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 -Delta u=lambda(+)(u+)(p-1)-lambda-(u-)(q-1), where 1 <= p < q < 2,lambda( +) > 0,lambda - >= 0. The equation is characterized by the sub-linear 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 mono-tonicity 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.