Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Blow-up estimates and a priori bounds for the positive solutions of a class of superlinear indefinite elliptic problems

Julián López-Gómez, Juan Carlos Sampedro

Abstract

In this paper we find out some new blow-up estimates for the positive explosive solutions of a paradigmatic class of elliptic boundary value problems of superlinear indefinite type. These estimates are obtained by combining the scaling technique of Guidas-Spruck together with a generalized De Giorgi-Moser weak Harnack inequality found, very recently, by Sirakov. In a further step, based on a comparison result of Amann and López-Gómez, we will show how these bounds provide us with some sharp a priori estimates for the classical positive solutions of a wide variety of superlinear indefinite problems. It turns out that this is the first general result where the decay rates of the potential in front of the nonlinearity $a(x)$ do not play any role for getting a priori bounds for the positive solutions when $N\geq 3$.

Blow-up estimates and a priori bounds for the positive solutions of a class of superlinear indefinite elliptic problems

Abstract

In this paper we find out some new blow-up estimates for the positive explosive solutions of a paradigmatic class of elliptic boundary value problems of superlinear indefinite type. These estimates are obtained by combining the scaling technique of Guidas-Spruck together with a generalized De Giorgi-Moser weak Harnack inequality found, very recently, by Sirakov. In a further step, based on a comparison result of Amann and López-Gómez, we will show how these bounds provide us with some sharp a priori estimates for the classical positive solutions of a wide variety of superlinear indefinite problems. It turns out that this is the first general result where the decay rates of the potential in front of the nonlinearity do not play any role for getting a priori bounds for the positive solutions when .
Paper Structure (4 sections, 10 theorems, 103 equations)

This paper contains 4 sections, 10 theorems, 103 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Proof of Theorem \ref{['BUS']}
  4. Proof of Theorem \ref{['APB']}

Key Result

Theorem 1.1

Suppose $\Omega_{-}=\emptyset$ and $\Gamma\subset \partial\Omega$ is a relatively open subset of $\partial\Omega$ of class $\mathcal{C}^{1,1}$ regularity. Let $\mathcal{D}$ be a subdomain of $\mathbb{R}^N$ such that $\bar{\mathcal{D}}\subset \Omega\cup\Gamma$ and either Then, for every subset of positive solutions, $\mathscr{S}\subset \mathbb{R}_{+}\times W^{2,p}(\Omega)$, of with $\lambda$-proj

Theorems & Definitions (12)

  • Theorem 1.1: Sirakov, 2020
  • Theorem 1.2: Sirakov, 2022
  • Theorem 1.3
  • Theorem 1.4: Amann and López-Gómez, 1998
  • Theorem 1.5
  • Lemma 2.1
  • proof
  • Proposition 2.2
  • Theorem 2.3
  • Theorem 2.4
  • ...and 2 more