Uniqueness and radial symmetry of least energy solution for a semilinear Neumann problem

2008 ◽  
Vol 24 (3) ◽  
pp. 473-482
Author(s):  
Zheng-ping Wang ◽  
Huan-song Zhou
Keyword(s):  
Neumann Problem ◽  
Radial Symmetry ◽  
Energy Solution ◽  
Least Energy Solution ◽  
Least Energy

scholarly journals Berestycki-Lions conditions on ground state solutions for a Nonlinear Schrödinger equation with variable potentials

2019 ◽  
Vol 9 (1) ◽  
pp. 496-515 ◽  
Author(s):  
Sitong Chen ◽  
Xianhua Tang
Keyword(s):  
Ground State ◽  
Radial Symmetry ◽  
Ground State Solution ◽  
Bounded Sequence ◽  
Energy Solution ◽  
Related Literature ◽  
Ground State Solutions ◽  
Nonlinear Schrödinger ◽  
Least Energy Solution ◽  
Least Energy

Abstract This paper is dedicated to studying the nonlinear Schrödinger equations of the form $$\begin{array}{} \displaystyle \left\{ \begin{array}{ll} -\triangle u+V(x)u=f(u), & x\in \mathbb{R}^N; \\ u\in H^1(\mathbb{R}^N), \end{array} \right. \end{array}$$ where V ∈ 𝓒1(ℝN, [0, ∞)) satisfies some weak assumptions, and f ∈ 𝓒(ℝ, ℝ) satisfies the general Berestycki-Lions assumptions. By introducing some new tricks, we prove that the above problem admits a ground state solution of Pohožaev type and a least energy solution. These results generalize and improve some ones in [L. Jeanjean, K. Tanka, Indiana Univ. Math. J. 54 (2005), 443-464], [L. Jeanjean, K. Tanka, Proc. Amer. Math. Soc. 131 (2003) 2399-2408], [H. Berestycki, P.L. Lions, Arch. Rational Mech. Anal. 82 (1983) 313-345] and some other related literature. In particular, our assumptions are “almost” necessary when V(x) ≡ V∞ > 0, moreover, our approach could be useful for the study of other problems where radial symmetry of bounded sequence either fails or is not readily available, or where the ground state solutions of the problem at infinity are not sign definite.

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

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 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.

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.

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