scholarly journals Spiraling solutions of nonlinear Schrödinger equations

2021 ◽  
pp. 1-34
Author(s):  
Oscar Agudelo ◽  
Joel Kübler ◽  
Tobias Weth
Keyword(s):  
Manuscripta Math ◽  
Axially Symmetric ◽  
Screw Motion ◽  
Nonlinear Schrödinger ◽  
Cahn Equation ◽  
New Family ◽  
Symmetry Properties ◽  
Variational Structure ◽  
Image Position ◽  
Sign Changing Solutions

We study a new family of sign-changing solutions to the stationary nonlinear Schrödinger equation \[ -\Delta v +q v =|v|^{p-2} v, \qquad \text{in}\,{ {\mathbb{R}^{3}},} \] with $2 < p < \infty$ and $q \ge 0$ . These solutions are spiraling in the sense that they are not axially symmetric but invariant under screw motion, i.e., they share the symmetry properties of a helicoid. In addition to existence results, we provide information on the shape of spiraling solutions, which depends on the parameter value representing the rotational slope of the underlying screw motion. Our results complement a related analysis of Del Pino, Musso and Pacard in their study (2012, Manuscripta Math., 138, 273–286) for the Allen–Cahn equation, whereas the nature of results and the underlying variational structure are completely different.

Many synchronized vector solutions for a Bose–Einstein system

2020 ◽  
Vol 150 (6) ◽  
pp. 3293-3320
Author(s):  
Wei Long ◽  
Zhongwei Tang ◽  
Sudan Yang
Keyword(s):  
Positive Integer ◽  
Positive Solution ◽  
Weight Functions ◽  
Coupling Constant ◽  
Nonlinear Schrödinger ◽  
Nonlinear Schrödinger System ◽  
Image Position ◽  
Bose Einstein ◽  
Sign Changing Solutions ◽  
Schrödinger System

AbstractThis paper is concerned with the following nonlinear Schrödinger system in ${\mathbb R}^3$$$\left\{ {\beging{matrix}{ {-\Delta u + (1 + \alpha P(x))u = \mu u^3 + \beta uv^2,} \hfill & {x\in {\open R}^3,} \hfill \cr {-\Delta v + (1 + \alpha Q(x))u = \nu v^3 + \beta u^2v,} \hfill & {x\in {\open R}^3,} \hfill \cr {u,v > 0,} \hfill & {x\in {\open R}^3,} \hfill \cr } } \right.$$ where $\beta \in {\mathbb R}$ is a coupling constant, $\mu ,\nu $ are positive constants, P,Q are weight functions decaying exponentially to zero at infinity, α can be regarded as a parameter. This type of system arises, in particular, in models in Bose–Einstein condensates theory and Kerr-like photo refractive media.We prove that, for any positive integer k > 1, there exists a suitable range of α such that the above problem has a non-radial positive solution with exactly k maximum points which tend to infinity as $\alpha \to +\infty $ (or $0^+$). Moreover, we also construct prescribed number of sign-changing solutions.

scholarly journals Positive solutions of the nonlinear Schrödinger equation with the fractional Laplacian

2012 ◽  
Vol 142 (6) ◽  
pp. 1237-1262 ◽  
Author(s):  
Patricio Felmer ◽  
Alexander Quaas ◽  
Jinggang Tan
Keyword(s):  
Schrödinger Equation ◽  
Positive Solutions ◽  
Nonlinear Schrödinger Equation ◽  
Schrodinger Equation ◽  
Fractional Laplacian ◽  
Nonlinear Schrodinger Equation ◽  
Nonlinear Schrödinger ◽  
Symmetry Properties ◽  
Existence Of Positive Solutions ◽  
Image Position

We study the existence of positive solutions for the nonlinear Schrödinger equation with the fractional LaplacianFurthermore, we analyse the regularity, decay and symmetry properties of these solutions.

Existence and multiplicity of positive solutions for the nonlinear Schrödinger–Poisson equations

2013 ◽  
Vol 143 (4) ◽  
pp. 745-764 ◽  
Author(s):  
Ching-yu Chen ◽  
Yueh-cheng Kuo ◽  
Tsung-fang Wu
Keyword(s):  
Positive Solutions ◽  
Nonlinear Schrödinger ◽  
Poisson Equations ◽  
Existence And Multiplicity ◽  
Multiplicity Of Positive Solutions ◽  
Image Position

We study the existence and multiplicity of positive solutions for the following nonlinear Schrödinger–Poisson equations: where 2 < p ≤ 3 or 4 ≤ p < 6, λ > 0 and Q ∈ C(ℝ3). We show that the number of positive solutions is dependent on the profile of Q(x).

