On elliptic system involving critical Sobolev-Hardy exponents (Q839823)

The authors deal with the following nonlinear elliptic system \[ \begin{cases} - \Delta u - \frac{\mu}{|x|^2} u = au+bv +( \alpha +1) \frac{u | u |^{\alpha-1} |v |^{\beta + 1}}{|x|^s} &\text{in } \Omega \\ - \Delta v - \frac{\mu}{|x|^2} v = bu+cv +( \beta +1) \frac{| u |^{\alpha+1} v|v |^{\beta - 1}}{|x|^s} &\text{in } \Omega \\ u=v=0 &\text{on } \partial \Omega, \end{cases}\tag{\(*\)} \] where \(\Omega \subset\mathbb R^N\), \(N \geq 4\), is a bounded regular domain such that \(0 \in \text{int}(\Omega)\), \(a,b,c\in\mathbb R\), \( 0\leq s< 2\), with \(\alpha + \beta=\frac{4-2s}{N-2} \) and \(0 \leq \mu < \overline{\mu}:=(\frac{N-2}{2})^2\). Following the method introduced by \textit{C. O. Alves, D. C. de Morais Filho} and \textit{M. A. S. Souto} in [Nonlinear Anal., Theory Methods Appl. 42, No.~5(A), 771--787 (2000; Zbl 0958.35037)] and by \textit{D. Kang} and \textit{S. Peng} in [Appl. Math. Lett. 18, No.~10, 1094--1100 (2005; Zbl 1086.35044)], the authors obtain the following existence results relating the eigenvalues \(\mu_k\), \(k \in\mathbb N^*\) of the operator \(- \Delta-\mu \frac{1}{| x |^2} \) to the eigenvalues \(\lambda_1\) and \(\lambda_2 \) of the matrix \(A = \left( \begin{smallmatrix} a&b\\ b&c \end{smallmatrix}\right)\), under the assumption that the matrix \(A\) is positive definite. More precisely {\parindent=6mm \begin{itemize}\item[(1)] If \(N \geq 4\) and \(0 \leq \mu \leq \overline{\mu} -1 \), then the system \((*)\) has a solution for all \(\lambda_2 < \mu_1 \). \item[(2)] If \(N \geq 4\) and \(\overline{\mu} - 1 < \mu < \overline{\mu}\), then the system \((*)\) has a solution for all \(\mu^*< \lambda_1\leq \lambda_2< \mu_1\), where \[ \mu^*= \min_{\varphi\in H_0^1(\Omega)} \frac{\int_\Omega \frac{|\nabla\varphi(x) |^2}{|x|^{2\sigma}}\,dx} {\int_\Omega\frac{|\varphi(x)|^2}{|x|^{2\sigma}}\,dx} \quad \text{with}\quad \sigma= \sqrt{\overline{\mu}}+ \sqrt{\overline{\mu}- \mu}. \] \item[(3)] Finally, for \(N\geq 5\), \(0 \leq \mu < \overline{\mu}-(\frac{N+2}{N})^2\), if \( \lambda_1\) and \(\lambda_2 \) satisfy one of the two conditions given in Theorem 1.2, then the system \((*)\) admits at least one solution. \end{itemize}}