Ground State Solutions Research Articles

Concentration of ground state solutions for critical generalized quasilinear equation with steep potential well

In this work, we are looking at the generalized quasilinear Schrödinger equations with critical growth and a steep potential well: where and is an even function and satisfies some suitable conditions (such as ). By using a change of variable and variational methods, for big enough , the existence of a ground state solution is obtained and its concentration behavior is also discussed.

Variational problems for the system of nonlinear Schrödinger equations with derivative nonlinearities

We consider the Cauchy problem of the system of nonlinear Schrödinger equations with derivative nonlinearlity. This system was introduced by Colin and Colin (Differ Int Equ 17:297–330, 2004) as a model of laser-plasma interactions. We study existence of ground state solutions and the global well-posedness of this system by using the variational methods. We also consider the stability of traveling waves for this system. These problems are proposed by Colin–Colin as the open problems. We give a subset of the ground-states set which satisfies the condition of stability. In particular, we prove the stability of the set of traveling waves with small speed for 1-dimension.

On a class of elliptic equations with critical perturbations in the hyperbolic space

We study the existence and non-existence of positive solutions for the following class of nonlinear elliptic problems in the hyperbolic space − Δ B N u − λ u = a ( x ) u p − 1 + ε u 2 ∗ − 1 in B N , u ∈ H 1 ( B N ) , where B N denotes the hyperbolic space, 2 < p < 2 ∗ : = 2 N N − 2 , if N ⩾ 3 ; 2 < p < + ∞, if N = 2, λ < ( N − 1 ) 2 4 , and 0 < a ∈ L ∞ ( B N ). We first prove the existence of a positive radially symmetric ground-state solution for a ( x ) ≡ 1. Next, we prove that for a ( x ) ⩾ 1, there exists a ground-state solution for ε small. For proof, we employ “conformal change of metric” which allows us to transform the original equation into a singular equation in a ball in R N . Then by carefully analysing the energy level using blow-up arguments, we prove the existence of a ground-state solution. Finally, the case a ( x ) ⩽ 1 is considered where we first show that there is no ground-state solution, and prove the existence of a bound-state solution (high energy solution) for ε small. We employ variational arguments in the spirit of Bahri–Li to prove the existence of high energy-bound-state solutions in the hyperbolic space.

Magnetic ground state of NdB4 : Interplay between anisotropic exchange interactions and hidden order on a Shastry-Sutherland lattice

The neodymium tetraboride NdB4, crystalizing into a tetragonal crystal structure, has the magnetic ions located on a Shastry-Sutherland lattice. The compound undergoes several magnetic phase transitions and exhibits a complex multi-k ground state that has so far been elusive. Here we report the solution and quantitative refinement of the magnetic ground state of NdB4 based on neutron diffraction data. The magnetic structure consists of two separate components; one of them is confined within the ab plane and makes an orthogonal all-in-all-out spin configuration maintaining translation symmetry of the crystal. Another is pointing along the c axis, and this component forms an anharmonic spin density wave consisting of three-up-two-down sequences with fivefold periodicity. The in-plane and out of plane spin structures represent distinct magnetic instabilities which normally would not coexist. Their simultaneous presence in the same phase is an interesting phenomenon associated with anisotropic exchange interactions and coupling to a symmetry breaking nonmagnetic order parameter undetectable by the available diffraction data. Published by the American Physical Society 2024

Non-radial ground state solutions for fractional Schrödinger–Poisson systems in $$\mathbb {R}^{2}$$

Non-radial ground state solutions for fractional Schrödinger–Poisson systems in $$\mathbb {R}^{2}$$

Second-order Sobolev gradient flows for computing ground state of ultracold Fermi gases

Based on density functional theory, the ground state for ultracold Fermi gases with dipole–dipole interaction is a functional minimization problem with orthonormality constraints. Many methods such as direct minimization, self-consistent iteration method and first-order gradient flow had been proposed to solve such problem. In this paper, we introduce the second-order Sobolev gradient flows, solve them to the steady state and get the ground state of ultracold Fermi gases numerically. Two contributions have been made: firstly we extend the usual first-order gradient flows to the second-order gradient flows; secondly, we extend the usual L2 gradient to the Sobolev gradient and get the Sobolev gradient flows. We prove that the second-order Sobolev gradient flows have the properties of orthonormality preserving and ‘modified’ energy diminishing, which are desirable for the computation of the ground state solution. Several numerical techniques have been used to discretize the time-dependent second-order Sobolev gradient flows. These include explicit leap-frog method and stabilized semi-implicit method for temporal discretization and Fourier pseudo-spectral method for spatial variables. Extensive numerical examples for the ground state are reported to demonstrate the power of the numerical methods.

