0,x∈(−1,1),u(−1)=u(1)=0 is a typical example, where l⩾0 and p>1. This problem always has the unique positive even solution. It is well-known that if l=0, then there is no positive non-even solution, and if l>0 is sufficiently large, then there exist positive non-even solutions. The result in this paper shows that if l(p−1)⩾4, then the Morse index of the positive least energy solution equals 1 and the Morse index of the positive even solution equals 2, and hence the positive least energy solution is non-even and symmetry-breaking phenomena occur. It is also shown that if l⩾0 and p>1 are sufficiently small, then there is no positive non-even solution and the Morse index of the even positive solution equals 1.

Parametric resonant states of charged fermions in the magnetar’s crust

We have presently derived the positive-energy solutions to the Dirac equation minimally coupled to a depth-dependent spatially harmonic tangential magnetostatic field to the magnetar crust, similar to the one proposed by Wareing and Hollerbach. It turns out that, for ultra-relativistic fermions and time-intervals much less the characteristic time (comparable to the average Ohmic timescale in the crust), the corresponding linearly independent modes get their depth-dependent amplitudes expressed in terms of Mathieu’s functions and therefore, non-trivial resonances arise, leading to instabilities in the system, for computable ranges of the model parameters. In order to detail these features, we have also discussed the current density components, pointing out the regions for which the particle density has a double bounded modulation. Finally as the magnetic field induction is increasing, the instability range gets larger triggering the exponential growth of the amplitudes, once the imaginary part of the Mathieu Characteristic Exponent becomes more and more dominant.