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Multiparameter bifurcation and asymptotics for the singular Lane-Emden-Fowler equation with convection term

Marius Ghergu, Vicentiu Radulescu

TL;DR

The paper develops a multiparameter bifurcation framework for a singular elliptic equation with a convection term, focusing on a problem of Lane-Emden–Fowler type with 0<p≤2 and a gradient nonlinearity |∇u|^p. Using a sub–super-solution approach and an exponential change of variables to handle the gradient term, it derives precise existence and nonexistence thresholds in terms of a = lim_{s→∞} g(s), the first eigenvalue λ1, and the parameters μ and λ, with distinct regimes depending on p. It establishes finite or infinite bifurcation values for μ and λ, depending on p, and proves regularity and asymptotic blow-up of solution branches near critical thresholds. The results extend classical λ=0 theory to multiparameter settings with convection, offering a detailed map of solvability and qualitative behavior of solutions. They illuminate how convection influences singular elliptic problems and provide tools potentially applicable to nonlinear media and boundary-layer analyses.

Abstract

We establish several bifurcation results for the singular Lane-Emden-Fowler equation.

Multiparameter bifurcation and asymptotics for the singular Lane-Emden-Fowler equation with convection term

TL;DR

The paper develops a multiparameter bifurcation framework for a singular elliptic equation with a convection term, focusing on a problem of Lane-Emden–Fowler type with 0<p≤2 and a gradient nonlinearity |∇u|^p. Using a sub–super-solution approach and an exponential change of variables to handle the gradient term, it derives precise existence and nonexistence thresholds in terms of a = lim_{s→∞} g(s), the first eigenvalue λ1, and the parameters μ and λ, with distinct regimes depending on p. It establishes finite or infinite bifurcation values for μ and λ, depending on p, and proves regularity and asymptotic blow-up of solution branches near critical thresholds. The results extend classical λ=0 theory to multiparameter settings with convection, offering a detailed map of solvability and qualitative behavior of solutions. They illuminate how convection influences singular elliptic problems and provide tools potentially applicable to nonlinear media and boundary-layer analyses.

Abstract

We establish several bifurcation results for the singular Lane-Emden-Fowler equation.

Paper Structure

This paper contains 6 sections, 12 theorems, 121 equations, 1 figure.

Table of Contents

  1. Introduction and the main results
  2. Auxiliary results
  3. Proof of Theorem \ref{['th1']}
  4. Proof of Theorem \ref{['th2']}
  5. Proof of Theorem \ref{['th3']}
  6. Proof of Theorem \ref{['th4']}

Key Result

Theorem 1.1

Assume \lambda=1 and 1<p\leq 2. (i) If p=2 and a\geq \lambda_1, then (P) has no solutions; (ii) If p=2 and a<\lambda_1 or 1<p<2, then there exists \mu^*>0 such that (P) has at least one classical solution for \mu<\mu^* and no solutions exist if \;\mu>\mu^*.

Figures (1)

  • Figure 1: Bifurcation diagrams

Theorems & Definitions (12)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Lemma 2.1
  • Lemma 2.2
  • Lemma 2.3
  • Lemma 2.4
  • Lemma 2.5
  • Lemma 2.6
  • ...and 2 more