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>

In this paper, we prove the existence of infinitely many solutions to differential problems where both the equation and the conditions are Sturm–Liouville type. The approach is based on critical point theory. Copyright Springer Science+Business Media, LLC. 2012