Abstract
Abstract In this paper, we consider the analogous of the Hénon equation for the prescribed mean curvature problem in {{\mathbb{R}^{N}}} , both in the Euclidean and in the Minkowski spaces. Motivated by the studies of Ni and Serrin [W. M. Ni and J. Serrin, Existence and non-existence theorems for ground states for quasilinear partial differential equations, Att. Convegni Lincei 77 1985, 231–257], we have been interested in finding the relations between the growth of the potential and that of the local nonlinearity in order to prove the nonexistence of a radial ground state. We also present a partial result on the existence of a ground state solution in the Minkowski space.
Highlights
As is well known, the partial differential operator∇ ⋅ [ ∇( ⋅ ) ] √1 ± |∇( ⋅ )|2 is related with questions on mean curvature in Euclidean and Minkowski spaces, depending on whether the sign under the square root is + or −
In this paper, we consider the analogous of the Hénon equation for the prescribed mean curvature problem in RN, both in the Euclidean and in the Minkowski spaces
Convegni Lincei 77 (1985), 231–257], we have been interested in finding the relations between the growth of the potential and that of the local nonlinearity in order to prove the nonexistence of a radial ground state
Summary
∇ ⋅ [ ∇( ⋅ ) ] √1 ± |∇( ⋅ )|2 is related with questions on mean curvature in Euclidean and Minkowski spaces, depending on whether the sign under the square root is + or −. From [2, 15], in our case the presence of a potential in the equation perturbs the identity in a way that makes it useless We overcome this difficulty by means of a suitable exploitation of some arguments which are typical in the ODE theory, comparing the graphs of ground state and sign-changing solutions in order to achieve our conclusion by a contradiction argument. We point out that variational arguments as those in [8] work fine for problem (P) in the Minkowski space This fact allows us to prove the existence of a radial ground state solution when p + 1 is larger than the α-critical exponent related to 2, that is, 2∗α. The problem of finding ground state solutions for the Hénon prescribed mean curvature equation in the Euclidean space seems to be more complex and, at the present, it is completely open
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