Nonexistence and multiplicity of solutions for nonlinear elliptic systems in RN

In the whole space RN, we will study the following nonlinear elliptic system in two cases: −△u+V1(x)u=f(x,u,v),x∈RN,−△v+V2(x)v=g(x,u,v),x∈RN,u(x)→0,v(x)→0,|x|→∞.Case 1 : The periodic case (i.e., V1, V2, f and g are periodic in xi for i=1,…,N), we obtain the nonexistence of nontrivial solutions for the system. The existence result has been obtained in Chen and Ma (2013). Case 2 : The non-periodic case (i.e., V1, V2, f and g are non-periodic), we mainly focus on the case where the nonlinearities f and g are superlinear at infinity, and we obtain infinitely many nontrivial solutions of the system by variational methods. To the best of our knowledge, there is no work focusing on the system in Cases 1-2. The main novelties of this paper can be summarized as follows: (1) The system is defined in the whole space RN; (2) The potentials Vi(x) (i=1,2) can be sign-changing; (3) The nonexistence of nontrivial solutions in the periodic case is obtained; (4) Infinitely many nontrivial solutions in the non-periodic case are obtained.