On semiclassical states of a nonlinear Dirac equation

2013 ◽  
Vol 143 (4) ◽  
pp. 765-790 ◽  
Author(s):  
Y. H. Ding ◽  
C. Lee ◽  
B. Ruf
Keyword(s):  
Dirac Equation ◽  
Semiclassical Limit ◽  
Limit Problem ◽  
Energy Solution ◽  
Nonlinear Dirac Equation ◽  
Least Energy Solution ◽  
Semiclassical States ◽  
Image Position ◽  
Least Energy Solutions ◽  
Least Energy

We study the semiclassical limit of the least energy solutions to the nonlinear Dirac equation for x ∈ ℝ3. We prove that the equation has least energy solutions for all ħ > 0 small, and, in addition, that the solutions converge in a certain sense to the least energy solution of the associated limit problem as ħ → 0.

Least energy solutions for indefinite biharmonic problems via modified Nehari–Pankov manifold

2018 ◽  
Vol 20 (04) ◽  
pp. 1750047 ◽  
Author(s):  
Miaomiao Niu ◽  
Zhongwei Tang ◽  
Lushun Wang
Keyword(s):  
Asymptotic Behavior ◽  
Potential Function ◽  
Biharmonic Equation ◽  
Positive Function ◽  
Energy Solution ◽  
Zero Set ◽  
Least Energy Solution ◽  
Least Energy Solutions ◽  
Least Energy ◽  
Biharmonic Problems

In this paper, by using a modified Nehari–Pankov manifold, we prove the existence and the asymptotic behavior of least energy solutions for the following indefinite biharmonic equation: [Formula: see text] where [Formula: see text], [Formula: see text], [Formula: see text] is a parameter, [Formula: see text] is a nonnegative potential function with nonempty zero set [Formula: see text], [Formula: see text] is a positive function such that the operator [Formula: see text] is indefinite and non-degenerate for [Formula: see text] large. We show that both in subcritical and critical cases, equation [Formula: see text] admits a least energy solution which for [Formula: see text] large localized near the zero set [Formula: see text].

A symmetry-breaking phenomenon and asymptotic profiles of least-energy solutions to a nonlinear Schrödinger equation

2005 ◽  
Vol 135 (2) ◽  
pp. 357-392 ◽  
Author(s):  
Kazuhiro Kurata ◽  
Tatsuya Watanabe ◽  
Masataka Shibata
Keyword(s):  
Schrödinger Equation ◽  
Symmetry Breaking ◽  
Nonlinear Schrödinger Equation ◽  
Schrodinger Equation ◽  
Nonlinear Schrodinger Equation ◽  
Nonlinear Schrödinger ◽  
Least Energy Solution ◽  
Image Position ◽  
Least Energy Solutions ◽  
Least Energy

In this paper, we study a symmetry-breaking phenomenon of a least-energy solution to a nonlinear Schrödinger equation under suitable assumptions on V(x), where λ > 1, p > 2 and χA is the characteristic function of the set A = [−(l + 2), −l] ∪ [l,l + 2] with l > 0. We also study asymptotic profiles of least-energy solutions for the singularly perturbed problem for small ε > 0.

scholarly journals On the location of a peak point of a least energy solution for Hénon equation

2011 ◽  
Vol 30 (4) ◽  
pp. 1055-1081 ◽  
Author(s):  
Jaeyoung Byeon ◽  
◽  
Sungwon Cho ◽  
Junsang Park ◽  
◽  
...  
Keyword(s):  
Energy Solution ◽  
Peak Point ◽  
Least Energy Solution ◽  
Hénon Equation ◽  
Least Energy

Existence and asymptotics of ground states to the nonlinear Dirac equation with Coulomb potential

2021 ◽  
pp. 1-26
Author(s):  
Tianfang Wang ◽  
Wen Zhang ◽  
Jian Zhang
Keyword(s):  
Dirac Equation ◽  
Coulomb Potential ◽  
Ground State ◽  
Continuous Dependence ◽  
Energy Functional ◽  
Ground State Solution ◽  
State Solution ◽  
Nonlinear Dirac Equation ◽  
Relationship Of ◽  
Least Energy

