Concentrating phenomena in some elliptic Neumann problem: Asymptotic behavior of solutions
Long Wei
We investigate here the elliptic equation -div$(a(x)\nabla u)+a(x)u=0$ posed on a bounded smooth domain $\Omega$ in $\mathbb R^2$ with nonlinear Neumann boundary condition $\frac{\partial u}{\partial \nu}=\varepsilon e^u$, where $\varepsilon$ is a small parameter. We extend the work of Davila-del Pino-Musso [5] and show that if a family of solutions $u_\varepsilon$ for which $\varepsilon\int_{\partial Omega}e^{u_\varepsilon}$ is bounded, then it will develop up to subsequences a finite number of bubbles $\xi_i\in\partial Omega$, in the sense that $\varepsilon e^{u_\varepsilon}\to 2\pi\sum_{i=1}^k m_i\delta_{\xi_i}$ as $\varepsilon\rightarrow 0$ with $k, m_i \in \mathbb Z^+$. Location of blow-up points is characterized in terms of function $a(x)$.
keywords: Exponential Neumann nonlinearity concentration of solutions.
Infinitely many positive and sign-changing solutions for nonlinear fractional scalar field equations
Wei Long Shuangjie Peng Jing Yang
We consider the following nonlinear fractional scalar field equation $$ (-\Delta)^s u + u = K(|x|)u^p,\ \ u > 0 \ \ \hbox{in}\ \ \mathbb{R}^N, $$ where $K(|x|)$ is a positive radial function, $N\ge 2$, $0 < s < 1$, and $1 < p < \frac{N+2s}{N-2s}$. Under various asymptotic assumptions on $K(x)$ at infinity, we show that this problem has infinitely many non-radial positive solutions and sign-changing solutions, whose energy can be made arbitrarily large.
keywords: Fractional Laplacian reduction method. nonlinear scalar field equation
A fourth order elliptic equation with a singular nonlinearity
Zongming Guo Long Wei
In this paper, we study the structure of solutions of a fourth order elliptic equation with a singular nonlinearity. For different boundary values $\kappa$, we establish the global bifurcation branches of solutions to the equation. More precisely, we show that $\kappa=1$ is a critical boundary value to change the structure of solutions to this problem.
keywords: singular nonlinearities Fourth order elliptic equations branches of solutions. blow-up arguments
Conserved quantities, global existence and blow-up for a generalized CH equation
Long Wei Zhijun Qiao Yang Wang Shouming Zhou

In this paper, we study conserved quantities, blow-up criterions, and global existence of solutions for a generalized CH equation. We investigate the classification of self-adjointness, conserved quantities for this equation from the viewpoint of Lie symmetry analysis. Then, based on these conserved quantities, blow-up criterions and global existence of solutions are presented.

keywords: Nonlinear self-adjointness conservation law global existence blow-up

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