scholarly journals Some hemivariational inequalities in the Euclidean space

2019 ◽  
Vol 9 (1) ◽  
pp. 958-977 ◽  
Author(s):  
Giovanni Molica Bisci ◽  
Dušan Repovš
Keyword(s):  
Weak Solution ◽  
Euclidean Space ◽  
Sobolev Space ◽  
Hemivariational Inequalities ◽  
Existence Of Weak Solutions ◽  
Lipschitz Continuous ◽  
Variational Structure ◽  
Locally Lipschitz ◽  
Sign Changing Solutions ◽  
Continuous Functionals

Abstract The purpose of this paper is to study the existence of weak solutions for some classes of hemivariational problems in the Euclidean space ℝd (d ≥ 3). These hemivariational inequalities have a variational structure and, thanks to this, we are able to find a non-trivial weak solution for them by using variational methods and a non-smooth version of the Palais principle of symmetric criticality for locally Lipschitz continuous functionals, due to Krawcewicz and Marzantowicz. The main tools in our approach are based on appropriate theoretical arguments on suitable subgroups of the orthogonal group O(d) and their actions on the Sobolev space H1(ℝd). Moreover, under an additional hypotheses on the dimension d and in the presence of symmetry on the nonlinear datum, the existence of multiple pairs of sign-changing solutions with different symmetries structure has been proved. In connection to classical Schrödinger equations a concrete and meaningful example of an application is presented.

Symmetrized Kronecker Products of Group Representations

1974 ◽  
Vol 26 (02) ◽  
pp. 328-339 ◽  
Author(s):  
P. H. Butler ◽  
R. C. King
Keyword(s):  
Irreducible Representation ◽  
Simple Groups ◽  
Kronecker Products ◽  
Group Representations ◽  
Symmetry Properties ◽  
Image Position

Certain phases are associated with the Kronecker squares and cubes of representations of the finite and of the compact semi-simple groups. These phases are important in giving the symmetry properties of the 1 — jm and 3 — jm symbols of the groups [4; 9]. It is our primary purpose to evaluate these phases. The Frobenius-Schur invariant [12, p. 142] for an irreducible representation of group G (1.1)

Radial solutions concentrating on spheres of nonlinear Schrödinger equations with vanishing potentials

2006 ◽  
Vol 136 (5) ◽  
pp. 889-907 ◽  
Author(s):  
A. Ambrosetti ◽  
D. Ruiz
Keyword(s):  
Perturbation Technique ◽  
Nonlinear Schrödinger Equations ◽  
Schrodinger Equations ◽  
Schrödinger Equations ◽  
Radial Solutions ◽  
Nonlinear Schrödinger ◽  
Nonlinear Schrodinger Equations ◽  
Image Position ◽  
Vanishing Potentials

We prove the existence of radial solutions of concentrating on a sphere for potentials which might be zero and might decay to zero at infinity. The proofs use a perturbation technique in a variational setting, through a Lyapunov–Schmidt reduction.

scholarly journals SYMMETRY AND EXISTENCE OF SOLUTIONS OF SEMI-LINEAR ELLIPTIC SYSTEMS IN HYPERBOLIC SPACE

2014 ◽  
Vol 57 (3) ◽  
pp. 519-541
Author(s):  
HAIYANG HE
Keyword(s):  
Positive Solution ◽  
Hyperbolic Space ◽  
Existence Of Solutions ◽  
Elliptic Systems ◽  
Ground State Solution ◽  
Decay Estimates ◽  
Symmetry Properties ◽  
Formula Group ◽  
Solution Pair ◽  
Image Position

Abstract(0.1) \begin{equation}\label{eq:0.1} \left\{ \begin{array}{ll} \displaystyle -\Delta_{\mathbb{H}^{N}}u=|v|^{p-1}v x, \\ \displaystyle -\Delta_{\mathbb{H}^{N}}v=|u|^{q-1}u, \\ \end{array} \right. \end{equation} in the whole Hyperbolic space ℍN. We establish decay estimates and symmetry properties of positive solutions. Unlike the corresponding problem in Euclidean space ℝN, we prove that there is a positive solution pair (u, v) ∈ H1(ℍN) × H1(ℍN) of problem (0.1), moreover a ground state solution is obtained. Furthermore, we also prove that the above problem has a radial positive solution.

scholarly journals Axially symmetric solutions of the Allen-Cahn equation with finite Morse index

2020 ◽  
Vol 373 (5) ◽  
pp. 3649-3668
Author(s):  
Changfeng Gui ◽  
Kelei Wang ◽  
Jucheng Wei
Keyword(s):  
Morse Index ◽  
Axially Symmetric ◽  
Cahn Equation
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