Sublinear eigenvalue problems on compact Riemannian manifolds with applications in Emden–Fowler equations

Abstract

Let (M;g) be a compact Riemannian manifold without boundary, with dimM 3; and f : R! R a continuous function which is sublinear at innity. By various variational approaches, existence of multiple solutions of the eigenvalue problem g! + ( )! = ~ K(; )f(!); 2 M; !2 H 2 1 (M); is established for certain eigenvalues > 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; ~ K are smooth positive functions. These multiplicity results are then applied to solve Emden{Fowler equations which involve sublinear terms at innity.