Ground states of a prescribed mean curvature equation

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We study the existence of radial ground state solutions for the problem- div (frac(∇ u, sqrt(1 + | ∇ u |2))) = uq, u > 0 in RN,u (x) → 0 as | x | → ∞,N ≥ 3, q > 1. It is known that this problem has infinitely many ground states when q ≥ frac(N + 2, N - 2), while no solutions exist if q ≤ frac(N, N - 2). A question raised by Ni and Serrin in [W.-M. Ni, J. Serrin, Existence and non-existence theorems for ground states for quasilinear partial differential equations, Atti Convegni Lincei 77 (1985) 231-257] is whether or not ground state solutions exist for frac(N, N - 2) < q < frac(N + 2, N - 2). In this paper we prove the existence of a large, finite number of ground states with fast decay O (| x |2 - N) as | x | → + ∞ provided that q lies below but close enough to the critical exponent frac(N + 2, N - 2). These solutions develop a bubble-tower profile as q approaches the critical exponent. © 2007 Elsevier Inc. All rights reserved.




del Pino, M., & Guerra, I. (2007). Ground states of a prescribed mean curvature equation. Journal of Differential Equations, 241(1), 112–129.

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