Solutions of elliptic problems with nonlinearities of linear growth

In this paper, we study existence of nontrivial solutions to the elliptic equation -Delta u = f(x, u) in Omega, u = 0 on partial derivative Omega and to the elliptic system -Delta u = del V-u(x, u) in Omega, u = 0 on partial derivative Omega, where Omega is a bounded domain in R-N with smooth boundary partial derivative Omega, f is an element of C-1((Omega) over bar X R, R), f(x, 0) = 0, V is an element of C-2 ((Omega) over bar X R-m, R) with m >= 2 and del V-u(x, 0) = 0. Nontrivial solutions are obtained in the case in which the nonlinearities have linear growth. That is, for some c > 0, vertical bar f(x, u)vertical bar <= c vertical bar u vertical bar for x is an element of Omega and u is an element of R, and -cI(m) <= del V-2(u)(x, u) <= cI(m) for x is an element of Omega and u is an element of R-m, where I-m is the m X m identity matrix. In sharp contrast to the existing results in the literature, we do not make any assumptions at infinity on the asymptotic behaviors of the nonlinearity f and del V-u.