Concentration on minimal submanifolds for a Yamabe-type problem

Abstract

We construct solutions to a Yamabe type problem on a Riemannian manifold M without boundary and of dimension greater than 2, with nonlinearity close to higher critical Sobolev exponents. These solutions concentrate their mass around a non degenerate minimal submanifold of M , provided a certain geometric condition involving the sectional curvatures is satisfied. A connection with the solution of a class of P.D.E.'s on the submanifold with a singular term of attractive or repulsive type is established.

Supercritical elliptic problems on the round sphere and nodal solutions to the Yamabe problem in projective spaces

“…A fruitful approach consists in reducing the supercritical problem to a more general elliptic critical or subcritical problem, either by considering rotational symmetries or by means of maps preserving the Laplace operator or by a combination of both, see [17] and the references therein. In case of closed Riemannian manifolds, these reduction methods also apply and have been combined with the Lyapunov-Schmidt reduction method in order to obtain sequences of positive and sign-changing solutions to similar supercritical problems, such that they blow-up or concentrate at minimal submanifolds of M [16,21,25,32,39].…”

“…This method has been further generalized to supercritical exponents in [6] to prove a similar result on general closed Riemannian manifolds, including the round sphere. Other results concerning the existence and concentration of positive solutions along minimal submanifolds for the supercritical and slightly supercritical can be found in [21,32]. However, very little is known about the existence, multiplicity and blow-up of nodal solutions for the supercritical problem on the sphere (1.2).…”

Concentration at sub-manifolds for an elliptic Dirichlet problem near high critical exponents

“…The first main ingredient in proving our main theorem is the construction of a very accurate approximate solution in powers of and , in a neighborhood of the scaled sub‐manifold . It is worth mentioning that concentration at higher dimensional sets for some related problem with Neumann boundary conditions or on manifolds has been extensively studied in the last decade, see [, , , , , ] and some references therein. In most of the above‐mentioned problems the profile has an exponential decay which is crucial in the construction of very accurate approximate solutions via an iterative scheme of Picard's type.…”

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 +∞.…”