Semiclassical Ground State Research Articles

Existence of Semiclassical Ground State Solutions for a Class of N-Laplace Choquard Equation with Critical Exponential Growth

Existence of Semiclassical Ground State Solutions for a Class of N-Laplace Choquard Equation with Critical Exponential Growth

Semiclassical solutions for a critical Choquard–Poisson system with competitive potentials

In this article, we establish the existence, concentration behaviour and decay estimation of semiclassical ground state solutions for a critical Choquard–Poisson system with multiple competitive potentials by variational methods. Moreover, we study the multiplicity of solutions by using the Lusternik–Schnirelmann theory.

A 3D-Schrödinger operator under magnetic steps with semiclassical applications

We define a Schrödinger operator on the half-space with a discontinuous magnetic field having a piecewise-constant strength and a uniform direction. Motivated by applications in the theory of superconductivity, we study the infimum of the spectrum of the operator. We give sufficient conditions on the strength and the direction of the magnetic field such that the aforementioned infimum is an eigenvalue of a reduced model operator on the half-plane. We use the Schrödinger operator on the half-space to study a new semiclassical problem in bounded domains of the space, considering a magnetic Neumann Laplacian with a piecewise-constant magnetic field. We then make precise the localization of the semiclassical ground state near specific points at the discontinuity jump of the magnetic field.

Semiclassical Ground State Solutions for a Class of Kirchhoff-Type Problem with Convolution Nonlinearity

Semiclassical Ground State Solutions for a Class of Kirchhoff-Type Problem with Convolution Nonlinearity

Singularly perturbed quasilinear Choquard equations with nonlinearity satisfying Berestycki\u2013Lions assumptions

In the present paper, we consider the following singularly perturbed problem: {−ε2Δu+V(x)u−ε2Δ(u2)u=ε−α(Iα∗G(u))g(u),x∈RN;u∈H1(RN),\\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}$$ \\textstyle\\begin{cases} -\\varepsilon ^{2}\\Delta u+V(x)u-\\varepsilon ^{2}\\Delta (u^{2})u= \\varepsilon ^{-\\alpha }(I_{\\alpha }*G(u))g(u), \\quad x\\in \\mathbb{R}^{N}; \\\\ u\\in H^{1}(\\mathbb{R}^{N}), \\end{cases} $$\\end{document} where varepsilon >0 is a parameter, Nge 3, alpha in (0, N), G(t)=int _{0}^{t}g(s),mathrm{d}s, I_{alpha }: mathbb{R}^{N}rightarrow mathbb{R} is the Riesz potential, and Vin mathcal{C}(mathbb{R}^{N}, mathbb{R}) with 0<min_{xin mathbb{R}^{N}}V(x)< lim_{|y|to infty }V(y). Under the general Berestycki–Lions assumptions on g, we prove that there exists a constant varepsilon _{0}>0 determined by V and g such that for varepsilon in (0,varepsilon _{0}] the above problem admits a semiclassical ground state solution hat{u}_{varepsilon } with exponential decay at infinity. We also study the asymptotic behavior of {hat{u}_{varepsilon }} as varepsilon to 0.

Ground state solutions to a class of critical Schrödinger problem

Abstract We consider the following critical nonlocal Schrödinger problem with general nonlinearities − ε 2 a + ε b ∫ R 3 | ∇ u | 2 Δ u + V ( x ) u = f ( u ) + u 5 , x ∈ R 3 , u ∈ H 1 ( R 3 ) , $$\begin{array}{} \displaystyle \left\{\begin{array}{} &amp;-\left(\varepsilon^{2}a+\varepsilon b\displaystyle\int\limits_{\mathbb{R}^{3}}|\nabla u|^{2}\right){\it\Delta} u+V(x)u=f(u)+u^{5}, &amp;x \in \mathbb{R}^{3},\\ &amp;u\in H^{1}(\mathbb{R}^{3}), \end{array}\right. \end{array}$$ (SKε ) and study the existence of semiclassical ground state solutions of Nehari-Pohožaev type to (SK ε ), where f(u) may behave like |u| q–2 u for q ∈ (2, 4] which is seldom studied. With some decay assumption on V, we establish an existence result which improves some exiting works which only handle q ∈ (4, 6). With some monotonicity condition on V, we also get a ground state solution v̄ ε and analysis its concentrating behaviour around global minimum x ε of V as ε → 0. Our results extend some related works.

Existence and Concentration of Semi-classical Ground State Solutions for Chern–Simons–Schrödinger System

In this paper, we consider the equation $$\begin{aligned} -\varepsilon ^{2}\Delta u+ V(x)u+\left( A_{0}(u)+A_{1}^{2}(u)+A_{2}^{2}(u)\right) u=f(u) \ \ \ \ \mathrm {in} ~ H^{1}({\mathbb {R}}^{2}), \end{aligned}$$ where $$\varepsilon $$ is a small parameter, V is the external potential, $$A_i(i=0,1,2)$$ is the gauge field and $$f\in C({\mathbb {R}}, {\mathbb {R}})$$ is 5-superlinear growth. By using variational methods and analytic technique, we prove that this system possesses a ground state solution $$u_\varepsilon $$ . Moreover, our results show that, as $$\varepsilon \rightarrow 0$$ , the global maximum point $$x_\varepsilon $$ of $$u_\varepsilon $$ must concentrate at the global minimum point $$x_0$$ of V.

