Existence and stability of ground-state solutions of a Schrödinger—KdV system

2003 ◽  
Vol 133 (5) ◽  
pp. 987-1029 ◽  
Author(s):  
John Albert ◽  
Jaime Angulo Pava
Keyword(s):  
Time Dependence ◽  
Ground State ◽  
Nonlinear Waves ◽  
Ground States ◽  
Energy Functional ◽  
Parameter Family ◽  
Existence And Stability ◽  
Image Position ◽  
Range Of Values ◽  
Two Parameter

We consider the coupled Schrödinger–Korteweg–de Vries system which arises in various physical contexts as a model for the interaction of long and short nonlinear waves. Ground states of the system are, by definition, minimizers of the energy functional subject to constraints on conserved functionals associated with symmetries of the system. In particular, ground states have a simple time dependence because they propagate via those symmetries. For a range of values of the parameters α, β, γ, δi, ci, we prove the existence and stability of a two-parameter family of ground states associated with a two-parameter family of symmetries.

POSITIVE GROUND STATES FOR A CLASS OF SUPERLINEAR -LAPLACIAN COUPLED SYSTEMS INVOLVING SCHRÖDINGER EQUATIONS

2019 ◽  
Vol 109 (2) ◽  
pp. 193-216 ◽  
Author(s):  
J. C. DE ALBUQUERQUE ◽  
JOÃO MARCOS DO Ó ◽  
EDCARLOS D. SILVA
Keyword(s):  
Large Class ◽  
Ground State ◽  
Variational Approach ◽  
Ground States ◽  
Nehari Manifold ◽  
Coupled Systems ◽  
Coupling Term ◽  
Ground State Solutions ◽  
Image Position ◽  
The Nehari Manifold

We study the existence of positive ground state solutions for the following class of $(p,q)$-Laplacian coupled systems $$\begin{eqnarray}\left\{\begin{array}{@{}lr@{}}-\unicode[STIX]{x1D6E5}_{p}u+a(x)|u|^{p-2}u=f(u)+\unicode[STIX]{x1D6FC}\unicode[STIX]{x1D706}(x)|u|^{\unicode[STIX]{x1D6FC}-2}u|v|^{\unicode[STIX]{x1D6FD}}, & x\in \mathbb{R}^{N},\\ -\unicode[STIX]{x1D6E5}_{q}v+b(x)|v|^{q-2}v=g(v)+\unicode[STIX]{x1D6FD}\unicode[STIX]{x1D706}(x)|v|^{\unicode[STIX]{x1D6FD}-2}v|u|^{\unicode[STIX]{x1D6FC}}, & x\in \mathbb{R}^{N},\end{array}\right.\end{eqnarray}$$ where $1<p\leq q<N$. Here the coefficient $\unicode[STIX]{x1D706}(x)$ of the coupling term is related to the potentials by the condition $|\unicode[STIX]{x1D706}(x)|\leq \unicode[STIX]{x1D6FF}a(x)^{\unicode[STIX]{x1D6FC}/p}b(x)^{\unicode[STIX]{x1D6FD}/q}$, where $\unicode[STIX]{x1D6FF}\in (0,1)$ and $\unicode[STIX]{x1D6FC}/p+\unicode[STIX]{x1D6FD}/q=1$. Using a variational approach based on minimization over the Nehari manifold, we establish the existence of positive ground state solutions for a large class of nonlinear terms and potentials.

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.

Ground States of some Fractional Schrödinger Equations on ℝN

2014 ◽  
Vol 58 (2) ◽  
pp. 305-321 ◽  
Author(s):  
Xiaojun Chang
Keyword(s):  
Variational Methods ◽  
Ground State ◽  
Schrodinger Equation ◽  
Fractional Laplacian ◽  
Ground States ◽  
Schrödinger Equations ◽  
Ground State Solutions ◽  
Fractional Schrödinger Equations ◽  
Image Position ◽  
Fractional Schrodinger Equation

AbstractIn this paper, we study a time-independent fractional Schrödinger equation of the form (−Δ)su + V(x)u = g(u) in ℝN, where N ≥, s ∈ (0,1) and (−Δ)s is the fractional Laplacian. By variational methods, we prove the existence of ground state solutions when V is unbounded and the nonlinearity g is subcritical and satisfies the following geometry condition:

scholarly journals On Asymptotic Properties of Semi-relativistic Hartree Equation with combined Hartree-type nonlinearities

2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Qingxuan Wang ◽  
Binhua Feng ◽  
Yuan Li ◽  
Qihong Shi
Keyword(s):  
Ground State ◽  
Blow Up ◽  
Asymptotic Properties ◽  
Ground States ◽  
Threshold Value ◽  
Hartree Equation ◽  
Existence And Stability ◽  
Blow Up Rate

