Suppose that \(F: {\mathbb{R}}\) \(n\to {\mathbb{R}}\) is a strictly convex function. Then it is well-known [\textit{T. Radò}, On the problem of Plateau (1933; Zbl 0007.11804)] that the Dirichlet-problem associated to the variational integral \(\int F(Du)dx\) and for smooth boundary data has a unique solution in the class of Lipschitz-functions. On the other hand if F is not convex but of growth \(m>2\), i.e. \[ (1)\quad c_ 0(| p| \quad m-1)\leq F(p)\leq c_ 1(| p| \quad m+1), \] then the techniques of \textit{M. Giaquinta} and \textit{E. Giusti} [Acta Math. 148, 31- 46 (1982; Zbl 0494.49031)] show that local minimizers in the space \(H^{1,m}\) are locally bounded and Hölder continuous. In the present note the author constructs examples of unbounded local minimizers with F strictly convex but satisfying a growth condition of the form \[ c_ 0(| p|^ 2-1)\leq F(p)\leq c_ 1(| p| \quad m+1) \] for some exponent \(m>2\). Thus (1) is a necessary condition for a local regularity.