Normalized solutions for nonlinear coupled fractional systems: low and high perturbations in the attractive case
The authors study coupled nonlocal systems of the form \begin{align*} \begin{cases} (-\Delta)^su-\lambda_1u=\mu_1 |u|^\alpha u+\beta |u|^{\frac{\alpha-2}{2}}u|v|^{\frac{\alpha+2}{2}} & \text{in } \mathbb{R}^N,\\ (-\Delta)^sv-\lambda_2v=\mu_2 |v|^\alpha v+\beta |u|^{\frac{\alpha+2}{2}}|v|^{\frac{\alpha-2}{2}}v & \text{in } \mathbb{R}^N, \end{cases} \end{align*} with \begin{align*} \int_{\mathbb{R}^N} u^2\,dx = b_1^2 \quad\text{and}\quad \int_{\mathbb{R}^N} v^2\,dx = b_2^2, \end{align*} where \((-\Delta)^s\) is the fractional Laplacian, \(0<s<1\), \(\mu_1, \mu_2>0\), \(N>2s\), and \(\frac{4s}{N}<\alpha\leq \frac{2s}{N-2s}\). In the case of low perturbations of the coupling parameter, by using two-dimensional linking arguments, the existence of \(\beta_1>0\) is shown such that when \(0<\beta<\beta_1\), then the system has a positive radial solution. In the second part, in the case of high perturbations of the coupling parameter, the existence of \(\beta_2>0\) is shown such that the system has a mountain-pass type solution for all \(\beta>\beta_2\).
scientific article; zbMATH DE number 7356253
scientific article