Entire solutions of quasilinear symmetric systems

Abstract

We study the following quasilinear elliptic system for all i = 1,··· ,m −div(� ' (|∇ui| 2 )∇ui) = Hi(u) in R n where u = (ui) m=1 : R n → R m and the nonlinearity Hi(u) ∈ C 1 (R m ) → R is a general nonlinearity. Several celebrated operators such as the prescribed mean curvature, the Laplacian and the p-Laplacian operators fit in the above form, for appropriate �. We establish a Hamiltonian identity of the following form for all xn ∈ R Z Rn−1 m X i=1 � 1 2 � |∇u i| 2 � − � ' |∇ui| 2 � |@xnui| 2 � − ˜ H(u)