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### Open Access Journals

CPAA

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.

CPAA

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 $\gamma
\in (0, 1)$.

CPAA

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

Let Ω be a bounded domain in

with smooth boundary, we study the following Neumann boundary value problem

$\mathbb{R}^2 $ |

$\left\{ \begin{gathered} \begin{gathered} - \Delta \upsilon + \upsilon = 0\;\;\;\;\;\;\;\;\;\;\;\;\;\;{\text{in}}\;\;\Omega {\text{,}} \hfill \\ \frac{{\partial \upsilon }}{{\partial \nu }} = {e^\upsilon } - s{\phi _1} - h\left( x \right)\;\;\;{\text{on}}\;\partial \Omega \hfill \\ \end{gathered} \end{gathered} \right.$ |

where

denotes the outer unit normal vector to

,

,

is a large parameter and

is a positive first Steklov eigenfunction. We construct solutions of this problem which exhibit multiple boundary concentration behavior around maximum points of

on the boundary as

.

$ν$ |

$\partial \Omega$ |

$h∈ C^{0,α}(\partial \Omega)$ |

$s>0$ |

$\phi_1$ |

$\phi_1$ |

$s\to+∞$ |

## Year of publication

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