Abstract: We investigate the bi-harmonic problem $$\left\{\begin{array}{ll} \Delta^{2}u - \alpha\nabla \cdot (f(\nabla u)) - \beta\Delta_{p}u = g(x,u) &\hbox{in}\ \ \Omega,\\[2mm] \frac{\partial u}{\partial n}=0, \frac{\partial(\Delta u)}{\partial n}=0 &\hbox{on}\ \ \partial\Omega,\end{array} \right. $$ where $\Delta^{2}u = \Delta(\Delta u), \Delta_{p}u =\div\left(|\nabla u|^{p-2}\nabla u\right)$ with $p > 2.$ $\Omega$ is a bounded smooth domain in $\mathbb{R}^{N},$ $N \geq 1.$ By using a special function space with the constraint $\int_{\Omega}u {\rm d}x = 0$, under suitable assumptions on $f$ and $g(x,u)$, we show the existence and multiplicity of sign-changing solutions to the above problem via the Mountain pass theorem and the Fountain theorem. Recent results from the literature are extended.

Key words: Bi-harmonic, sign-changing solution, Fountain theorem