The Existence Of Nontrivial Solutions For Two Class Of Strongly Indefinite Elliptic Systems

The strongly indefinite elliptic system is related to the reaction-difusionsystems appeared in chemical and biological phenomena. In this paper wewill consider the existence of nontrivial solutions for two class of stronglyindefinite elliptic systems by using variational methods.In the second chapter we consider a class of noncooperative ellipticsystemswhereΩ R^2Nis a bounded and smooth domain. Assume that the sys-tem is asymptotically linear and resonant at infinity. By using the infinitedimensional Morse theory established by Kryszewski, Szulkin and penal-ized functional technique, we obtain the existence of nontrivial solutionsfor this elliptic system.In the third chapter we consider a class of Hamiltonian elliptic systemdefined on the whole space where N≥3, V （x）∈C（R^N, R）, the nonlinearity f, g∈C（R^N×R, R） areperiodic with respect to x and superlinear at infinity. By using the gen-eralized linking theorem established by Kryszewski, Szulkin and a mono-tonicity trick, we obtain the existence of nontrivial ground state for thesystem.