An infinite-dimensional linking theorem and applications

In this paper, we establish a new infinite-dimensional linking theorem without (PS)-type assumptions. The new theorem needs a weaker linking geometry and produces bounded (PS) sequences. The abstract result will be applied to the study of the existence of solutions of the strongly indefinite partial differential systems.For the first application, we consider the system (delta)u=u in (omega), (delta)u/(delta)(eta)=Hr(x,u,v) on (delta)(omega) (delta)v=v in (omega), (delta)v/(delta)(eta)=h(alpha)(x,u,v) on (delta)(omega) where (omega) is a bounded domain in RN with smooth boundary(delta)/(delta)(eta) is the outer normal derivative, H : (delta)(omega)×R×R (right arrow)is a positive C1-function. One nontrivial solution is obtained. The second application, we will solve the eigenvalue problem of the system Av=-f(x,v,w),Bw=(beta)g(x,v,w) , (beta)>O, where A,B are self-adjoint operators on L^2(omega),(omega)(subset)RN is not necessarily bounded; f,g are Caratheodory functions on (omega)×R2. We get infinitely many solutions. We deal with asymptotically linear cases for both systems.