The nodal set of solutions to some elliptic problems: Singular nonlinearities

Abstract

This paper deals with solutions to the equation−Δu=λ+(u+)q−1−λ−(u−)q−1in B1 where λ+,λ−>0, q∈(0,1), B1=B1(0) is the unit ball in RN, N≥2, and u+:=max⁡{u,0}, u−:=max⁡{−u,0} are the positive and the negative part of u, respectively. We extend to this class of singular equations the results recently obtained in [25] for sublinear and discontinuous equations, 1≤q<2, namely: (a) the finiteness of the vanishing order at every point and the complete characterization of the order spectrum; (b) a weak non-degeneracy property; (c) regularity of the nodal set of any solution: the nodal set is a locally finite collection of regular codimension one manifolds up to a residual singular set having Hausdorff dimension at most N−2 (locally finite when N=2). As an intermediate step, we establish the regularity of a class of not necessarily minimal solutions.The proofs are based on a priori bounds, monotonicity formulæ for a 2-parameter family of Weiss-type functionals, blow-up arguments, and the classification of homogeneous solutions.