On the Nodal Set of Solutions to Some Sublinear Equations Without Homogeneity

We investigate the structure of the nodal set of solutions to an unstable Alt-Philips type problem \(\begin{aligned} -\Delta u = \lambda _+(u^+)^{p-1}-\lambda _-(u^-)^{q-1}, \end{aligned}\) $$\begin{array}{c}\hfill -\mathrm{\Delta }u={\lambda }_{+}{(u^{+})}^{p-1}-{\lambda }_{-}{(u^{-})}^{q-1},\end{array}$$ where \(1 \le p<q<2\) $1\le p<q<2$ , \(\lambda _+ >0\) ${\lambda }_{+}>0$ , \(\lambda _- \ge 0\) ${\lambda }_{-}\ge 0$ . The equation is characterized by the sublinear inhomogeneous character of the right hand-side, which makes it 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.