The Emden-Fowler equation on a spherical cap of Sn

Let S-n subset of Rn+ 1, n >= 3, be the unit sphere, and let S-Theta( )subset of S-n be a geodesic ball with geodesic radius Theta is an element of (0,pi). We study the bifurcation diagram {(Theta, parallel to U parallel to(infinity))} subset of R-2 of the radial solutions of the Emden-Fowler equation oil S-Theta Delta U-Sn + U-p = 0 in S-Theta, U = 0 on partial derivative S-Theta, U > 0 in S-Theta, where p > 1. Among other things, we prove the following: For each p > p(S) := (n - 2)/(n + 2), there exists (Theta) under bar is an element of (0, pi) such that the problem has a radial solution for Theta is an element of ((Theta) under bar pi) and has no radial solution for Theta is an element of (0, (Theta) under bar). Moreover, this solution is unique in the space of radial functions if Theta is close to pi. If p(S) < p < p(JL), then there exists Theta* E ((Theta) under bar pi) such that the problem has infinitely many radial solutions for Theta = Theta*, where pJL = {1 + 4/n - 4 2 root n - 1 if n >= 11, infinity if 2 <= n <= 10. Asymptotic behaviors of the bifurcation diagram as p -> infinity and p down arrow 1 are also studied. (C) 2018 Elsevier Ltd. All rights reserved.