scholarly journals Local minimizers in absence of ground states for the critical NLS energy on metric graphs

2020 ◽  
pp. 1-29 ◽  
Author(s):  
Dario Pierotti ◽  
Nicola Soave ◽  
Gianmaria Verzini
Keyword(s):  
State Energy ◽  
Ground States ◽  
Energy Functional ◽  
Negative Energy ◽  
Topological Properties ◽  
Local Minimizers ◽  
Metric Graphs ◽  
Global Minimizers ◽  
Underlying Graph ◽  
Mass Constraint

We consider the mass-critical non-linear Schrödinger equation on non-compact metric graphs. A quite complete description of the structure of the ground states, which correspond to global minimizers of the energy functional under a mass constraint, is provided by Adami, Serra and Tilli in [R. Adami, E. Serra and P. Tilli. Negative energy ground states for the L2-critical NLSE on metric graphs. Comm. Math. Phys. 352 (2017), 387–406.] , where it is proved that existence and properties of ground states depend in a crucial way on both the value of the mass, and the topological properties of the underlying graph. In this paper we address cases when ground states do not exist and show that, under suitable assumptions, constrained local minimizers of the energy do exist. This result paves the way to the existence of stable solutions in the time-dependent equation in cases where the ground state energy level is not achieved.

scholarly journals Negative Energy Ground States for the L 2-Critical NLSE on Metric Graphs

2016 ◽  
Vol 352 (1) ◽  
pp. 387-406 ◽  
Author(s):  
Riccardo Adami ◽  
Enrico Serra ◽  
Paolo Tilli
Keyword(s):  
Ground States ◽  
Negative Energy ◽  
Metric Graphs

Count and Symmetry of Global and Local Minimizers of the Cahn-Hilliard Energy Over Cylindrical Domains

2012 ◽  
Vol 12 (1) ◽  
Author(s):  
Arnaldo Simal do Nascimento ◽  
João Biesdorf ◽  
Janete Crema
Keyword(s):  
Three Dimensional ◽  
Unique Continuation ◽  
Energy Functional ◽  
Circular Cylinders ◽  
Cylindrical Domain ◽  
Local Minimizers ◽  
Global Minimizers ◽  
Cylindrical Domains ◽  
Γ Convergence ◽  
Global And Local

AbstractWe address the problem of minimization of the Cahn-Hilliard energy functional under a mass constraint over two and three-dimensional cylindrical domains. Although existence is presented for a general case the focus is mainly on rectangles, parallelepipeds and circular cylinders. According to the symmetry of the domain the exact numbers of global and local minimizers are given as well as their geometric profile and interface locations; all are onedimensional increasing/decreasing and odd functions for domains with lateral symmetry in all axes and also for circular cylinders. The selection of global minimizers by the energy functional is made via the smallest interface area criterion.The approach utilizes Γ−convergence techniques to prove existence of an one-parameter family of local minimizers of the energy functional for any cylindrical domain. The exact numbers of global and local minimizers as well as their geometric profiles are accomplished via suitable applications of the unique continuation principle while exploring the domain geometry in each case and also the preservation of global minimizers through the process of Γ−convergence.

scholarly journals On the existence of bound and ground states for some coupled nonlinear Schrödinger–Korteweg–de Vries equations

2017 ◽  
Vol 6 (4) ◽  
pp. 407-426 ◽  
Author(s):  
Eduardo Colorado
Keyword(s):  
Positive Solutions ◽  
Ground States ◽  
Bound State ◽  
Energy Functional ◽  
Nonlinear Schrödinger ◽  
De Vries ◽  
Mass Constraint ◽  
General Power ◽  
Multiplicity Of Positive Solutions ◽  
Approach Working

AbstractWe show the existence of positive bound and ground states for a system of coupled nonlinear Schrödinger–Korteweg–de Vries equations. More precisely, we prove that there exists a positive radially symmetric ground state if either the coupling coefficient satisfies {\beta>\Lambda} (for an appropriate constant {\Lambda>0}) or if {\beta>0} under appropriate conditions on the other parameters of the problem. We also prove that there exists a positive radially symmetric bound state if either {0<\beta} is sufficiently small or if {0<\beta<\Lambda} under some appropriate conditions on the parameters. These results give a classification of positive solutions as well as the multiplicity of positive solutions. Furthermore, we study systems with more general power nonlinearities and systems with more than two nonlinear Schrödinger–Korteweg–de Vries equations. Our variational approach (working on the full energy functional without the {L^{2}}-mass constraint) improves many previously known results and also allows us to show new results for some range of parameters not considered in the past.

