Oscillating solutions for prescribed mean curvature equations: Euclidean and Lorentz-Minkowski cases

This paper deals with the prescribed mean curvature equations − div 1± ∇u |∇u| 2 = g(u) in RN , both in the Euclidean case, with the sign “+”, and in the Lorentz-Minkowski case, with the sign “−”, for N 1 under the assumption g(0) > 0. We show the existence of oscillating solutions, namely with an unbounded sequence of zeros. Moreover these solutions are periodic, if N = 1, while they are radial symmetric and decay to zero at infinity with their derivatives, if N 2.