In this paper we study the Dirac equation with Coulomb potential − i α · ∇ u + a β u − μ | x | u = f ( x , | u | ) u , x ∈ R 3 where a is a positive constant, μ is a positive parameter, α = ( α 1 , α 2 , α 3 ), α i and β are 4 × 4 Pauli–Dirac matrices. The Dirac operator is unbounded from below and above so the associate energy functional is strongly indefinite. Under some suitable conditions, we prove that the problem possesses a ground state solution which is exponentially decay, and the least energy has continuous dependence about μ. Moreover, we are able to obtain the asymptotic property of ground state solution as μ → 0 + , this result can characterize some relationship of the above problem between μ > 0 and μ = 0.

Least energy solution for a scalar field equation with a singular nonlinearity

2020 ◽  
pp. 1-17
Author(s):  
Jaeyoung Byeon ◽  
Sun-Ho Choi ◽  
Yeonho Kim ◽  
Sang-Hyuck Moon
Keyword(s):  
Field Equation ◽  
Scalar Field ◽  
Existence Result ◽  
Nonnegative Solution ◽  
Energy Solution ◽  
Existence Of A Solution ◽  
Singular Nonlinearity ◽  
Scalar Field Equation ◽  
Least Energy Solution ◽  
Least Energy

Abstract We are concerned with a nonnegative solution to the scalar field equation $$\Delta u + f(u) = 0{\rm in }{\open R}^N,\quad \mathop {\lim }\limits_{|x|\to \infty } u(x) = 0.$$ A classical existence result by Berestycki and Lions [3] considers only the case when f is continuous. In this paper, we are interested in the existence of a solution when f is singular. For a singular nonlinearity f, Gazzola, Serrin and Tang [8] proved an existence result when $f \in L^1_{loc}(\mathbb {R}) \cap \mathrm {Lip}_{loc}(0,\infty )$ with $\int _0^u f(s)\,{\rm d}s < 0$ for small $u>0.$ Since they use a shooting argument for their proof, they require the property that $f \in \mathrm {Lip}_{loc}(0,\infty ).$ In this paper, using a purely variational method, we extend the previous existence results for $f \in L^1_{loc}(\mathbb {R}) \cap C(0,\infty )$ . We show that a solution obtained through minimization has the least energy among all radially symmetric weak solutions. Moreover, we describe a general condition under which a least energy solution has compact support.

Least energy solutions for elliptic equations in unbounded domains

1996 ◽  
Vol 126 (1) ◽  
pp. 195-208 ◽  
Author(s):  
Manuel A. del Pino ◽  
Patricio L. Felmer
Keyword(s):  
Critical Point ◽  
Unbounded Domain ◽  
Elliptic Equations ◽  
Unbounded Domains ◽  
Asymptotic Results ◽  
Semilinear Elliptic ◽  
Superlinear Growth ◽  
Image Position ◽  
Least Energy Solutions ◽  
Least Energy

In this paper we study the existence of least energy solutions to subcritical semilinear elliptic equations of the formwhere Ω is an unbounded domain in RN and f is a C1 function, with appropriate superlinear growth. We state general conditions on the domain Ω so that the associated functional has a nontrivial critical point, thus yielding a solution to the equation. Asymptotic results for domains stretched in one direction are also provided.

Least Energy Solutions and Group Invariant Solutions of the Hénon Equation

2013 ◽  
Vol 13 (3) ◽  
Author(s):  
Ryuji Kajikiya
Keyword(s):  
Unit Ball ◽  
Radial Symmetry ◽  
Coefficient Function ◽  
Invariant Solutions ◽  
Group Invariant ◽  
Least Energy Solution ◽  
Hénon Equation ◽  
Group Invariant Solutions ◽  
Least Energy Solutions ◽  
Least Energy

AbstractIn this paper we study the generalized Hénon equation in the unit ball, where the coefficient function may or may not change its sign. We prove that the least energy solution is not radial and moreover we show the existence of a group invariant positive solution without radial symmetry.

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