Electron. J. Diff. Equ., Vol. 2013 (2013), No. 109, pp. 1-16.

Existence of solutions for critical elliptic systems with boundary singularities

Jianfu Yang, Linli Wu

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.

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Jianfu Yang
Department of Mathematics, Jiangxi Normal University
Nanchang, Jiangxi 330022, China
email: jfyang_2000@yahoo.com
  Linli Wu
Department of Mathematics, Jiangxi Normal University
Nanchang, Jiangxi 330022, China
email: llwujxsd@sina.com

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