Ground State Solutions Research Articles

Quasi-one-dimensional harmonically trapped quantum droplets

We theoretically consider effectively one-dimensional quantum droplets in a symmetric Bose-Bose mixture confined in a parabolic trap. We systematically investigate ground and excited families of localized trapped modes which bifurcate from eigenstates of the quantum harmonic oscillator as the number of particles departs from zero. Families of nonlinear modes have nonmonotonous behavior of chemical potential on the number of particles and feature bistability regions. Excited states are unstable close to the linear limit, but become stable when the number of particles is large enough. In the limit of large density, we derive a modified Thomas-Fermi distribution. Smoothly decreasing the trapping strength down to zero, one can dynamically transform the ground-state solution to the solitonlike quantum droplet, while excited trapped states break in several moving quantum droplets.

On nonlinear fractional Schrödinger equations with indefinite and Hardy potentials

This paper is concerned with a class of fractional Schrödinger equation with Hardy potential ( − Δ ) s u + V ( x ) u − κ | x | 2 s u = f ( x , u ) , x ∈ R N , where s ∈ ( 0 , 1 ) and κ ⩾ 0 is a parameter. Under some suitable conditions on the potential V and the nonlinearity f, we prove the existence of ground state solutions when the parameter κ lies in a given range by using the non-Nehari manifold method. Moreover, we investigate the continuous dependence of ground state energy about κ. Finally, we are able to explore the asymptotic behavior of ground state solutions when κ tends to 0.

Axial Collective Mode of a Dipolar Quantum Droplet

In this work, we investigate the ground state properties and collective excitations of a dipolar Bose–Einstein condensate that self-binds into a quantum droplet, stabilized by quantum fluctuations. We demonstrate that a sum rule approach can accurately determine the frequency of the low energy axial excitation, using properties of the droplet obtained from the ground state solutions. This excitation corresponds to an oscillation in the length of the filament-shaped droplet. Additionally, we evaluate the static polarizabilities, which quantify change in the droplet dimensions in response to a change in harmonic confinement.

On the Existence of Ground State Solutions to a Quasilinear Schrödinger Equation involving p-Laplacian

On the Existence of Ground State Solutions to a Quasilinear Schrödinger Equation involving p-Laplacian

Existence of ground state solutions to some Nonlinear Schrödinger equations on lattice graphs

Existence of ground state solutions to some Nonlinear Schrödinger equations on lattice graphs

Ground State and Sign-Changing Solutions for Critical Schrödinger–Poisson System with Lower Order Perturbation

Ground State and Sign-Changing Solutions for Critical Schrödinger–Poisson System with Lower Order Perturbation

Positive and nodal ground state solutions for a critical Schrödinger-Poisson system with indefinite potentials

We consider the Schrödinger-Poisson system{−Δu+V(x)u+K(x)ϕu=a(x)|u|p−2u+|u|4u,x∈R3,−Δϕ=K(x)u2,x∈R3, where 4<p<6 and the potentials V, a are allowed to change their signs. Under some reasonable assumptions on V, K and a, we apply the constraint minimization argument to establish the existence of positive ground state solutions and ground state nodal solutions.

Multiple positive solutions for a Schrödinger-Poisson-Slater equation with critical growth

We consider the multiplicity of positive solutions for a Schrödinger-Poisson-Slater equation of the type−△u+(u2⁎1|4πx|)u=μ|u|p−1u+g(x)|u|4u,inR3, where μ>0, 3<p<5, and g∈C(R3,R+). Using Ekeland's variational principle and the well-known arguments of the concentration-compactness principle of Lions (1984) [24,25], when g has one local maximum point, we obtain a positive ground-state solution for all μ>0, while for g with k strict local maximum points, we prove that the equation has at least k distinct positive solutions for μ>0 small. The proof uses a minimization procedure on the Nehari manifold.

Ground state solutions of a Klein–Gordon–Maxwell system for two scalar fields

In this article, we consider a Klein–Gordon–Maxwell system for two scalar fields: We show that there exists a ground state solution under the condition and suitable constraints on and for .

