Multiple existence of solutions for a nonlinear elliptic problem on a Riemannian manifold

Multiple existence of solutions for a singularly perturbed nonlinear elliptic problem on a Riemannian manifold

“…As far as we know, their results are the first multiple existence results for (P) on a Riemannian manifold. In [7], Hirano showed that the number of solutions of (P) is affected by the topology of suitable subset of M . More recently, Micheletti and Pistoia [8] studied the role of the scalar curvature for the multiple existence of positive solutions of (P) with f (t) = t p−1 .…”

Positive solutions for a singularly perturbed nonlinear elliptic problem on manifolds via Morse theory

“…Recently, in [8][9][10] it has been proved that the existence of positive solutions is strongly related to the geometry of M; that is, stable critical points of the scalar curvature S g generate positive solutions with one ore more peaks as " goes to zero. Previously Benci et al [11] (see also [12,13]) pointed out that the topology of M has effect on the number of solutions of (1.1), that is (1.1) has at least cat M nonconstant solutions for " small enough. Here cat M is the Lusternik Schnirelmann category of M. Moreover, in [11] the Poincare´polynomial is considered (Definition 2.1) and the authors assume that all the solution of the problem (1.1) are nondegenerate.…”

Some Generic Properties of Non Degeneracy for Critical Points of Functionals and Applications

“…Here (M, g) is a smooth connected compact Riemannian manifold of dimension n ≥ 3 embedded in R N . In [1,9,20] it is shown that the number of solutions is influenced by the topology of M . In [3,13,14] there are some results about the effect of the geometry of M in finding solutions, more precisely the role of the scalar curvature S g of (M, g).…”