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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 $N\ge 4;V,K,g$ are periodic in $x_j$ for $1\le j\le N$ and 0 is in a gap of the spectrum of $-\Delta +V$; $K\>0$. 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.

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 ≥ 4; are periodic in x for 1 ≤ ≤ and 0 is in a gap of the spectrum of -Δ + ; . If $0<g(x,u)u\le {c\left|u\right|}^{2^{*}}$ for an appropriate constant , we show that this equation has a nontrivial solution.