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Lack of interior $L^q$ bounds for stable solutions to elliptic equations

Salvador Villegas

Abstract

We consider stable solutions of semilinear elliptic equations of the form $-Δu=f(u)$ in a bounded domain $Ω\subset\mathbb{R}^N$. In a well-known paper \cite{cfrs}, Cabré, Figalli, Ros-Oton and Serra obtained interior estimates for the $W^{1,2}$-norm of $u$ in terms of the $L^1$-norm of $u$ and proved interior Hölder regularity for dimensions $N\leq 9$. All these results rely on the assumption that $f$ is nonnegative. We show that, for general nonlinearities $f\in C^\infty(\mathbb{R})$, it is impossible, in any dimension $N\geq 1$, to obtain an interior $L^q$ estimate in terms of the $L^p$-norm of $u$ whenever $1\leq p<q\leq \infty$.

Lack of interior $L^q$ bounds for stable solutions to elliptic equations

Abstract

We consider stable solutions of semilinear elliptic equations of the form in a bounded domain . In a well-known paper \cite{cfrs}, Cabré, Figalli, Ros-Oton and Serra obtained interior estimates for the -norm of in terms of the -norm of and proved interior Hölder regularity for dimensions . All these results rely on the assumption that is nonnegative. We show that, for general nonlinearities , it is impossible, in any dimension , to obtain an interior estimate in terms of the -norm of whenever .
Paper Structure (2 sections, 5 theorems, 33 equations)

This paper contains 2 sections, 5 theorems, 33 equations.

Table of Contents

  1. Introduction and results
  2. Proof of the lack of interior $L^q$ bounds

Key Result

Theorem 1.1

Let $u\in C^\infty(\overline{B_1})$ be a stable solution of $-\Delta u=f(u)$ in $B_1\subset \mathbb{R}^N$, for some nonnegative function $f\in C^1(\mathbb{R})$ . Then, for some dimensional constants $\gamma>0$ and $C$. In addition, where $\alpha>0$ and $C$ are dimensional constants.

Theorems & Definitions (6)

  • Theorem 1.1: cfrs, Theorem 1.2.
  • Theorem 1.2
  • Theorem 1.3
  • Proposition 1.4
  • Lemma 2.1
  • proof