Electronic Journal of Differential Equations,
Vol. 2013 (2013), No. 109, pp. 1-16.

Title: Existence of solutions for critical elliptic systems with
       boundary singularities
        
Authors: Jianfu Yang (Jiangxi Normal Univ., Nanchang, China)
         Linli Wu    (Jiangxi Normal Univ., Nanchang, China)

Abstract:
 This article concerns the existence of positive solutions to  the
 nonlinear elliptic system involving critical Hardy-Sobolev exponent
 $$\displaylines{
 -\Delta u= \frac{2\lambda\alpha}{\alpha+\beta}
 \frac{u^{\alpha-1} v^\beta}{|\pi(x)|^s}- u^p,  \quad \hbox{in } \Omega,\cr
 -\Delta v= \frac{2\lambda\beta}{\alpha+\beta}
 \frac{u^\alpha v^{\beta-1}}{|\pi(x)|^s}- v^p,   \quad \hbox{in }  \Omega,\cr
 u>0,\quad v>0, \quad \hbox{in }  \Omega,\cr
 u=v=0, \quad \hbox{on }   \partial\Omega,
 }$$
 where $N\geq 4$ and $\Omega$ is a $C^1$ bounded domain in $\mathbb{R}^N$,
 $0<s<2$,  $\alpha+\beta=2^*(s)=\frac{2(N-s)}{N-2}$, $\alpha,\beta>1$,
 $\lambda>0$ and $1\leq p<\frac{N}{N-2}$.

 Let $\mathcal{P}$ be a linear subspace of $\mathbb{R}^N$ such that
 $k = \dim_{\mathbb{R}}\mathcal{P}\geq 2$, and $\pi$ be the orthogonal
 projection on $\mathcal{P}$ with respect to the Euclidean structure.
 We consider mainly  the case when  $\mathcal{P}^\bot\cap \Omega =\emptyset$
 and $\mathcal{P}^\bot\cap\partial\Omega \neq \emptyset$.
 We show that there exists $\lambda^*>0$ such that the system above
 possesses at least one positive solution for $0<\lambda<\lambda^*$
 provided that at each point $x\in \mathcal{P}^\bot\cap\partial\Omega$
 the principal curvatures of $\partial\Omega$ at $x$ are non-positive,
 but not all vanish.

Submitted October 16, 2012. Published April 29, 2013.
Math Subject Classifications: 35J25, 25J50, 35J57.
Key Words: Existence; compactness; critical Hardy-Sobolev exponent; 
           nonlinear system.