CPAA
Positive solutions for singular elliptic equations with critical Hardy-Sobolev exponent
Xiaomei Sun Wenyi Chen
In this paper, we consider the following semilinear elliptic equations with critical Hardy-Sobolev exponent:

$ -\Delta u+\lambda\frac{u}{|x-a|^2}-\gamma\frac{u}{|x|^2} =\frac{Q(x)}{|x|^s}|u|^{2^*(s)-2}u+g(x,u), u>0$ in $\Omega,$

$ \frac{\partial u}{\partial\nu}+\alpha(x)u=0 $ on $\partial\Omega. $

By variational method, the existence of positive solution is obtained.

keywords: concentration compactness principle. Critical Hardy-Sobolev exponent
CPAA
Uniqueness for the solution of semi-linear elliptic Neumann problems in $\mathbb R^3$
Guangyue Huang Wenyi Chen
Considering the positive solution of the following nonlinear elliptic Neumann problem

$\Delta_0 u-\lambda u+f(u)=0, u>0,\ $ in $\Omega,\quad \frac{\partial u}{\partial\nu}=0\ $ on $\partial\Omega$

where $\Omega$ is convex and $f(u)$ defined by (2). We prove that for $1< p_i < 5$, $i=1,\cdots, K$ and $\lambda$ small, the only solution to the above problem is constant. This can be seen as a generalization of Theorem 1 in [7].

keywords: nonlinear elliptic equation. Neumann problem Uniqueness

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