Ground state solutions for magnetic Schrödinger equations with polynomial growth

Abstract In this article, we investigate the following nonlinear magnetic Schrödinger equations: ( − i ∇ + A ( x ) ) 2 u + V ( x ) u = f 1 ( x , ∣ v ∣ 2 ) v , ( − i ∇ + A ( x ) ) 2 v + V ( x ) v = f 2 ( x , ∣ u ∣ 2 ) u , \left\{\begin{array}{l}{\left(-i\nabla +A\left(x))}^{2}u+V\left(x)u={f}_{1}\left(x,{| v| }^{2})v,\\ {\left(-i\nabla +A\left(x))}^{2}v+V\left(x)v={f}_{2}\left(x,{| u| }^{2})u,\end{array}\right. where V V is the electric potential and A A is the magnetic potential. Assuming that the nonlinear function f i ( i = 1 , 2 ) {f}_{i}\left(i=1,2) satisfies three types of polynomial growth assumptions: super-quadratic, asymptotically quadratic, and local super-quadratic at ∣ x ∣ → ∞ | x| \to \infty , we prove the existence of the Nehari-Pankov type ground state solutions using critical point theory together with the non-Nehari manifold method. The resulting problem engages two major difficulties: the first one is that the associated functional is strongly indefinite, and the second lies in verifying the link geometry and showing the boundedness of Cerami sequences. Our results extend and complement the present ones in the literature.

Ground state solutions of Nehari-Pankov type for a indefinite Choquard equation with the Hardy potential and critical nonlinearity

In this paper, we are concerned with the following nonlinear Choquard-type equation with the Hardy potential and critical nonlinearity−Δu+(V(x)−ϑ|x|2)u=1p¯(Iα⁎F(x,u))f(x,u),x∈RN∖{0}, where N≥7, N−4<α<N, p¯=N+αN−2, f(x,u)=p¯|u|p¯−2u+g(x,u), F(x,u)=∫0uf(x,s)ds, the potential V(x) is 1-periodic and 0 lies in a gap of the spectrum of the Schrödinger operator −Δ+V. Under some mild assumptions on the subcritical term g, we obtain the existence of ground state solutions of Nehari-Pankov type for the above problem for sufficiently small ϑ>0.

Normalized ground state solutions for the Schrödinger systems with critical exponential growth in [formula omitted]

In this paper, we study the existence of normalized solutions to the following nonlinear Schrödinger systems with critical exponential growth:{−Δu+λ1u=f1(u)+βr1|u|r1−2u|v|r2,inR2,−Δv+λ2v=f2(v)+βr2|u|r1|v|r2−2v,inR2,∫R2u2dx=a2and∫R2v2dx=b2,u,v∈H1(R2), where 0<a,b<1,β>0,r1,r2>1 and r1+r2∈(4,+∞), f1,f2∈C(R,R) have critical exponential growth in the sense of Trudinger-Moser inequality. λ1,λ2∈R will arise as Lagrange multipliers. Under some suitable assumptions on f1,f2, we prove the existence of positive normalized ground state solutions for the problem when β>0 is sufficiently large via variational method. Our results improve and extend the previous results.

Ground state solutions of fractional equations with Coulomb potential and critical exponent

AbstractIn this paper, we are concerned with ground state solutions to fractional equations with Coulomb potential and critical exponent. The existence of ground state solutions is established under certain conditions by proposing a new analytical method.

Positive solutions for parametric equations with unbalanced growth and indefinite perturbation

We consider a nonlinear Dirichlet problem driven by the double phase operator. The reaction has the combined effects of parametric concave term and of an indefinite convex one (“concave-convex” problem with indefinite weight). Using the Nehari method we prove the existence of two bounded, positive ground state solutions.

Ground state solutions for the Hamilton–Choquard elliptic system with critical exponential growth

Ground state solutions for the Hamilton–Choquard elliptic system with critical exponential growth

Existence, uniqueness, and asymptotic behaviors of ground state solutions of Kirchhoff‐type equation with fourth‐order dispersion

In this paper, we focus on the following Schrödinger–Kirchhoff‐type problem with fourth‐order dispersion: where are constants and . We make use of Nehari manifold technique together with concentration‐compactness principle to prove that the above equation has at least a ground state solution for if , 6, and 7, and for if . Moreover, we also investigate the asymptotic behaviors of ground state solutions when some coefficients tend to zero. Among them, a uniqueness result about ground state solutions is obtained by implicit function theorem, and a blow‐up result is established by Pohozaev identity if dimension .

Ground state solutions for weighted biharmonic Kirchhoff problem via the Nehari manifold approach

In this article, we delve into the investigation of the following non-local problem within the unit ball B 1 of R 4 : g ( ∫ B 1 w β ( x ) | Δu | 2 ) Δ w β 2 u = | u | q − 2 u + f ( x , u ) in B 1 , u = ∂ n u = 0 on ∂ B 1 , where Δ w β 2 . = Δ ( w β ( x ) Δ . ) is the weighted Biharmonic operator, where w β ( x ) represents a logarithmically singular weight, and the non-linearity involves a combination of a reaction source f ( x , u ) , critical in the context of Adams' type exponential inequality and a polynomial function. The Kirchhoff function g is characterized by being positive and continuous. Employing the Nehari manifold method, the quantitative deformation lemma and degree theory results, we establish the existence of a ground state solution.

