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Multiplicity and asymptotics of positive solutions for critical-concave Kirchhoff equation

Zhi-Yun Tang, Gui-Dong Li, Yong-Yong Li

Abstract

This paper focuses on the critical Kirchhoff equation with concave perturbation \begin{align*} \begin{cases} \displaystyle -\Big(a+b\int_Ω|\nabla u|^2dx\Big)Δu=|u|^4u+λ|u|^{q-2}u\ \ &\mbox{in}\ Ω, \displaystyle u=0\ \ &\mbox{on}\ \partialΩ, \end{cases} \end{align*} where $Ω$ is a smooth bounded domain in $\mathbb{R}^3$, $a,b,λ>0$ and $1<q<2$. By the constrained minimization methods, the mountain pass theorem and the concentration-compactness principle, we verify the multiplicity of positive solutions for $λ>0$ small enough. Moreover, we analyse the asymptotic behaviour of positive solutions as $b\rightarrow0$ and $λ\rightarrow0$, respectively. This work is a counterpart of [A. Ambrosetti et al., J.~Funct.~Anal. 1994] for the Kirchhoff equation. It is noteworthy that we don't require that $b>0$ is small enough here, which is imposed in the existing literatures to make refined estimates for the mountain pass level.

Multiplicity and asymptotics of positive solutions for critical-concave Kirchhoff equation

Abstract

This paper focuses on the critical Kirchhoff equation with concave perturbation \begin{align*} \begin{cases} \displaystyle -\Big(a+b\int_Ω|\nabla u|^2dx\Big)Δu=|u|^4u+λ|u|^{q-2}u\ \ &\mbox{in}\ Ω, \displaystyle u=0\ \ &\mbox{on}\ \partialΩ, \end{cases} \end{align*} where is a smooth bounded domain in , and . By the constrained minimization methods, the mountain pass theorem and the concentration-compactness principle, we verify the multiplicity of positive solutions for small enough. Moreover, we analyse the asymptotic behaviour of positive solutions as and , respectively. This work is a counterpart of [A. Ambrosetti et al., J.~Funct.~Anal. 1994] for the Kirchhoff equation. It is noteworthy that we don't require that is small enough here, which is imposed in the existing literatures to make refined estimates for the mountain pass level.
Paper Structure (5 sections, 10 theorems, 97 equations)

This paper contains 5 sections, 10 theorems, 97 equations.

Table of Contents

  1. Introduction and main results
  2. Proof of Theorem \ref{['T1']}
  3. Existence of positive solution with negative energy
  4. Existence of positive solution with positive energy
  5. Proof of Theorems \ref{['T2']} and \ref{['T3']}

Key Result

Theorem 1.1

Assume that $\Omega\subset \mathbb{R}^3$ is a bounded domain with smooth boundary, $a>0$ and $1<q<2$. Then there exists some $\Lambda>0$ (independent of $b$) such that Eq. K1 has at least two positive solutions $u_{b,\lambda}^1$ and $u_{b,\lambda}^2$ for any $b>0$ and $\lambda\in(0,\Lambda)$, where

Theorems & Definitions (10)

  • Theorem 1.1
  • Theorem 1.3
  • Theorem 1.5
  • Lemma 2.1
  • Theorem 2.2
  • Lemma 2.3
  • Lemma 2.4
  • Lemma 2.5
  • Lemma 2.6
  • Theorem 2.7