Semiclassical ground state solutions for critical Schrödinger-Poisson systems with lower perturbations

In this paper, we study the following singularly perturbed Schrödinger-Poisson system{−ε2△u+V(x)u+ϕu=f(u)+u5,x∈R3,−ε2△ϕ=u2,x∈R3, where ε is a small positive parameter, V∈C(R3,R) and f∈C(R,R) satisfies neither the usual Ambrosetti-Rabinowitz type condition nor any monotonicity condition on f(u)/u3. By using some new techniques and subtle analysis, we prove that there exists a constant ε0>0 determined by V and f such that for ε∈(0,ε0] the above system admits a semiclassical ground state solution vˆε with exponential decay at infinity. We also study the asymptotic behavior of {vˆε} as ε→0. In particular, our results can be applied to the nonlinearity f(u)∼|u|q−2u for q∈[3,4], and extend the previous work that only deals with the case in which f(u)∼|u|q−2u for q∈(4,6).

Spin-valley density wave in moiré materials

We introduce and study a minimum two-orbital Hubbard model on a triangular lattice, which captures the key features of both the trilayer ABC-stacked graphene-boron nitride heterostructure and twisted transition metal dichalcogenides in a broad parameter range. Our model comprises first- and second-nearest neighbor hoppings with valley-contrasting flux that accounts for trigonal warping in the band structure. For the strong-coupling regime with one electron per site, we derive a spin-orbital exchange Hamiltonian and find the semiclassical ground state to be a spin-valley density wave. We show that a relatively small second-neighbor exchange interaction is sufficient to stabilize the ordered state against quantum fluctuations. Effects of spin- and valley Zeeman fields as well as thermal fluctuations are also examined.

Singularly perturbed Choquard equations with nonlinearity satisfying Berestycki-Lions assumptions

AbstractIn the present paper, we consider the following singularly perturbed problem:$$\begin{array}{} \displaystyle \left\{ \begin{array}{ll} -\varepsilon^2\triangle u+V(x)u=\varepsilon^{-\alpha}(I_{\alpha}*F(u))f(u), &amp; x\in \mathbb{R}^N; \\ u\in H^1(\mathbb{R}^N), \end{array} \right. \end{array}$$whereε&gt; 0 is a parameter,N≥ 3,α∈ (0,N),F(t) =$\begin{array}{} \int_{0}^{t} \end{array}$f(s)dsandIα: ℝN→ ℝ is the Riesz potential. By introducing some new tricks, we prove that the above problem admits a semiclassical ground state solution (ε∈ (0,ε0)) and a ground state solution (ε= 1) under the general “Berestycki-Lions assumptions” on the nonlinearityfwhich are almost necessary, as well as some weak assumptions on the potentialV. Whenε= 1, our results generalize and improve the ones in [V. Moroz, J. Van Schaftingen, T. Am. Math. Soc. 367 (2015) 6557-6579] and [H. Berestycki, P.L. Lions, Arch. Rational Mech. Anal. 82 (1983) 313-345] and some other related literature. In particular, we propose a new approach to recover the compactness for a (PS)-sequence, and our approach is useful for many similar problems.

Existence and concentration of semiclassical ground state solutions for the generalized Chern–Simons–Schrödinger system in [formula omitted

This paper is concerned with the following singularly perturbed problem in H1(R2)−ε2Δu+V(x)u+A0(u(x))u+∑j=12Aj2(u(x))u=f(u),ε(∂1A2(u(x))−∂2A1(u(x)))=−12u2,∂1A1(u(x))+∂2A2(u(x))=0,εΔA0(u)=∂1(A2|u|2)−∂2(A1|u|2),where ε is a small parameter, V∈C(R2,R) and f∈C(R,R). By using some new variational and analytic techniques joined with the manifold of Pohoz̆aev–Nehari type, we prove that there exists a constant ε0>0 determined by V and f such that for ε∈(0,ε0], the above problem admits a semiclassical ground state solution vˆε with exponential decay at infinity. We also establish a new concentration behaviour of {vˆε} as ε→0. In particular, our results are available to the nonlinearity f(u)∼|u|s−2u for s∈(4,6], which extend the existing results concerning the case f(u)∼|u|s−2u for s>6.

Semiclassical ground state solutions for a Choquard type equation in ℝ2with critical exponential growth

In this paper we study a nonlocal singularly perturbed Choquard type equation$$-\varepsilon^2\Delta u +V(x)u =\vr^{\mu-2}\left[\frac{1}{|x|^{\mu}}\ast \big(P(x)G(u)\big)\right]P(x)g(u)$$in ℝ2, whereεis a positive parameter, \hbox{$\frac{1}{|x|^\mu}$} with 0 &lt;μ&lt; 2 is the Riesz potential, ∗ is the convolution operator,V(x),P(x) are two continuous real functions andG(s) is the primitive function ofg(s). Suppose that the nonlinearitygis of critical exponential growth in ℝ2in the sense of the Trudinger-Moser inequality, we establish some existence and concentration results of the semiclassical solutions of the Choquard type equation in the whole plane. As a particular case, the concentration appears at the maximum point set ofP(x) ifV(x) is a constant.

