Niveau: Supérieur, Licence, Bac+2
MULTIPLE SOLUTIONS FOR NONLINEAR ELLIPTIC EQUATIONS ON COMPACT RIEMANNIAN MANIFOLDS JEROME VETOIS Abstract. Let (M, g) be a smooth, compact Riemannian n-manifold, and h be a Holder continuous function on M . We prove the existence of multiple changing sign solutions for equations like ∆gu + hu = |u| 2??2 u, where ∆g is the Laplace–Beltrami operator and the exponent 2? = 2n/ (n? 2) is critical from the Sobolev viewpoint. 1. Introduction 1.1. Statement of the results Let (M, g) be a smooth, compact Riemannian manifold of dimension n ≥ 3 and h be a Holder continuous function on M , namely a function which belongs to C0,? (M) for some real number ? in (0, 1). We consider equations like ∆gu+ hu = |u| 2??2 u , (1.1) where ∆g = ? divg? is the Laplace–Beltrami operator, and 2? = 2n/ (n? 2). If H21 (M) stands for the Sobolev space of all functions in L2 (M) with one derivative in L2 (M), then 2? is the critical exponent for the embeddings of H21 (M) into Lebesgue spaces. We provide H21 (M) with the scalar product ?u, v?H21 (M) = ∫ M ??u,?v?g dvg + ? ∫ M uvdvg , (1.2) where ? is a positive constant to be chosen large later on.
- like ∆g
- multiple solutions
- positive solution
- elliptic equations
- critical elliptic
- unique positive
- solution u∞
- changing sign
- nodal solutions
- euclidean space