Sublinear eigenvalue problems on compact Riemannian manifolds with applications in Emden–Fowler equations
Volume 191 / 2009
Abstract
Let $(M,g)$ be a compact Riemannian manifold without boundary, with $\dim M\geq 3,$ and $f:\mathbb R \to \mathbb R$ a continuous function which is {sublinear} at infinity. By various variational approaches, existence of multiple solutions of the eigenvalue problem $$-{\mit\Delta}_{g} \omega+\alpha (\sigma) \omega= \tilde K(\lambda,\sigma)f(\omega),\ \quad \sigma\in M,\, \omega\in H_1^2(M),$$ is established for certain eigenvalues $\lambda>0$, depending on further properties of $f$ and on explicit forms of the function $\tilde K.$ Here, ${\mit\Delta}_{g}$ stands for the Laplace–Beltrami operator on $(M,g),$ and $\alpha,$ $\tilde K$ are smooth positive functions. These multiplicity results are then applied to solve Emden–Fowler equations which involve sublinear terms at infinity.