A priori estimates and positivity for semiclassical ground states for systems of critical Schrödinger equations in dimension two

ABSTRACTWe consider in the whole plane the following Hamiltonian coupling of Schrödinger equationswhere V0>0, f,g have critical growth in the sense of Moser. We prove that the (nonempty) set S of ground-state solutions is compact in up to translations. Moreover, for each (u,v)∈S, one has that u,v are uniformly bounded in and uniformly decaying at infinity. Then we prove that actually the ground state is positive and radially symmetric. We apply those results to prove the existence of semiclassical ground-state solutions to the singularly perturbed systemwhere V∈𝒞(ℝ2) is a Schrödinger potential bounded away from zero. Namely, as the adimensionalized Planck constant 𝜀→0, we prove the existence of minimal energy solutions which concentrate around the closest local minima of the potential with some precise asymptotic rate.

Semiclassical ground state solutions for a Schrödinger equation in with critical exponential growth

Consider a Schrödinger equation urn:x-wiley:0025584X:media:mana201400267:mana201400267-math-0002where and are two continuous real functions on , ε is a positive parameter, the nonlinearity f is assumed to be of critical exponential growth in the sense of the Trudinger‐Moser inequality. By truncating the potentials and , we are able to establish some new existence and concentration results for critical Schrödinger equation in by variational methods. As a particular case, we observe that the concentration appears at the maximum set of the nonlinear potential which complements the results in , .

Existence of semiclassical ground state solutions for a generalized Choquard equation

In this paper, we study a generalized quasilinear Choquard equation−εpΔpu+V(x)|u|p−2u=εμ−N(∫RNQ(y)F(u(y))|x−y|μ)Q(x)f(u)in RN, where Δp is the p-Laplacian operator, 1<p<N, V and Q are two continuous real functions on RN, 0<μ<N, F(s) is the primate function of f(s) and ε is a positive parameter. Under suitable assumptions on p,μ and f, we establish a new concentration behavior of solutions for the quasilinear Choquard equation by variational methods. The results are also new for the semilinear case p=2.

On semiclassical ground state solutions for Hamiltonian elliptic systems

In this paper, we study the following Hamiltonian elliptic system with gradient termfor , where , is a small positive parameter, is a constant vector, and . Suppose that has at least one maximum, we prove that the system has ground state solutions for all sufficiently small . Moreover, we show that these solutions converge to the ground state solutions of the associated limit problem and concentrate to the maxima of in certain sense as .

Existence of semiclassical ground-state solutions for semilinear elliptic systems

This paper is concerned with the following semilinear elliptic equations of the formwhere ε is a small positive parameter, and where f and g denote superlinear and subcritical nonlinearity. Suppose that b(x) has at least one maximum. We prove that the system has a ground-state solution (ψε, φε) for all sufficiently small ε &gt; 0. Moreover, we show that (ψε, φε) converges to the ground-state solution of the associated limit problem and concentrates to a maxima point of b(x) in certain sense, as ε → 0. Furthermore, we obtain sufficient conditions for nonexistence of ground-state solutions.

Quantum signatures of the self-trapping transition in attractive lattice bosons

We consider the Bose-Hubbard model describing attractive bosonic particles hopping across the sites of a translation-invariant lattice, and compare the relevant ground-state properties with those of the corresponding symmetry-breaking semiclassical nonlinear theory. The introduction of a suitable measure allows us to highlight many correspondences between the nonlinear theory and the inherently linear quantum theory, characterized by the well-known self-trapping phenomenon. In particular we demonstrate that the localization properties and bifurcation pattern of the semiclassical ground-state can be clearly recognized at the quantum level. Our analysis highlights a finite-number effect.

Least-Action Tunneling Transmission Coefficient for Polyatomic Reactions.

We present a new least-action variational approximation for tunneling in polyatomic reactions based on the procedure developed by Garrett and Truhlar for atom-diatom reactions. (63) The method calculates the semiclassical ground-state tunneling probability at every tunneling energy by minimizing the value of imaginary action integral along a family of paths ranging from the minimum energy path to the straight path. The method is illustrated by applications to two hydrogen-atom abstraction reactions from methane using analytical potential energy surfaces.

Semiclassical droplet states in matrix quantum Hall effect

We derive semiclassical ground state solutions that correspond to the quantum Hall states earlier found in the Maxwell-Chern-Simons matrix theory. They realize the Jain composite-fermion construction and their density is piecewise constant as that of phenomenological wave functions. These results support the matrix theory as a possible effective theory of the fractional Hall effect. A crucial role is played by the constraint limiting the degeneracy of matrix states: we find its explicit gauge invariant form and clarify its physical interpretation.