2018/03/18 by Nicola Soave, Susanna Terracini, Soave, Nicola +1 · 1 citation
Computer Science · Mathematics · #Advanced Mathematical Modeling in Engineering #Analysis of PDEs (math.AP) #FOS: Mathematics #Nonlinear Partial Differential Equations #Numerical methods in inverse problems
paper · pdf · doi:10.48550/arxiv.1803.06637
openalex publication_date 2018/03/18 · openalex created_date 2022/10/04 · openalex updated_date 2026/07/28
This paper deals with solutions to the equation \-
Delta u =\n
lambda+
left(u+
right)q-1 -
lambda-
left(u-
right)q-1
quad\n
textin B1 where \λ+,\λ- > 0, q \∈ (0,1),\nB1=B1(0) is the unit ball in \ℝN, N \≥ 2, and u+:=\n\max u,0 , u-:= \max -u,0 are the positive and the negative part of\nu, respectively. We extend to this class of \singular equations the\nresults recently obtained in citeSoTe2018 for \sublinear and\ndiscontinuous equations, 1\≤ q<2, namely: (a) the finiteness of the\nvanishing order at every point and the complete characterization of the order\nspectrum; (b) a weak non-degeneracy property; (c) regularity of the nodal set\nof any solution: the nodal set is a locally finite collection of regular\ncodimension one manifolds up to a residual singular set having Hausdorff\ndimension at most N-2 (locally finite when N=2). As an intermediate step,\nwe establish the regularity of a class of \not necessarily minimal\nsolutions.\n The proofs are based on a priori bounds, monotonicity formul ae for a\n2-parameter family of Weiss-type functionals, blow-up arguments, and the\nclassification of homogenous solutions.\n