A variational approach to uniqueness of ground states for certain quasilinear PDEs

2010 ◽  
Vol 3 (3) ◽  
Author(s):  
Martial Agueh
Keyword(s):  
Variational Approach ◽  
Ground States ◽  
Quasilinear Pdes

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.

Polarons and molecules in a mixed-dimensional Fermi gas

2021 ◽  
pp. 2150354
Author(s):  
Baihua Gong ◽  
Xin-Hui Zhang
Keyword(s):  
Effective Mass ◽  
Impurity Atom ◽  
Variational Approach ◽  
Interaction Strength ◽  
Ground States ◽  
Fermi Gas ◽  
Energy Difference ◽  
Two Dimensional ◽  
Spatial Constraint ◽  
Polaron Energy

We address the problem of a single impurity atom moving in a two-dimensional (2D) layer immersed in a 3D Fermi gas. Using a variational approach, we show that in this mixed-dimensional (MD) system, there is a transition between polaron and molecule ground states, similar to the case of pure 3D. Moreover, we find that the attractive polaron energy in MD is higher than that in 3D, while molecular energy in MD is lower than that in 3D, which leads to a shift of the critical interaction strength of transition. Further analysis shows that the energy difference between 3D and MD systems is attributed to the increment of the effective mass of the impurity, which is induced by the spatial constraint on the impurity.

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.

Ground states for the ising model with an external field on the Cayley tree

2018 ◽  
Vol 2018 (3) ◽  
pp. 147-155
Author(s):  
M.M. Rakhmatullaev ◽  
M.A. Rasulova
Keyword(s):  
Ising Model ◽  
External Field ◽  
Cayley Tree ◽  
Ground States

scholarly journals Multiple positive solutions for nonlocal elliptic problems involving the Hardy potential and concave–convex nonlinearities

2020 ◽  
pp. 2050008
Author(s):  
Shaya Shakerian
Keyword(s):  
Variational Methods ◽  
Bounded Domain ◽  
Positive Solutions ◽  
Variational Approach ◽  
Nehari Manifold ◽  
Multiple Positive Solutions ◽  
Hardy Potential ◽  
Smooth Bounded Domain ◽  
Existence And Multiplicity ◽  
The Nehari Manifold

In this paper, we study the existence and multiplicity of solutions for the following fractional problem involving the Hardy potential and concave–convex nonlinearities: [Formula: see text] where [Formula: see text] is a smooth bounded domain in [Formula: see text] containing [Formula: see text] in its interior, and [Formula: see text] with [Formula: see text] which may change sign in [Formula: see text]. We use the variational methods and the Nehari manifold decomposition to prove that this problem has at least two positive solutions for [Formula: see text] sufficiently small. The variational approach requires that [Formula: see text] [Formula: see text] [Formula: see text], and [Formula: see text], the latter being the best fractional Hardy constant on [Formula: see text].

scholarly journals Influence of growth flux solvent on anneal-tuning of ground states in CaFe2As2

2018 ◽  
Vol 2 (4) ◽  
Author(s):  
Connor Roncaioli ◽  
Tyler Drye ◽  
Shanta R. Saha ◽  
Johnpierre Paglione
Keyword(s):  
Ground States
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