<p style='text-indent:20px;'>We consider the semi-relativistic Hartree equation with combined Hartree-type nonlinearities given by</p><p style='text-indent:20px;'><disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ i\partial_t \psi = \sqrt{-\triangle+m^2}\, \psi+\beta(\frac{1}{|x|^\alpha}\ast |\psi|^2)\psi-(\frac{1}{|x|}\ast |\psi|^2)\psi\ \ \ \text{on $\mathbb{R}^3$.} $\end{document} </tex-math></disp-formula></p><p style='text-indent:20px;'>where <inline-formula><tex-math id="M1">\begin{document}$ 0&lt;\alpha&lt;1 $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M2">\begin{document}$ \beta&gt;0 $\end{document}</tex-math></inline-formula>. Firstly we study the existence and stability of the maximal ground state <inline-formula><tex-math id="M3">\begin{document}$ \psi_\beta $\end{document}</tex-math></inline-formula> at <inline-formula><tex-math id="M4">\begin{document}$ N = N_c $\end{document}</tex-math></inline-formula>, where <inline-formula><tex-math id="M5">\begin{document}$ N_c $\end{document}</tex-math></inline-formula> is a threshold value and can be regarded as "Chandrasekhar limiting mass". Secondly, we analyse blow-up behaviours of maximal ground states <inline-formula><tex-math id="M6">\begin{document}$ \psi_\beta $\end{document}</tex-math></inline-formula> when <inline-formula><tex-math id="M7">\begin{document}$ \beta\rightarrow 0^+ $\end{document}</tex-math></inline-formula>, and the optimal blow-up rate with respect to <inline-formula><tex-math id="M8">\begin{document}$ \beta $\end{document}</tex-math></inline-formula> will be calculated.</p>

The embedding problem for Markov chains with three states

1980 ◽  
Vol 87 (2) ◽  
pp. 285-294 ◽  
Author(s):  
Halina Frydman
Keyword(s):  
Markov Chains ◽  
Stochastic Matrix ◽  
Sufficient Conditions ◽  
Parameter Family ◽  
Necessary And Sufficient Conditions ◽  
Embedding Problem ◽  
Stochastic Matrices ◽  
Image Position ◽  
Necessary And Sufficient ◽  
Two Parameter

In this paper we consider the embedding problem for Markov chains with three states. A non-singular stochastic matrix P is called embeddable if there exists a two-parameter family of stochastic matricessatisfyingand such thatThough extensive characterizations of embeddable n × n stochastic matrices have been given in (l), (2), (3), (6), and further characterizations of embeddable 3 × 3 stochastic matrices in (4), they do not provide, except in the case of 2 × 2 stochastic matrices, easily applicable necessary and sufficient conditions for embeddability.

NON-NEHARI-MANIFOLD METHOD FOR ASYMPTOTICALLY LINEAR SCHRÖDINGER EQUATION

2014 ◽  
Vol 98 (1) ◽  
pp. 104-116 ◽  
Author(s):  
X. H. TANG
Keyword(s):  
Schrödinger Equation ◽  
Ground State ◽  
Schrodinger Equation ◽  
Main Idea ◽  
Energy Functional ◽  
Nehari Manifold ◽  
Asymptotically Linear ◽  
Cerami Sequence ◽  
Image Position ◽  
Semilinear Schrödinger Equation

AbstractWe consider the semilinear Schrödinger equation$$\begin{eqnarray}\left\{\begin{array}{@{}l@{}}-\triangle u+V(x)u=f(x,u),\quad x\in \mathbb{R}^{N},\\ u\in H^{1}(\mathbb{R}^{N}),\end{array}\right.\end{eqnarray}$$ where $f(x,u)$ is asymptotically linear with respect to $u$, $V(x)$ is 1-periodic in each of $x_{1},x_{2},\dots ,x_{N}$ and $\sup [{\it\sigma}(-\triangle +V)\cap (-\infty ,0)]<0<\inf [{\it\sigma}(-\triangle +V)\cap (0,\infty )]$. We develop a direct approach to find ground state solutions of Nehari–Pankov type for the above problem. The main idea is to find a minimizing Cerami sequence for the energy functional outside the Nehari–Pankov manifold ${\mathcal{N}}^{-}$ by using the diagonal method.

scholarly journals Dual ground state solutions for the critical nonlinear Helmholtz equation

2019 ◽  
Vol 150 (3) ◽  
pp. 1155-1186 ◽  
Author(s):  
Gilles Evéquoz ◽  
Tolga Yeşil
Keyword(s):  
Helmholtz Equation ◽  
Critical Points ◽  
Ground State ◽  
Variational Approach ◽  
Ground States ◽  
Minimal Energy ◽  
Ground State Solutions ◽  
Dual Functional ◽  
Asymptotically Periodic ◽  
Image Position

AbstractUsing a dual variational approach, we obtain nontrivial real-valued solutions of the critical nonlinear Helmholtz equation $$-\Delta u-k^2u = Q(x) \vert u \vert ^{2^*-2}u,\quad u\in W^{2,2^*}({\open R}^{N})$$for N ⩾ 4, where 2* : = 2N/(N − 2). The coefficient $Q \in L^{\infty }({\open R}^{N}){\setminus }\{0\}$ is assumed to be nonnegative, asymptotically periodic and to satisfy a flatness condition at one of its maximum points. The solutions obtained are so-called dual ground states, that is, solutions arising from critical points of the dual functional with the property of having minimal energy among all nontrivial critical points. Moreover, we show that no dual ground state exists for N = 3.

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