CPAA
A result on Hardy-Sobolev critical elliptic equations with boundary singularities
Jinhui Chen Haitao Yang
Communications on Pure & Applied Analysis 2007, 6(1): 191-201 doi: 10.3934/cpaa.2007.6.191
In this note, a Hardy-Sobolev critical elliptic equation with boundary singularities and sublinear perturbation is studied. We obtain a result on the existence of classical solution and the multiplicity of weak solutions by making use of sub-super solutions and variational methods.
keywords: Hardy-Sobolev critical exponent sublinear perturbation. boundary singularity
CPAA
On the existence and asymptotic behavior of large solutions for a semilinear elliptic problem in $R^n$
Haitao Yang
Communications on Pure & Applied Analysis 2005, 4(1): 187-198 doi: 10.3934/cpaa.2005.4.197
In this paper, we give an existence result for nonradial large solutions of the semilinear elliptic equation $\Delta u =p(x)f(u)$ in $R^N (N\ge 3)$, where $f$ is assumed to satisfy $(f_1)$ and $(f_2)$ below. The asymptotic behavior of the large solutions at infinity are also studied in the sublinear case that $f(u)$ behaves like $u^{\gamma}$ at $\infty$ for $\gamma \in (0, 1)$.
keywords: Large solution asymptotic behavior. sublinear
CPAA
On the blow-up boundary solutions of the Monge -Ampére equation with singular weights
Haitao Yang Yibin Chang
Communications on Pure & Applied Analysis 2012, 11(2): 697-708 doi: 10.3934/cpaa.2012.11.697
We consider the Monge-Ampére equations det$D^2 u = K(x) f(u)$ in $\Omega$, with $u|_{\partial\Omega}=+\infty$, where $\Omega$ is a bounded and strictly convex smooth domain in $R^N$. When $f(u) = e^u$ or $f(u)= u^p$, $p>N$, and the weight $K(x)\in C^\infty (\Omega )$ grows like a negative power of $d(x)=dist(x, \partial \Omega)$ near $\partial \Omega$, we show some results on the uniqueness, nonexistence and exact boundary blow-up rate of strictly convex solutions for this problem. Existence of such solutions will be also studied in a more general case.
keywords: Monge-Ampére equation uniqueness singular weight blow-up solution boundary behavior.
DCDS
Boundary bubbling solutions for a planar elliptic problem with exponential Neumann data
Haitao Yang Yibin Zhang
Discrete & Continuous Dynamical Systems - A 2017, 37(10): 5467-5502 doi: 10.3934/dcds.2017238
Let Ω be a bounded domain in
$\mathbb{R}^2 $
with smooth boundary, we study the following Neumann boundary value problem
$\left\{ \begin{gathered} \begin{gathered} - \Delta \upsilon + \upsilon = 0\;\;\;\;\;\;\;\;\;\;\;\;\;\;{\text{in}}\;\;\Omega {\text{,}} \hfill \\ \frac{{\partial \upsilon }}{{\partial \nu }} = {e^\upsilon } - s{\phi _1} - h\left( x \right)\;\;\;{\text{on}}\;\partial \Omega \hfill \\ \end{gathered} \end{gathered} \right.$
where
$ν$
denotes the outer unit normal vector to
$\partial \Omega$
,
$h∈ C^{0,α}(\partial \Omega)$
,
$s>0$
is a large parameter and
$\phi_1$
is a positive first Steklov eigenfunction. We construct solutions of this problem which exhibit multiple boundary concentration behavior around maximum points of
$\phi_1$
on the boundary as
$s\to+∞$
.
keywords: Nonlinear Neumann boundary condition boundary bubbling solutions Lyapunov-Schmidt reduction procedure

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