We established the existence of weak solutions of the fourth-order elliptic equation of the form <Equation ID="Equa1"> <EquationSource Format="TEX">$$\begin{aligned} \Delta ^2 u -\Delta u + a(x)u=\lambda b(x) f(u) + \mu g (x, u), \qquad x \in \mathbb{R }^N, u \in H^2(\mathbb{R }^N), \end{aligned}$$</EquationSource> </Equation>where <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$$\lambda $$</EquationSource> </InlineEquation> is a positive parameter, <InlineEquation ID="IEq4">...</inlineequation></equationsource></inlineequation></equationsource></equation>