Multiplicity of Radial Ground States for the Scalar Curvature Equation Without Reciprocal Symmetry
- PDF / 640,943 Bytes
- 20 Pages / 439.37 x 666.142 pts Page_size
- 5 Downloads / 185 Views
Multiplicity of Radial Ground States for the Scalar Curvature Equation Without Reciprocal Symmetry Francesca Dalbono1 · Matteo Franca2
· Andrea Sfecci3
Received: 22 March 2020 / Revised: 20 August 2020 / Accepted: 2 September 2020 © The Author(s) 2020
Abstract We study existence and multiplicity of positive ground states for the scalar curvature equation n+2
Δu + K (|x|) u n−2 = 0, x ∈ Rn , n > 2, when the function K : R+ → R+ is bounded above and below by two positive constants, i.e. 0 < K ≤ K (r ) ≤ K for every r > 0, it is decreasing in (0, R) and increasing in (R, +∞) for a certain R > 0. We recall that in this case ground states have to be radial, so the problem is reduced to an ODE and, then, to a dynamical system via Fowler transformation. We provide a smallness non perturbative (i.e. computable) condition on the ratio K /K which guarantees the existence of a large number of ground states with fast decay, i.e. such that u(|x|) ∼ |x|2−n as |x| → +∞, which are of bubble-tower type. We emphasize that if K (r ) has a unique critical point and it is a maximum the radial ground state with fast decay, if it exists, is unique. Keywords Scalar curvature equation · Ground states · Fowler transformation · Invariant manifold · Bubble tower solutions · Phase plane analysis · Multiplicity results Mathematics Subject Classification 35J60 · 37D10 · 34C37
Francesca Dalbono is partially supported by the PRIN project 2017JPCAPN “Qualitative and quantitative aspects of nonlinear PDEs. Francesca Dalbono, Matteo Franca and Andrea Sfecci are partially supported by the GNAMPA project “Dinamiche non autonome, analisi reale e applicazioni”.
B
Matteo Franca [email protected] Francesca Dalbono [email protected] Andrea Sfecci [email protected]
1
Dipartimento di Matematica e Informatica, Università degli Studi di Palermo, Via Archirafi 34, 90123 Palermo, Italy
2
Dipartimento di Matematica, Università degli Studi di Bologna, Piazza di porta san Donato 5, 40126 Bologna, Italy
3
Dipartimento di Matematica e Geoscienze, Università degli Studi di Trieste, Via Valerio 12/1, 34127 Trieste, Italy
123
Journal of Dynamics and Differential Equations
1 Introduction In this paper we study the scalar curvature equation n+2
Δu + K (|x|) u n−2 = 0, x ∈ Rn , n > 2, Rn ,
(1)
where x ∈ n > 2, K ∈ and 0 < K ≤ K (|x|) ≤ K whenever |x| > 0 for suitable positive constants 0 < K < K . In particular, we will focus our attention on radially symmetric positive solutions with fast decay. Radial solutions of Eq. (1) solve C1
n+2
(u r n−1 ) + K (r ) r n−1 u n−2 = 0 ,
r ∈ (0, ∞).
(2)
We are interested in studying multiplicity of grounds states u of (2) having fast decay, where with ground state (GS) we mean a positive regular solution u of (2) defined for any r ≥ 0 and for fast decay we mean that u(r )r 2−n has positive finite limit as r → +∞. The existence of GS with fast decay has been the subject of many papers for its intrinsic mathematical interest, but also for the relevant applications it finds in differential geometry,
Data Loading...
