Existence and multiplicity of solutions for a class of elliptic systems in RnRn

Using minimax methods we study the existence and multiplicity of nontrivial solutions for semilinear elliptic systems of the form −Δui+ai(x)ui=fi(x,u1,…,um)forx∈Rnandi=1,…,m. The potentials ai(x)ai(x) for i=1,…,mi=1,…,m may change sign, control the growth of nonlinearities and can be “large” at the infinity. We treat both superquadratic situation and nonquadratic situation at infinity on the nonlinearities fi(x,u1,…,um)fi(x,u1,…,um) for i=1,…,mi=1,…,m.