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We investigate the structure of the nodal set of solutions to \[ -\Delta u =\lambda_+(u^+)^{p-1}-\lambda_-(u^-)^{q-1} \] where $1 \le p<q<2$, $\lambda_+>0$, $\lambda_- \ge 0$. The equation is characterized by the sublinearinhomogeneous character of the right hand-side, which makes difficult to adaptin a standard way classical tools from free-boundary problems, such asmonotonicity formulas and blow-up arguments. Our main results are: the localbehavior of solutions close to the nodal set; the complete classification ofthe admissible vanishing orders, and estimates on the Hausdorff dimension ofthe singular set, for local minimizers; the existence of degenerate (notlocally minimal) solutions.

On the nodal set of solutions to some sublinear equations without homogeneity