Solutions to a singularly perturbed supercritical elliptic equation on a Riemannian manifold concentrating at a submanifold

Abstract

Abstract. Given a smooth Riemannian manifold (M, g) we investigate the existence of positive solutions to the equationwhich concentrate at some submanifold of M as ε → 0, for supercritical nonlinearities. We obtain a posive answer for some manifolds, which include warped products.Using one of the projections of the warped product or some harmonic morphism, we reduce this problem to a problem of the formwith the same exponent p, on a Riemannian manifold (M, h) of smaller dimension, so that p turns out to be subcritical for this last problem. Then, applying Lyapunov-Schmidt reduction, we establish existence of a solution to the last problem which concentrates at a point as ε → 0.

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].…”

Semiclassical states for a static supercritical Klein–Gordon–Maxwell–Proca system on a closed Riemannian manifold

“…For the proof of the following statement we refer to Lemma 4.1 of [3] (see also Proposition 3.1 of [12]). Proposition 2.4.…”

“…Finally, we prove (4.4). Following the proof of Lemma 5.1 in [3], we need only to prove that G ′ ε W ε,ξ(y) + φ ε,ξ(y) [Z l ε,ξ(y) ] = o(1), that is On the other hand, from Remark 5.2, the proof of Lemma 5.4, and the estimates (2.8) and (2.9), for some t ∈ (1, 3/2) we obtain…”

“…This approach was introduced by Ruf and Srikanth in [13], where a Hopf map is used to obtain the reduction. Reductions may also be performed by means of other maps which preserve the Laplace-Beltrami operator, or by considering warped products, or by a combination of both, see [3,14] and the references therein. We describe these reductions in the following two subsections.…”

“…There exist ε 0 > 0 and C > 0 such that, for any ξ ∈ M and any ε ∈ (0, ε 0 ), the inequality R ε,ξ ε ≤ Cε holds true.Proof. See Lemma 4.2 in[3].…”