scholarly journals Existence of ground states for aggregation-diffusion equations

2019 ◽  
Vol 17 (03) ◽  
pp. 393-423 ◽  
Author(s):  
J. A. Carrillo ◽  
M. G. Delgadino ◽  
F. S. Patacchini
Keyword(s):  
Free Energy ◽  
Ground States ◽  
Energy Functional ◽  
Multi Agent Systems ◽  
Linear Diffusion ◽  
Sharp Condition ◽  
Macroscopic Models ◽  
Energy Functionals ◽  
Continuous Description ◽  
Degenerate Diffusions

We analyze free energy functionals for macroscopic models of multi-agent systems interacting via pairwise attractive forces and localized repulsion. The repulsion at the level of the continuous description is modeled by pressure-related terms in the functional making it energetically favorable to spread, while the attraction is modeled through nonlocal forces. We give conditions on general entropies and interaction potentials for which neither ground states nor local minimizers exist. We show that these results are sharp for homogeneous functionals with entropies leading to degenerate diffusions while they are not sharp for fast diffusions. The particular relevant case of linear diffusion is totally clarified giving a sharp condition on the interaction potential under which the corresponding free energy functional has ground states or not.

scholarly journals ROTATING FERMIONS IN TWO DIMENSIONS: A THOMAS–FERMI APPROACH

2001 ◽  
Vol 15 (19n20) ◽  
pp. 2799-2810
Author(s):  
SANKALPA GHOSH ◽  
M. V. N. MURTHY ◽  
SUBHASIS SINHA
Keyword(s):  
Ground State ◽  
Analytical Approach ◽  
State Energy ◽  
Critical Size ◽  
Energy Functional ◽  
Two Dimensions ◽  
Fermionic System ◽  
Thomas Fermi ◽  
Fermionic Systems ◽  
Analytical Expressions

Properties of confined mesoscopic systems have been extensively studied numerically over recent years. We discuss an analytical approach to the study of finite rotating fermionic systems in two dimension. We first construct the energy functional for a finite fermionic system within the Thomas–Fermi approximation in two dimensions. We show that for specific interactions the problem may be exactly solved. We derive analytical expressions for the density, the critical size as well as the ground state energy of such systems in a given angular momentum sector.

GROUND STATES IN THREE-DIMENSIONAL ±J EDWARDS–ANDERSON SPIN GLASSES WITH FREE BOUNDARIES

2000 ◽  
Vol 11 (03) ◽  
pp. 589-592
Author(s):  
FRAUKE LIERS ◽  
MICHAEL JÜNGER
Keyword(s):  
Spin Glass ◽  
Spin Glasses ◽  
State Energy ◽  
Infinite System ◽  
Three Dimensional ◽  
Finite Size ◽  
Ground States ◽  
Free Boundaries ◽  
Finite Size Corrections

By an exact calculation of the ground states for the ±J Edwards–Anderson spin glass, one can extrapolate the ground state energy to infinite system sizes. We calculate the exact ground states for the three-dimensional spin glass with free boundaries for system sizes up to 10 and fit different finite-size functions. We cannot decide, only from the quality of the fit, which fitting function to choose. Relying on the literature values for the extrapolated energy, we find the finite-size corrections to vary as 1/L.

scholarly journals Multiple Solutions to a Nonlinear Curl–Curl Problem in $${\mathbb {R}}^3$$

2019 ◽  
Vol 236 (1) ◽  
pp. 253-288
Author(s):  
Jarosław Mederski ◽  
Jacopo Schino ◽  
Andrzej Szulkin
Keyword(s):  
Ground State ◽  
Maxwell Equations ◽  
Bound States ◽  
Ground States ◽  
Point Theory ◽  
Energy Functional ◽  
Multiplicity Results ◽  
Infinite Dimensional ◽  
Indefinite Problems ◽  
Least Energy

