Multiple non-trivial solutions of the Dirichlet problem for the prescribed mean curvature equation

We prove the existence of three non-trivial solutions of the prescribed mean curvature problem \begin{equation*} -{\rm div } \Big({\nabla u}/{ \sqrt{1+{|\nabla u|}^2}}\Big) = \lambda f(x,u) \mbox{\, in $\Omega$}, \qquad u=0 \mbox{\, on $\partial \Omega$}, \end{equation*} in a bounded domain $\Omega\subset\RR^N$, assuming that the potential $F(x,s)=\int_0^s f(x,t)\,dt$ is subquadratic at $s=0$ and $\lambda >0$ is small. This yields an extension to a genuine PDE setting of some recent results of K.C. Chang and T. Zhang. Two further solutions are obtained when $F(x,s)$ is superlinear at infinity. The case where $F(x,s)$ is even is discussed too.