Blow-up solutions for asymptotically critical elliptic equations on Riemannian manifolds

Abstract

Given (M, g) a smooth, compact Riemannian n-manifold, we consider equations like ∆ g u + hu = u 2 * −1−ε , where h is a C 1-function on M , the exponent 2 * = 2n/ (n − 2) is critical from the Sobolev viewpoint, and ε is a small real parameter such that ε → 0. We prove the existence of blowing-up families of positive solutions in the subcritical and supercritical case when the graph of h is distinct at some point from the graph of n−2 4(n−1) Scal g .

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

“…That is, they reduce the problem to finding the critical point of a functional on the finite-dimensional manifold M . Our purpose in this paper is to obtain an analogous result to the result obtained in [8] for a more general class of functions f . In case that f (t) = t p−1 , the uniqueness of the least energy solution of problem (1.2) up to translations plays a crucial role.…”

“…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 . They showed that, for each C 1 -suitable critical set K, there exists a solution u ε of (P) under the assumption that the limiting problem…”

“…In general, we cannot expect such uniqueness. Moreover, the Lyapunov-Schmidt reduction procedure is involved and it is not clear if the procedure works for general cases such that the set of the positive solutions of (1.2) is degenerate [8]. In this paper, we will show the multiple existence of positive solutions of (P) under some assumptions on f using homology group arguments.…”

From periodic ODE’s to supercritical PDE’s

“…In the present paper we build solutions to problem (1), which concentrate along an (m − 1)-dimensional submanifold of M as p goes to +∞. Moreover, for any integer k between 0 and (m − 3) solutions which concentrate along a k-dimensional minimal submanifold of M as p approaches the critical exponents 2 * m,k − 1 have been found in [14,11,4,7]. Therefore, it is natural to ask if it is possible to find solutions which concentrate along an (m − 2)-dimensional minimal submanifold of M as p approaches +∞.…”

“…Indeed, it would be a geodesic if r 0 was a critical point of the warping function f (see [3]). In general r 0 is only a critical point of the function V f ,κ defined in (14). Remark 1.5.…”