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

Abstract: We investigate the structure of the nodal set of solutions to an unstable Alt-Phillips type problem [ -\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 sublinear inhomogeneous character of the right hand-side, which makes difficult to adapt in a standard way classical tools from free-boundary problems, such as monotonicity formulas and blow-up arguments. Our main results are: the local behavior of solutions close to the nodal set; the complete classification of the admissible vanishing orders, and estimates on the Hausdorff dimension of the singular set, for local minimizers; the existence of degenerate (not locally minimal) solutions.