A new infinite-dimensional linking theorem for parameter-dependent functionals and periodic Schrödinger equations

We present a new infinite-dimensional linking theorem for parameter-dependent functionals. This theorem replaces some semi-continuous assumptions in the classical infinite-dimensional linking theorem with new assumptions and insures the existence of bounded and variant Palais–Smale sequences for a strongly indefinite functional. As an application of this theorem, we obtain nontrivial solutions for strongly indefinite periodic Schrödinger equations with sign-changing and asymptotically linear nonlinearities.