Liouville theorems for elliptic systems and applications

Let u be a solution of the system of PDE L ( u ) = f ( u ) in R N , where L is a quasilinear second order elliptic operator in divergence form and f a given function. Our aim is to find uniform bounds for all possible solutions u of the system. In this paper we prove some bounds which are universal, in the sense that they are related only to the zeros of the nonlinearity f. Among others, the results apply to Allen–Cahn equation, Ginzburg–Landau systems, Gross–Pitaevskii systems and Lichnerowiczʼs type equations.