Symmetric solutions for some elliptic equation on spherical cap of \(\mathrm{S}^n\) (Q6611874)

This paper studied the existence of positive and symmetric solutions for the problem\N\[\N\begin{cases} \Delta_{{\mathbb S}^n}w+\lambda w+w^p=0 \quad \text{in}\ D_\beta\\\Nw>0 \quad \text{in}\ D_\beta\\\Nw=0\quad \text{on}\ \partial D_\beta, \end{cases}\N\]\Nwhere \(\lambda\in R\) and \(P>1\) are parameters, \(D_\beta\) is the spherical cap including the North pole \((0,\ldots, 0, 1)\) on \({\mathcal S}^n\) whose geodesic radius is denoted by \(\beta\in (0,\pi)\), and \(\Delta_{{\mathbb S}^n}\) is the Laplace-Beltrami operator on \({\mathcal S}^n\). The paper investigated non-critical case for \(n\)-dimensional space, and for the sub-critical case, it proves the existence and nonexistence result if the spherical cap is contained in the hemisphere.