FITZHUGH-NAGUMO SYSTEM WITH ZERO MASS AND CRITICAL GROWTH

We show existence of a nontrivial nonnegative solution for the system -Delta u=K(x)f(u)+gamma vertical bar u vertical bar(2*) -2u - v, -Delta v = u - v in Double-struck capital R-N. Since the function f can verify f '(0)=0, this type of system is known in the literature as zero mass. We analyze three types of problems with K being periodic, asymptotically periodic and with a vanishing property at infinity. In the first place we consider N >= 3, and we prove existence results considering the function f with polynomial growth which can be subcritical, corresponding to gamma = 0, or critical, in case gamma = 1. Finally, we consider specifically N = 2 with gamma = 0 and f with possible critical exponential behavior.