A remark on doubly critical elliptic systems
The paper deals with the weakly coupled nonlinear elliptic system \[ \begin{cases} -\Delta u-\frac{\lambda_1}{|x|^2}u-|u|^{2^*-2}u=\nu h(x)\alpha|u|^{\alpha-2}|v|^{\beta}u,\,\,\,x\in\mathbb{R}^N,\\ -\Delta v-\frac{\lambda_2}{|x|^2}v-|v|^{2^*-2}v=\nu h(x)\beta|u|^{\alpha}|v|^{\beta-2}v,\,\,\,x\in\mathbb{R}^N,\\ u(x),\,\,v(x)\in D^{1,2}(\mathbb{R}^N),\end{cases} \leqno(1) \] where \(N\geq 3\), \(\lambda_1,\,\lambda_2\in (0,\Lambda_N)\), \(\Lambda_N:=\frac{(N-2)^2}{4}\), \(2^*=\frac{2N}{N-2}\) is the critical Sobolev exponent, \(\nu>0\) is a parameter, \(\alpha,\,\beta>1\), \(\alpha+\beta\leq 2^*\) and \(D^{1,2}(\mathbb{R}^N)\) is the completion of \(C_0^{\infty}(\mathbb{R}^N)\) with respect to the norm \(\|u\|=\left(\int_{\mathbb{R}^N}|\nabla u|^2\,dx\right)^{1/2}\). By using some variational methods, the authors investigate the existence and properties of the positive solutions of problem \((1)\).
scientific article