Existence and multiplicity of solutions to magnetic Kirchhoff equations in Orlicz-Sobolev spaces

In this paper, we study the existence and multiplicity of weak solutions to a general type of Kirchhoff equations in magnetic fractional Orlicz-Sobolev spaces. Specifically, we appeal to Critical Point Theory to prove the existence of non-trivial solutions under the so-called Ambrosetti-Rabinowitz condition. We also state the existence of ground-state solutions. Moreover, multiplicity results which yield the existence of an unbounded sequence of solutions are also provided. Finally, we show existence under a weak-type Ambrosetti-Rabinowitz condition formulated in the framework of Orlicz spaces.

Existence and concentration behavior of positive solutions to Schrödinger-Poisson-Slater equations

Abstract This article is directed to the study of the following Schrödinger-Poisson-Slater type equation: − ε 2 Δ u + V ( x ) u + ε − α ( I α ∗ ∣ u ∣ 2 ) u = λ ∣ u ∣ p − 1 u in R N , -{\varepsilon }^{2}\Delta u+V\left(x)u+{\varepsilon }^{-\alpha }\left({I}_{\alpha }\ast | u{| }^{2})u=\lambda | u{| }^{p-1}u\hspace{1em}\hspace{0.1em}\text{in}\hspace{0.1em}\hspace{0.33em}{{\mathbb{R}}}^{N}, where ε , λ &gt; 0 \varepsilon ,\lambda \gt 0 are parameters, N ⩾ 2 N\geqslant 2 , ( α + 6 ) / ( α + 2 ) &lt; p &lt; 2 ∗ − 1 \left(\alpha +6)\hspace{0.1em}\text{/}\hspace{0.1em}\left(\alpha +2)\lt p\lt {2}^{\ast }-1 , I α {I}_{\alpha } is the Riesz potential with 0 &lt; α &lt; N 0\lt \alpha \lt N , and V ∈ C ( R N , R ) V\in {\mathcal{C}}\left({{\mathbb{R}}}^{N},{\mathbb{R}}) . By using variational methods, we prove that there is a positive ground state solution for the aforementioned equation concentrating at a global minimum of V V in the semi-classical limit, and then we found that this solution satisfies the property of exponential decay. Finally, the multiplicity and concentration behavior of positive solutions for the aforementioned problem is investigated by the Ljusternik-Schnirelmann theory. Our article improves and extends some existing results in several directions.

Ground state solution for a periodic p&amp;q‐Laplacian equation involving critical growth without the Ambrosetti–Rabinowitz condition

We study the ground state solutions for the following p&amp;q‐Laplacian equation where is a parameter large enough, with denotes the ‐Laplacian operator, and . Under some assumptions for the periodic potential , weight function and nonlinearity without the Ambrosetti–Rabinowitz condition, we show the above equation has a ground state solution.

Multiplicity and concentration of positive solutions for critical Choquard equations with concave perturbation

This paper focuses on three kinds of critical Choquard equations with concave perturbation λh(x)|u|q−2u, where λ>0, q∈(1,2) and h satisfies some mild conditions. For the upper critical and lower critical cases, by the Nehari manifold methods we obtain two positive solutions for λ>0 small enough, one of which is a ground state solution. For the doubly critical case, by minimization techniques and the mountain pass theorem, we derive two positive solutions for λ>0 small enough, one of which is a ground state positive solution. Moreover, the asymptotic behaviors of positive solutions as λ→0+ are verified.

The ground states of Hénon equations for [formula omitted]-Laplacian in [formula omitted] involving upper weighted critical exponents

In this paper, we investigate the existence of nontrivial ground state solutions of the Hénon equations with p-Laplacian in RN involving weighted upper critical exponents including Sobolev, Hardy–Sobolev and Hénon–Sobolev critical exponents. The Nehari manifold method and the mountain pass theorem are applied to get the nontrivial ground states for the problems under a subcritical weighted perturbation. Regularity and non-existence of the nontrivial solutions are also discussed.

