Uniqueness of positive solutions to some coupled nonlinear Schrödinger equations
Juncheng Wei Wei Yao
We study the uniqueness of positive solutions of the following coupled nonlinear Schrödinger equations: \begin{eqnarray*} \Delta u_1-\lambda_1 u_1+\mu_1u_1^3+\beta u_1u_2^2=0\quad in\quad R^N,\\ \Delta u_2-\lambda_2u_2+\mu_2u_2^3+\beta u_1^2u_2=0\quad in\quad R^N, \\ u_1>0, u_2>0, u_1, u_2 \in H^1 (R^N), \end{eqnarray*} where $N\leq3$, $\lambda_1,\lambda_2,\mu_1,\mu_2$ are positive constants and $\beta\geq 0$ is a coupling constant. We prove first the uniqueness of positive solution for sufficiently small $\beta > 0$. Secondly, assuming that $\lambda_1=\lambda_2$, we show that $u_1=u_2\sqrt{\beta-\mu_1}/\sqrt{\beta-\mu_2}$ when $\beta > \max\{\mu_1,\mu_2\}$ and thus obtain the uniqueness of positive solution using the corresponding result of scalar equation. Finally, for $N=1$ and $\lambda_1=\lambda_2$, we prove the uniqueness of positive solution when $0\leq \beta\notin [\min\{\mu_1,\mu_2\},\max\{\mu_1,\mu_2\}]$ and thus give a complete classification of positive solutions.
keywords: uniqueness. Coupled nonlinear Schrödinger equations
On the Gierer-Meinhardt system with precursors
Juncheng Wei Matthias Winter
We consider the following Gierer-Meinhardt system with a precursor $ \mu (x)$ for the activator $A$ in $\mathbb{R}^1$:

$A_t=$ε2$A^{''}- \mu (x) A+\frac{A^2}{H} \mbox{ in } (-1, 1),$
$\tau H_t=D H^{''}-H+ A^2 \mbox{ in } (-1, 1),$
$ A' (-1)= A' (1)= H' (-1) = H' (1) =0.$

Such an equation exhibits a typical Turing bifurcation of the second kind, i.e., homogeneous uniform steady states do not exist in the system.
   We establish the existence and stability of $N-$peaked steady-states in terms of the precursor $\mu(x)$ and the diffusion coefficient $D$. It is shown that $\mu (x)$ plays an essential role for both existence and stability of spiky patterns. In particular, we show that precursors can give rise to instability. This is a new effect which is not present in the homogeneous case.

keywords: precursor. mathematical biology singular perturbation Pattern formation
On least energy solutions to a semilinear elliptic equation in a strip
Henri Berestycki Juncheng Wei
We consider the following semilinear elliptic equation on a strip:

$\Delta u-u + u^p=0 \ \mbox{in} \ \R^{N-1} \times (0, L),$
$ u>0, \frac{\partial u}{\partial \nu}=0 \ \mbox{on} \ \partial (\R^{N-1} \times (0, L)) $

where $ 1< p\leq \frac{N+2}{N-2}$. When $ 1 < p <\frac{N+2}{N-2}$, it is shown that there exists a unique L * >0 such that for L $\leq $L * , the least energy solution is trivial, i.e., doesn't depend on $x_N$, and for L >L * , the least energy solution is nontrivial. When $N \geq 4, p=\frac{N+2}{N-2}$, it is shown that there are two numbers L * < L ** such that the least energy solution is trivial when L $\leq$L *, the least energy solution is nontrivial when L $\in$(L *,L **], and the least energy solution does not exist when L >L **. A connection with Delaunay surfaces in CMC theory is also made.

keywords: Critical Sobolev Exponent. Strip Semilinear Elliptic Equations Least Energy Solutions Unbounded Domains
DCDS-A Special Volume Qualitative properties of solutions of nonlinear elliptic equations and systems. Preface
Susanna Terracini Juncheng Wei
The field of nonlinear elliptic equations/systems has experienced a new burst of activities in recent years. This includes the resolution of De Giorgi's conjecture for Allen-Cahn equation, the classification of stable/finite Morse index solutions for Lane-Emden equation, the regularity of interfaces of elliptic systems with large repelling parameter, Caffarelli-Silvestre extension of fractional laplace equations, the analysis of Toda type systems, etc. This special volume touches several aspects of these new activities.

