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-Philips type problem

$$\begin{aligned} -\Delta u = \lambda _+(u^+)^{p-1}-\lambda _-(u^-)^{q-1}, \end{aligned}$$

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 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.