Elliptic equations with critical exponent on spherical caps of \(S^{3}\) (Q2479606)

The authors deal with a problem on a geodesic ball \(\widetilde{\mathbb{B}}\) centered at the North pole in \(S^3\): \[ \begin{cases} -\Delta_{S^3} u= u^5+\lambda u\quad &\text{in }\widetilde{\mathbb{B}},\\ u> 0\quad &\text{in }\widetilde{\mathbb{B}},\\ u= 0\quad &\text{on }\partial\widetilde{\mathbb{B}},\end{cases}\tag{1} \] where \(\Delta_{S^3}\) is the Laplace-Beltrami operator on \(\widetilde{\mathbb{B}}\). Let \(\theta_*\in (0,\pi)\) be the radius of \(\widetilde{\mathbb{B}}\), that is, the geodesic distance between the North pole and \(\partial\widetilde{\mathbb{B}}\). The main goal of the authors is to identify the range of values of the parameters \(\theta_*\) and \(\lambda\) for which there exists a solution of (1).