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Liouville type results and regularity of the extremal solutions of biharmonic equation with negative exponents
Zongming Guo Juncheng Wei
We first obtain Liouville type results for stable entire solutions of the biharmonic equation $-\Delta^2 u=u^{-p}$ in $\mathbb{R}^N$ for $p>1$ and $3 \leq N \leq 12$. Then we consider the Navier boundary value problem for the corresponding equation and improve the known results on the regularity of the extremal solution for $3 \leq N \leq 12$. As a consequence, in the case of $p=2$, we show that the extremal solution $ u^{*}$ is regular when $N =7$. This improves earlier results of Guo-Wei [21] ($N \leq 4$), Cowan-Esposito-Ghoussoub [2] ($N=5$), Cowan-Ghoussoub [4] ($N=6$).
keywords: Stable entire solutions biharmonic equations with singularity regularity of the extremal solutions.
Bifurcations of some elliptic problems with a singular nonlinearity via Morse index
Zongming Guo Zhongyuan Liu Juncheng Wei Feng Zhou
We study the boundary value problem

$\Delta u=\lambda |x|^\alpha f(u)$ in $\Omega, u=1$ on $\partial \Omega\qquad$ (1)

where $\lambda>0$, $\alpha \geq 0$, $\Omega$ is a bounded smooth domain in $R^N$ ($N \geq 2$) containing $0$ and $f$ is a $C^1$ function satisfying $\lim_{s \to 0^+} s^p f(s)=1$. We show that for each $\alpha \geq 0$, there is a critical power $p_c (\alpha)>0$, which is decreasing in $\alpha$, such that the branch of positive solutions possesses infinitely many bifurcation points provided $p > p_c (\alpha)$ or $p > p_c (0)$, and this relies on the shape of the domain $\Omega$. We get some important estimates of the Morse index of the regular and singular solutions. Moreover, we also study the radial solution branch of the related problems in the unit ball. We find that the branch possesses infinitely many turning points provided that $p>p_c (\alpha)$ and the Morse index of any radial solution (regular or singular) in this branch is finite provided that $0 < p \leq p_c (\alpha)$. This implies that the structure of the radial solution branch of (1) changes for $0 < p \leq p_c (\alpha)$ and $p > p_c (\alpha)$.

keywords: infinitely many bifurcation points singular nonlinearity Morse index MEMS. Branch of positive solutions
Solutions with interior bubble and boundary layer for an elliptic problem
Liping Wang Juncheng Wei
We study positive solutions of the equation $\varepsilon^2\Delta u - u + u^{\frac{n+2}{n-2}} = 0$, where $n=3,4,5$, and $\varepsilon > 0$ is small, with Neumann boundary condition in a smooth bounded domain $\Omega \subset R^n$. We prove that, along some sequence $\{\varepsilon_j \}$ with $ \varepsilon_j \to 0$, there exists a solution with an interior bubble at an innermost part of the domain and a boundary layer on the boundary $\partial\Omega$.
keywords: Semilinear elliptic problem blow up. critical Sobolev exponent
Toda system and interior clustering line concentration for a singularly perturbed Neumann problem in two dimensional domain
Juncheng Wei Jun Yang
We consider the equation $\varepsilon^2\Delta$ ũ-ũ+ũ$^p =0$ in a bounded, smooth domain $\Omega$ in $\R^2$ under homogeneous Neumann boundary conditions. Let $\Gamma$ be a segment contained in $\Omega$, connecting orthogonally the boundary, non-degenerate and non-minimal with respect to the curve length. For any given integer $N\ge 2$ and for small $\varepsilon$ away from certain critical numbers, we construct a solution exhibiting $N$ interior layers at mutual distances $O(\varepsilon|\ln\varepsilon|)$ whose center of mass collapse onto $\Gamma$ at speed $O(\varepsilon^{1+\mu})$ for small positive constant $\mu$ as $\varepsilon\to 0$. Asymptotic location of these layers is governed by a Toda system.
keywords: Gap condition Clustered layers singularly perturbed Neumann problem. Toda system
Infinite multiplicity for an inhomogeneous supercritical problem in entire space
Liping Wang Juncheng Wei
Let $K(x)$ be a positive function in $R^N, N \geq 3$ and satisfy $\lim\limits_{|x|\rightarrow \infty} K(x) = K_\infty$ where $K_\infty$ is a positive constant. When $p > \frac{N + 1}{N - 3}, N \geq 4$, we prove the existence of infinitely many positive solutions to the following supercritical problem: \begin{eqnarray*} \Delta u(x) + K(x)u^p = 0, u>0 \quad in \quad R^N, \lim_{|x|\rightarrow \infty} u(x) = 0. \end{eqnarray*} If in addition we have, for instance, $\lim\limits_{|x| \rightarrow \infty}|x|^\mu (K(x) - K_\infty ) = C_0 \neq 0, 0 < \mu \leq N - \frac{2p+2}{p-1}$, then this result still holds provided that $p > \frac{N + 2}{N - 2}$.
keywords: entire space. Supercritical infinite multiplicity
$2\pi$-Periodic self-similar solutions for the anisotropic affine curve shortening problem II
Meiyue Jiang Juncheng Wei
The existence of $2\pi$-periodic positive solutions of the equation $$ u'' + u = \displaystyle{\frac{a(x)}{u^3}} $$ is studied, where $a$ is a positive smooth $2\pi$-periodic function. Under some non-degenerate conditions on $a$, the existence of $2\pi$-periodic solutions to the equation is established.
keywords: anisotropic affine curve shortening problem. Self-similar solutions

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