In this paper we establish a variant and generalized weak linking theorem, which contains more delicate result and insures the existence of bounded Palais–Smale sequences of a strongly indefinite functional. The abstract result will be used to study the semilinear Schrödinger equation $-\Delta u+V\left(x\right)u={K\left(x\right)\left|u\right|}^{2^{*}-2}u+g(x,u),u\in W^{1,2}\left({𝐑}^N\right)$, where N ≥ 4; V,K,g are periodic in xj for 1 ≤ j ≤ N and 0 is in a gap of the spectrum of -Δ + V; K>0. If $0<g(x,u)u\le {c\left|u\right|}^{2^{*}}$ for an appropriate constant c, we show that this equation has a nontrivial solution.