Invariant criteria for existence of bounded positive solutions
Dong Ye Feng Zhou
Discrete & Continuous Dynamical Systems - A 2005, 12(3): 413-424 doi: 10.3934/dcds.2005.12.413
We consider semilinear elliptic equations $\Delta u \pm \rho(x)f(u) = 0$, or more generally $\Delta u + \varphi(x, u) = 0$, posed in $\R^N$ ($N\geq 3$). We prove that the existence of entire bounded positive solutions is closely related to the existence of bounded solution for $\Delta u + \rho(x) = 0$ in $\mathbb R^N$. Many sufficient conditions which are invariant under the isometry group of $\mathbb R^N$ are established. Our proofs use the standard barrier method, but our results extend many earlier works in this direction. Our ideas can also be applied for the existence of large solutions, for the exterior domain problem and for the system situations.
keywords: barrier methods entire bounded solution. Invariant criteria
Isolated singularities for elliptic equations with hardy operator and source nonlinearity
Huyuan Chen Feng Zhou
Discrete & Continuous Dynamical Systems - A 2018, 38(6): 2945-2964 doi: 10.3934/dcds.2018126
In this paper, we concern the isolated singular solutions for semi-linear elliptic equations involving Hardy-Leray potential
$- \Delta u + \frac{\mathit{\mu }}{{|x{|^2}}}u = {u^p}\;\;\;{\rm{in }}\;\;\;\Omega \setminus \{ 0\} ,\;\;\;u = 0\;\;\;{\rm{on}}\;\;\;\partial \Omega .\;\;\;\;\;\;\;\;\;\;\;\;\left( 1 \right)$
We classify the isolated singularities and obtain the existence and stability of positive solutions of (1). Our results are based on the study of nonhomogeneous Hardy problem in a new distributional sense.
keywords: Hardy potential isolated singularity classification
Bifurcations of some elliptic problems with a singular nonlinearity via Morse index
Zongming Guo Zhongyuan Liu Juncheng Wei Feng Zhou
Communications on Pure & Applied Analysis 2011, 10(2): 507-525 doi: 10.3934/cpaa.2011.10.507
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
Boundary blow-up solutions with interior layers and spikes in a bistable problem
Yihong Du Zongming Guo Feng Zhou
Discrete & Continuous Dynamical Systems - A 2007, 19(2): 271-298 doi: 10.3934/dcds.2007.19.271
We show that for small $\epsilon>0$, the boundary blow-up problem

$-\epsilon^2\Delta u= u (u-a(x))(1-u) \mbox{ in } \Omega, u|_{\partial\Omega}=\infty$

has solutions with sharp interior layers and spikes, apart from boundary layers. We also determine the location of these layers and spikes.

keywords: spike layer reduction method. bistable boundary blow-up Boundary and interior layer
Dynamics for the damped wave equations on time-dependent domains
Feng Zhou Chunyou Sun Xin Li
Discrete & Continuous Dynamical Systems - B 2018, 23(4): 1645-1674 doi: 10.3934/dcdsb.2018068

We consider the asymptotic dynamics of a damped wave equations on a time-dependent domains with homogeneous Dirichlet boundary condition, the nonlinearity is allowed to have a cubic growth rate which is referred to as the critical exponent. To this end, we establish the existence and uniqueness of strong and weak solutions satisfying energy inequality under the assumption that the spatial domains $\mathcal{O}_{t}$ in $\mathbb{R}^{3}$ are obtained from a bounded base domain $\mathcal{O}$ by a $C^{3}$-diffeomorphism $r(·, t)$. Furthermore, we establish the pullback attractor under a slightly weaker assumption that the measure of the spatial domains are uniformly bounded above.

keywords: Non-autonomous dynamical systems wave equation time-dependent domain critical exponent pullback attractor
Bubble tower solutions of slightly supercritical elliptic equations and application in symmetric domains
Yuxin Ge Ruihua Jing Feng Zhou
Discrete & Continuous Dynamical Systems - A 2007, 17(4): 751-770 doi: 10.3934/dcds.2007.17.751
We construct solutions of the semilinear elliptic problem

$\Delta u+ |u|^{p-1}u+$ε1/2 f = 0 in Ω
u=ε1/2 g on $\partial$Ω

in a bounded smooth domain $\Omega \subset \R^N$ $(N\geq 3)$, when the exponent $p$ is supercritical and close enough to $\frac{N+2}{N-2}$. As $p\rightarrow \frac{N+2}{N-2}$, the solutions have multiple blow up at finitely many points which are the critical points of a function whose definition involves Green's function. As applications, we will give some existence results, in particular, when $\O$ are symmetric domains perforated with the small hole and when $f=0$ and $g=0$.

keywords: Supercritical Sobolev exponent Green function Multiple blow up.

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