Concentrating phenomena in some elliptic Neumann problem: Asymptotic behavior of solutions
Long Wei
Communications on Pure & Applied Analysis 2008, 7(4): 925-946 doi: 10.3934/cpaa.2008.7.925
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
Discrete & Continuous Dynamical Systems - A 2016, 36(2): 917-939 doi: 10.3934/dcds.2016.36.917
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
Communications on Pure & Applied Analysis 2014, 13(6): 2493-2508 doi: 10.3934/cpaa.2014.13.2493
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
Discrete & Continuous Dynamical Systems - A 2017, 37(3): 1733-1748 doi: 10.3934/dcds.2017072

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
A perturbed fourth order elliptic equation with negative exponent
Zongming Guo Long Wei
Discrete & Continuous Dynamical Systems - B 2018, 22(11): 1-19 doi: 10.3934/dcdsb.2018132

By a new type of comparison principle for a fourth order elliptic problem in general domains, we investigate the structure of positive solutions to Navier boundary value problems of a perturbed fourth order elliptic equation with negative exponent, which arises in the study of the deflection of charged plates in electrostatic actuators in the modeling of electrostatic micro-electromechanical systems (MEMS). It is seen that the structure of solutions relies on the boundary values. The global branches of solutions to the Navier boundary value problems are established. We also show that the behaviors of these branches are relatively "stable" with respect to the Navier boundary values.

keywords: Perturbed fourth order elliptic equations Navier boundary value problems negative exponent branches of solutions

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