Multi-peak standing waves for nonlinear Schrödinger equations with a general nonlinearity
Jaeyoung Byeon Louis Jeanjean
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
Standing waves of nonlinear Schrödinger equations with optimal conditions for potential and nonlinearity
Soohyun Bae Jaeyoung Byeon
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
On the location of a peak point of a least energy solution for Hénon equation
Jaeyoung Byeon Sungwon Cho Junsang Park
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
Standing wave concentrating on compact manifolds for nonlinear Schrödinger equations
Jaeyoung Byeon Ohsang Kwon Yoshihito Oshita
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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