DCDS
Multi-peak standing waves for nonlinear Schrödinger equations with a general nonlinearity
Jaeyoung Byeon Louis Jeanjean
Discrete & Continuous Dynamical Systems - A 2007, 19(2): 255-269 doi: 10.3934/dcds.2007.19.255
We consider singularly perturbed elliptic equations $\varepsilon^2\Delta u - V(x) u + f(u)=0, x\in R^N, N \ge 3.$ For small $\varepsilon > 0,$ we glue together localized bound state solutions concentrating at isolated components of positive local minimum of $V$ under conditions on $f$ we believe to be almost optimal.
keywords: singularly perturbed elliptic problems. Variational methods
DCDS
The Hénon equation with a critical exponent under the Neumann boundary condition
Jaeyoung Byeon Sangdon Jin
Discrete & Continuous Dynamical Systems - A 2018, 38(9): 4353-4390 doi: 10.3934/dcds.2018190
For $n≥ 3$ and $p = (n+2)/(n-2), $ we consider the Hénon equation with the homogeneous Neumann boundary condition
$ -Δ u + u = |x|^{α}u^{p}, \; u > 0 \;\text{in} \; Ω,\ \ \frac{\partial u}{\partial ν} = 0 \; \text{ on }\;\partial Ω,$
where
$Ω \subset B(0,1) \subset \mathbb{R}^n, n ≥ 3$, $α≥ 0$ and $\partial^*Ω \equiv \partialΩ \cap \partial B(0,1) \ne \emptyset.$
It is well known that for
$α = 0,$
there exists a least energy solution of the problem. We are concerned on the existence of a least energy solution for
$α > 0$
and its asymptotic behavior as the parameter
$α$
approaches from below to a threshold
$α_0 ∈ (0,∞]$
for existence of a least energy solution.
keywords: Least energy solutions Hénon equation limiting profile Neumann boundary condition weighted equations
CPAA
Standing waves of nonlinear Schrödinger equations with optimal conditions for potential and nonlinearity
Soohyun Bae Jaeyoung Byeon
Communications on Pure & Applied Analysis 2013, 12(2): 831-850 doi: 10.3934/cpaa.2013.12.831
We consider the singularly perturbed nonlinear elliptic problem \begin{eqnarray*} \varepsilon^2 \Delta v - V(x)v + f(v) =0, v > 0, \lim_{|x|\to \infty} v(x) = 0. \end{eqnarray*} Under almost optimal conditions for the potential $V$ and the nonlinearity $f$, we establish the existence of single-peak solutions whose peak points converge to local minimum points of $V$ as $\varepsilon \to 0$. Moreover, we exhibit a threshold on the condition of $V$ at infinity between existence and nonexistence of solutions.
keywords: optimal conditions. variational method decaying potential standing waves Nonlinear Schrödinger equations
DCDS
On the location of a peak point of a least energy solution for Hénon equation
Jaeyoung Byeon Sungwon Cho Junsang Park
Discrete & Continuous Dynamical Systems - A 2011, 30(4): 1055-1081 doi: 10.3934/dcds.2011.30.1055
Let $\Omega $ be a smooth bounded domain. We are concerned about the following nonlinear elliptic problem:

$\Delta u + |x|^{\alpha}u^{p} = 0, \ u > 0 \quad$ in $\Omega$,
$\ u = 0 \quad$ on $\partial \Omega $,

where $\alpha > 0, p \in (1,\frac{n+2}{n-2}).$ In this paper, we show that for $n \ge 8$, a maximum point $x_{\alpha }$ of a least energy solution of above problem converges to a point $x_0 \in \partial^$*$ \Omega $ satisfying $H(x_0) = \min_$$\w \in \partial ^$*$ \Omega$$H( w )$ as $\alpha \to \infty,$ where $H$ is the mean curvature on $\partial \Omega $ and $\partial ^$*$\Omega \equiv \{ x \in \partial \Omega : |x| \ge |y|$ for any $y \in \Omega \}.$
keywords: asymptotic profile mean curvature. Hénon equation least energy solutions
CPAA
Standing wave concentrating on compact manifolds for nonlinear Schrödinger equations
Jaeyoung Byeon Ohsang Kwon Yoshihito Oshita
Communications on Pure & Applied Analysis 2015, 14(3): 825-842 doi: 10.3934/cpaa.2015.14.825
For $k =1,\cdots,K,$ let $M_k$ be a $q_k$-dimensional smooth compact framed manifold in $R^N$ with $q_k \in \{1,\cdots,N-1\} $. We consider the equation $-\varepsilon^2\Delta u + V(x)u - u^p = 0$ in $R^N$ where for each $k \in \{1,\cdots,K\}$ and some $m_k > 0,$ $V(x)=|\textrm{dist}(x,M_k)|^{m_k}+O(|\textrm{dist}(x,M_k)|^{m_k+1})$ as $\textrm{dist}(x,M_k) \to 0 $. For a sequence of $\varepsilon$ converging to zero, we will find a positive solution $u_{\varepsilon}$ of the equation which concentrates on $M_1\cup \dots \cup M_K$.
keywords: Concentration phenomena nondegeneracy nonlinear Schrödinger equation. infinite dimensional reduction

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