The Emden–Fowler equation on a spherical cap of Sn

Abstract Let S n ⊂ R n + 1 , n ≥ 3 , be the unit sphere, and let S Θ ⊂ S n be a geodesic ball with geodesic radius Θ ∈ ( 0 , π ) . We study the bifurcation diagram { ( Θ , ‖ U ‖ ∞ ) } ⊂ R 2 of the radial solutions of the Emden–Fowler equation on S Θ Δ S n U + U p = 0 in S Θ , U = 0 on ∂ S Θ , U > 0 in S Θ , where p > 1 . Among other things, we prove the following: For each p > p S ≔ ( n − 2 ) ∕ ( n + 2 ) , there exists Θ ∈ ( 0 , π ) such that the problem has a radial solution for Θ ∈ ( Θ , π ) and has no radial solution for Θ ∈ ( 0 , Θ ) . Moreover, this solution is unique in the space of radial functions if Θ is close to π . If p S p p JL , then there exists Θ ∗ ∈ ( Θ , π ) such that the problem has infinitely many radial solutions for Θ = Θ ∗ , where p JL = 1 + 4 n − 4 − 2 n − 1 if n ≥ 11 , ∞ if 2 ≤ n ≤ 10 . Asymptotic behaviors of the bifurcation diagram as p → ∞ and p ↓ 1 are also studied.

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