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In this paper, we give a new variational characterization of certain 4-manifolds. More precisely, let $R$ and $Ric$ denote the scalar curvature and Ricci curvature respectively of a Riemannian metric, we prove that if $(M^4,g)$ is compact and locally conformally flat and $g$ is the critical point of the functional $$F\left(g\right)={\int }_{M^4}(a{R}^2+{b\left|Ric\right|}^2)\mathrm{d}{v}_g,$$ where $$(a,b)\in {ℝ}^2\setminus L_1\cup L_2$$ $$L_1:3a+b=0;L_2:6a-b+1=0,$$ then $(M^4,g)$ is either scalar flat or a space form.