Local uniqueness of ground states for the generalized Choquard equation

We consider the generalized Choquard equation of the type -ΔQ+Q=I(|Q|p)|Q|p-2Q,\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\begin{aligned} -\\Delta Q + Q = I(|Q|^p)|Q|^{p-2} Q, \\end{aligned}$$\\end{document}for 3≤n≤5\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$3\\le n\\le 5$$\\end{document}, with Q∈Hrad1(Rn),\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$Q \\in H^1_{rad}(\\mathbb {R}^n),$$\\end{document} where the operator I is the classical Riesz potential defined by I(f)(x)=(-Δ)-1f(x)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$I(f)(x) =(-\\Delta )^{-1}f(x)$$\\end{document} and the exponent p∈(2,1+4/(n-2))\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$p \\in (2,1+4/(n-2))$$\\end{document} is energy subcritical. We consider Weinstein-type functional restricted to rays passing through the ground state. The corresponding real valued function of the path parameter has an appropriate analytic extension. We use the properties of this analytic extension in order to show local uniqueness of ground state solutions. The uniqueness of the ground state solutions for the case p=2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$p=2$$\\end{document}, i.e. for the case of Hartree–Choquard, is well known. The main difficulty for the case p>2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$p > 2$$\\end{document} is connected with a possible lack of control on the Lp\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$L^p$$\\end{document} norm of the ground states as well on the lack of Sturm’s comparison argument.

Relationship between variational problems with norm constraints and ground state of semilinear elliptic equations in R2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathbb {R}^2$$\\end{document}

In this paper, we investigate variational problems in R2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathbb {R}^2$$\\end{document} with the Sobolev norm constraints and with the Dirichlet norm constraints. We focus on property of maximizers of the variational problems. Concerning variational problems with the Sobolev norm constraints, we prove that maximizers are ground state solutions of corresponding elliptic equations, while we exhibit an example of a ground state solution which is not a maximizer of corresponding variational problems. On the other hand, we show that maximizers of maximization problems with the Dirichlet norm constraints and ground state solutions of corresponding elliptic equations are the same functions, up to scaling, under suitable setting.

Regularity for critical fractional Choquard equation with singular potential and its applications

Abstract We study the following fractional Choquard equation ( − Δ ) s u + u ∣ x ∣ θ = ( I α * F ( u ) ) f ( u ) , x ∈ R N , {\left(-\Delta )}^{s}u+\frac{u}{{| x| }^{\theta }}=({I}_{\alpha }* F\left(u))f\left(u),\hspace{1em}x\in {{\mathbb{R}}}^{N}, where N ⩾ 3 N\geqslant 3 , s ∈ 1 2 , 1 s\in \left(\phantom{\rule[-0.75em]{}{0ex}},\frac{1}{2},1\right) , α ∈ ( 0 , N ) \alpha \in \left(0,N) , θ ∈ ( 0 , 2 s ) \theta \in \left(0,2s) , and I α {I}_{\alpha } is the Riesz potential. The main purpose of this article is twofold. We first study the regularity of weak solutions for the aforementioned equation with critical nonlinearity, which extends the results of θ = 0 \theta =0 in Moroz-Van Schaftingen [Existence of groundstates for a class of nonlinear Choquardequations, Trans. Amer. Math. Soc. 367 (2015), no. 9, 6557–6579]. Then, as an application of the regularity results, we establish the existence of ground state solutions for above equation with the nonlinearity involving embedding top and bottom indices, which is related to the Hardy-Littlewood-Sobolev inequality and singular term 1 ∣ x ∣ θ \frac{1}{{| x| }^{\theta }} . It is worth noting that our approach is not involving the concentration-compactness principle.

Ground State Solutions of Nehari-Pohozaev Type for Schrödinger–Poisson–Slater Equation with Zero Mass and Critical Growth

Ground State Solutions of Nehari-Pohozaev Type for Schrödinger–Poisson–Slater Equation with Zero Mass and Critical Growth

Ground state solutions for generalized quasilinear Schrödinger equations

In this paper we consider the generalized quasilinear Schrödinger equations − div ( g 2 ( u ) ∇ u ) + g ( u ) g ′ ( u ) | ∇ u | 2 + V ( x ) u = h ( x , u ) , x ∈ R N , where V and h are periodic in x i , 1 ⩽ i ⩽ N. By using variational methods, we prove the existence of ground state solutions, i.e., nontrivial solutions with least possible energy.

Anderson localization and quenched induced dynamics of spin orbit coupled Bose–Einstein condensate in a random potential

We consider the localization and dynamical properties of a one dimensional spin orbit coupled Bose–Einstein condensate trapped by a disordered speckle potential. We numerically solve coupled Gross–Pitaevskii equation to obtain ground sate solutions. The effects of spin–orbit coupling and detuning parameter on localization are investigated. It is found that the increase of spin–orbit coupling delocalizes the condensate while the increase of detuning favors localization. After achieving the numerical ground state solutions, we examine the quench induced dynamics of the condensate by the complete cessation of the spin–orbit coupling. We show that at parameters where the ground state is not localized, the dynamics of the system is chaotic.