Shooting method with sign-changing nonlinearity

In this paper, we study the existence of solution to a nonlinear system: equation(0.1){−Δui=fi(u)in  Rn,ui>0in  Rn,i=1,2,…,L for sign changing nonlinearities fifi’s. A degree theory approach to shooting method for this type of problems is introduced independently by Li (2013) and Liu et al. (2006) for nonnegative fifi’s. However, many systems of nonlinear Schödinger type involve interaction with undetermined sign. Here, based on some new dynamic estimates, we are able to extend the degree theory approach to systems with sign-changing source terms.