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

TL;DR

This paper studies the structure and properties of the nodal set of solutions to a class of sublinear, inhomogeneous equations, revealing their local behavior, classification of vanishing orders, and the existence of degenerate solutions.

Contribution

It provides a detailed analysis of the nodal set for sublinear equations without homogeneity, including classification and dimension estimates, which are challenging due to the inhomogeneous nature.

Findings

Classification of admissible vanishing orders

Estimates on Hausdorff dimension of singular set

Existence of degenerate solutions

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.