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

Nicola Soave, Giorgio Tortone

Abstract

We investigate the structure of the nodal set of solutions to an unstable Alt-Phillips type problem \[ -Δu = λ_+(u^+)^{p-1}-λ_-(u^-)^{q-1} \] where $1 \le p<q<2$, $λ_+ >0$, $λ_- \ge 0$. 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.

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-Phillips type problem where , , . 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.
Paper Structure (9 sections, 19 theorems, 140 equations)

This paper contains 9 sections, 19 theorems, 140 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Existence and non-degeneracy
  4. Blow-up analysis
  5. Proof of Theorem \ref{['thm.mainblow']}-(i)
  6. Proof of Theorem \ref{['thm.mainblow']}-(ii)
  7. Classification of vanishing orders for local minimizer
  8. Homogeneous solutions in two dimensions and the Hausdorff dimension of the singular set
  9. Construction of degenerate solutions in two dimensions

Key Result

Theorem 1.1

Let $\Omega$ be a bounded regular domain of $\mathbb{R}^n$, let $\lambda_{+}>0$, $\lambda_- >0$ (resp. $\lambda_-=0$), and suppose that $1 \le p<q <2$. Then there exists a solution to eq0 (resp. equation1) obtained as minimizer of $J(\cdot\,,\Omega)$ in $\mathcal{A}$.

Theorems & Definitions (46)

  • Theorem 1.1
  • Definition 1.2
  • Definition 1.3
  • Theorem 1.4
  • Definition 1.5
  • Theorem 1.6
  • Remark 1.7
  • Theorem 1.8
  • Theorem 1.9
  • Theorem 1.10
  • ...and 36 more