The existence of positive solutions is proved for theprescribed mean curvature problem$$\displaystyle- \,{\rm div} \left( {\nabla u}/{\sqrt{1+\|\nabla u\|^2}}\right) =\lambda f(x,u) +g(x,u) \mbox{ in }\, \Omega,\hfill\quadu(x) = 0 \mbox{ on }\, \partial \Omega,$$where $\Omega\subset\mathbb R^N$ is a bounded smoothdomain, not necessarily radially symmetric.We assume that $\int_0^u f(x,s)\, ds$ is locallysubquadratic at$0$,$\int_0^u g(x,s)\, ds$ is superquadratic at $0$ and $\lambda>0$is sufficiently small. A multiplicity result is alsoobtained, when$\int_0^uf(x,s)\, ds$ has an oscillatory behaviour near $0$. We allow $f$ and $g$ to change sign in any neighbourhoodof$0$.