Existence and Concentration of Ground State Solutions for Chern–Simons–Schrödinger System with General Nonlinearity

Existence and Concentration of Ground State Solutions for Chern–Simons–Schrödinger System with General Nonlinearity

Ground State Solution for Schrödinger–KdV System with Periodic Potential

Ground State Solution for Schrödinger–KdV System with Periodic Potential

Electronic excited states in deep variational Monte Carlo

Obtaining accurate ground and low-lying excited states of electronic systems is crucial in a multitude of important applications. One ab initio method for solving the Schrödinger equation that scales favorably for large systems is variational quantum Monte Carlo (QMC). The recently introduced deep QMC approach uses ansatzes represented by deep neural networks and generates nearly exact ground-state solutions for molecules containing up to a few dozen electrons, with the potential to scale to much larger systems where other highly accurate methods are not feasible. In this paper, we extend one such ansatz (PauliNet) to compute electronic excited states. We demonstrate our method on various small atoms and molecules and consistently achieve high accuracy for low-lying states. To highlight the method’s potential, we compute the first excited state of the much larger benzene molecule, as well as the conical intersection of ethylene, with PauliNet matching results of more expensive high-level methods.

Normalized Ground State Solutions of Nonlinear Schrödinger Equations Involving Exponential Critical Growth

We are concerned with the following nonlinear Schr\"odinger equation \begin{eqnarray*} \begin{aligned} \begin{cases} -\Delta u+\lambda u=f(u) \ \ {\rm in}\ \mathbb{R}^{2},\\ u\in H^{1}(\mathbb{R}^{2}),~~~ \int_{\mathbb{R}^2}u^2dx=\rho, \end{cases} \end{aligned} \end{eqnarray*} where $\rho>0$ is given, $\lambda\in\mathbb{R}$ arises as a Lagrange multiplier and $f$ satisfies an exponential critical growth. Without assuming the Ambrosetti-Rabinowitz condition, we show the existence of normalized ground state solutions for any $\rho>0$. The proof is based on a constrained minimization method and the Trudinger-Moser inequality in $\mathbb{R}^2$.

Nonstationary homoclinic solutions for infinite-dimensional fractional reaction-diffusion system with two types of superlinear nonlinearity

This paper is dedicated to studying nonstationary homoclinic solutions with the least energy for a class of fractional reaction-diffusion system \begin{document}$ \begin{eqnarray*} \label{1.1} \left\{\begin{array}{lll} \partial_t u+ (-\Delta)^s u+V(x)u+W(x)v = H_v(t, x, u, v), \\ - \partial_t v + (-\Delta)^s v+V(x)v+W(x)u = H_u(t, x, u, v), \\ |u(t, x)|+|v(t, x)|\rightarrow 0, \ \ \text{as}\ \ |t|+|x|\rightarrow \infty, \end{array} \right. \end{eqnarray*} $\end{document} where $ 0&lt;s&lt;1, \ z = (u, v): \mathbb{R}\times \mathbb{R}^N\rightarrow \mathbb{R}^{2} $, which originate from a wide variety of fields such as theoretical physics, optimal control, chemistry and biology. We obtain ground state solutions of Nehari-Pankov type under mild conditions on the nonlinearity by further developing non-Nehari method with two types of superlinear nonlinearity. If in addition the corresponding functional is even, we also obtain infinitely many geometrically distinct solutions by using some arguments about deformation type and Krasnoselskii genus. Nevertheless, we need to overcome some difficulties: one is that the associated functional is strongly indefinite, the second is due to the absence of strict monotonicity condition, a key ingredient of seeking the ground state solution on suitable manifold, we need some new methods and techniques. The third lies that some delicate analysis are needed for the dropping of classical super-quadratic assumption on the nonlinearity and in verifying the link geometry and showing the boundedness of Cerami sequences.

Ground state solutions and decay estimation of Choquard equation with critical exponent and Dipole potential

&lt;p style='text-indent:20px;'&gt;In this paper, we study a class of Choquard equations with critical exponent and Dipole potential. We prove the existence of radial ground state solutions for Choquard equations by using the refined Sobolev inequality with the Morrey norm, and show that any nonnegative weak solutions of Choquard equations have additional decay properties in terms of ground state representation.&lt;/p&gt;