AbstractWe look for ground states and bound states $$E:{\mathbb {R}}^3\rightarrow {\mathbb {R}}^3$$E:R3→R3 to the curl–curl problem $$\begin{aligned} \nabla \times (\nabla \times E)= f(x,E) \qquad \text { in } {\mathbb {R}}^3, \end{aligned}$$∇×(∇×E)=f(x,E)inR3,which originates from nonlinear Maxwell equations. The energy functional associated with this problem is strongly indefinite due to the infinite dimensional kernel of $$\nabla \times (\nabla \times \cdot )$$∇×(∇×·). The growth of the nonlinearity f is controlled by an N-function $$\Phi :{\mathbb {R}}\rightarrow [0,\infty )$$Φ:R→[0,∞) such that $$\displaystyle \lim _{s\rightarrow 0}\Phi (s)/s^6=\lim _{s\rightarrow +\infty }\Phi (s)/s^6=0$$lims→0Φ(s)/s6=lims→+∞Φ(s)/s6=0. We prove the existence of a ground state, that is, a least energy nontrivial solution, and the existence of infinitely many geometrically distinct bound states. We improve previous results concerning ground states of curl–curl problems. Multiplicity results for our problem have not been studied so far in $${\mathbb {R}}^3$$R3 and in order to do this we construct a suitable critical point theory; it is applicable to a wide class of strongly indefinite problems, including this one and Schrödinger equations.

SOLUTIONS OF THE SCHRODINGER EQUATION WITH INVERSELY QUADRATIC YUKAWA PLUS WOODS-SAXON POTENTIAL USING NIKIFOROV-UVAROV METHOD

2015 ◽  
Vol 8 (2) ◽  
pp. 2094-2098
Author(s):  
Benedict Ita ◽  
A. I. Ikeuba ◽  
O. Obinna
Keyword(s):  
State Energy ◽  
Jacobi Polynomials ◽  
Bound State ◽  
Negative Energy ◽  
Saxon Potential ◽  
Energy Eigenvalues ◽  
Eigen Values ◽  
Uvarov Method ◽  
Special Case ◽  
Nu Method

The solutions of the SchrÓ§dinger equation with inversely quadratic Yukawa plus Woods-Saxon potential (IQYWSP) have been presented using the parametric Nikiforov-Uvarov (NU) method. The bound state energy eigenvalues and the corresponding un-normalized eigen functions are obtained in terms of Jacobi polynomials. Also, a special case of the potential has been considered and its energy eigen values obtained. The result of the work could be applied to molecules moving under the influence of IQYWSP potential as negative energy eigenvalues obtained indicate a bound state system.

Multiple Solutions of Sublinear Quasilinear Schrödinger Equations with Small Perturbations

2018 ◽  
Vol 62 (2) ◽  
pp. 471-488
Author(s):  
Liang Zhang ◽  
X. H. Tang ◽  
Yi Chen
Keyword(s):  
Critical Point Theory ◽  
Multiple Solutions ◽  
Point Theory ◽  
Energy Functional ◽  
Negative Energy ◽  
Quasilinear Schrödinger Equation ◽  
Small Perturbations ◽  
Bounded Smooth Domain ◽  
Bounded Smooth ◽  
Image Position

AbstractIn this paper, we consider the existence of multiple solutions for the quasilinear Schrödinger equation $$\left\{ {\matrix{ {-\Delta u-\Delta (\vert u \vert ^\alpha )\vert u \vert ^{\alpha -2}u = g(x,u) + \theta h(x,u),\;\;x\in \Omega } \hfill \cr {u = 0,\;\;x\in \partial \Omega ,} \hfill \cr } } \right.$$ where Ω is a bounded smooth domain in ℝN (N ≥ 1), α ≥ 2 and θ is a parameter. Under the assumption that g(x, u) is sublinear near the origin with respect to u, we study the effect of the perturbation term h(x, u), which may break the symmetry of the associated energy functional. With the aid of critical point theory and the truncation method, we show that this system possesses multiple small negative energy solutions.

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