Sublinear eigenvalue problems on compact Riemannian manifolds with applications in Emden–Fowler equations
Let (M, g) be a compact Riemannian manifold without boundary, with dimM ≥ 3, and f : R→ R a continuous function which is sublinear at infinity. By various variational approaches, existence of multiple solutions of the eigenvalue problem −∆gω + α(σ)ω = K(λ, σ)f(ω), σ ∈M, ω ∈ H 1 (M), is established for certain eigenvalues λ > 0, depending on further properties of f and on explicit forms of the function K. Here, ∆g stands for the Laplace–Beltrami operator on (M, g), and α, K are smooth positive functions. These multiplicity results are then applied to solve Emden–Fowler equations which involve sublinear terms at infinity.