A note on nonlinear fourth-order elliptic equations on <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$$\mathbb R ^N$$</EquationSource> </InlineEquation>

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"> <EquationSource Format="TEX">$$a(x)$$</EquationSource> </InlineEquation> and <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$$b(x)$$</EquationSource> </InlineEquation> are positive functions, while <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$$f : \mathbb{R }\rightarrow \mathbb{R }$$</EquationSource> </InlineEquation> is sublinear at infinity and superlinear at the origin. In particular, by using Ricceri’s recent three critical points theorem, we show that the problem has at least three solutions. Copyright Springer Science+Business Media New York 2013