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

Alexandru Kristály¹ ; Vicenţiu Rădulescu²

¹Department of Economics University of Babeş-Bolyai 400591 Cluj-Napoca, Romania
²Institute of Mathematics “Simion Stoilow" of the Romanian Academy 014700 Bucureşti, Romania and Department of Mathematics University of Craiova 200585 Craiova, Romania

Studia Mathematica, Tome 191 (2009) no. 3, pp. 237-246

Résumé

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.

Détail

DOI : 10.4064/sm191-3-5

Keywords: compact riemannian manifold without boundary dim geq mathbb mathbb continuous function which sublinear infinity various variational approaches existence multiple solutions eigenvalue problem mit delta omega alpha sigma omega tilde lambda sigma omega quad sigma omega established certain eigenvalues lambda depending further properties explicit forms function tilde here mit delta stands laplace beltrami operator alpha tilde smooth positive functions these multiplicity results applied solve emden fowler equations which involve sublinear terms infinity