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Existence and multiplicity of positive solutions to a critical elliptic equation with logarithmic perturbation

Qihan He, Yiqing Pan

TL;DR

This work addresses the existence and multiplicity of positive solutions for a critical elliptic equation with a logarithmic perturbation on a bounded smooth domain. Using a variational framework, the authors establish Mountain Pass geometry and construct a bubbling test function to compare the Mountain Pass level with a threshold tied to the Sobolev constant $S$, demonstrating the existence of at least two positive solutions: a least energy (ground/state) solution and a Mountain Pass solution. The analysis extends classical Brézis–Nirenberg results to include the sign-changing logarithmic term and handles a range of parameter regimes, including both positive and negative perturbations, with detailed PS and concentration-compactness arguments. The results advance understanding of multiplicity for critical growth problems with logarithmic perturbations and have potential implications in geometry and physics where such terms arise.

Abstract

We consider the existence and multiplicity of positive solutions for the following critical problem with logarithmic term: \begin{equation*}\label{eq11}\left\{ \begin{array}{ll} -Δu={μ\left|u\right|}^{{2}^{\ast }-2}u+ν|u|^{q-2}u+λu+θu\log {u}^{2}, &x\in Ω,\\ u=0, &x\in \partial Ω,\\ \end{array} \right.\end{equation*} where $Ω$ $\subset$ $\mathbb{R}^N$ is a bounded smooth domain, $ ν, λ\in \mathbb{R}$, $μ>0, θ<0$, $N\ge3$, ${2}^{\ast }=\frac{2N}{N-2}$ is the critical Sobolev exponent for the embedding $H^1_{0}(Ω)\hookrightarrow L^{2^\ast}(Ω)$ and $q\in (2, 2^*)$, and which can be seen as a Br$\acute{e}$zis-Nirenberg problem. Under some assumptions on the $μ, ν, λ, θ$ and $q$, we will prove that the above problem has at least two positive solutions: One is the least energy solution, and the other one is the Mountain pass solution. As far as we know, the existing results on the existence of positive solutions to a Br$\acute{e}$zis-Nirenberg problem are to find a positive solution, and no one has given the existence of at least two positive solutions on it. So our results is totally new on this aspect.

Existence and multiplicity of positive solutions to a critical elliptic equation with logarithmic perturbation

TL;DR

This work addresses the existence and multiplicity of positive solutions for a critical elliptic equation with a logarithmic perturbation on a bounded smooth domain. Using a variational framework, the authors establish Mountain Pass geometry and construct a bubbling test function to compare the Mountain Pass level with a threshold tied to the Sobolev constant , demonstrating the existence of at least two positive solutions: a least energy (ground/state) solution and a Mountain Pass solution. The analysis extends classical Brézis–Nirenberg results to include the sign-changing logarithmic term and handles a range of parameter regimes, including both positive and negative perturbations, with detailed PS and concentration-compactness arguments. The results advance understanding of multiplicity for critical growth problems with logarithmic perturbations and have potential implications in geometry and physics where such terms arise.

Abstract

We consider the existence and multiplicity of positive solutions for the following critical problem with logarithmic term: \begin{equation*}\label{eq11}\left\{ \begin{array}{ll} -Δu={μ\left|u\right|}^{{2}^{\ast }-2}u+ν|u|^{q-2}u+λu+θu\log {u}^{2}, &x\in Ω,\\ u=0, &x\in \partial Ω,\\ \end{array} \right.\end{equation*} where is a bounded smooth domain, , , , is the critical Sobolev exponent for the embedding and , and which can be seen as a Brzis-Nirenberg problem. Under some assumptions on the and , we will prove that the above problem has at least two positive solutions: One is the least energy solution, and the other one is the Mountain pass solution. As far as we know, the existing results on the existence of positive solutions to a Brzis-Nirenberg problem are to find a positive solution, and no one has given the existence of at least two positive solutions on it. So our results is totally new on this aspect.
Paper Structure (4 sections, 16 theorems, 134 equations)

This paper contains 4 sections, 16 theorems, 134 equations.

Table of Contents

  1. Introduction
  2. Preliminary results and the proofs of Theorems \ref{['buth1']} and \ref{['buth3']}
  3. Some important inequalities and energy estimates
  4. The proof of Theorem \ref{['th4']}

Key Result

Theorem 1.1

Assume that $N\geq 3$ and $q\in (2, 2^*)$. If $\nu\leq 0$ and $(\lambda,\mu,\theta)\in M_1\cup M_2$ or $\nu>0$ and $(\lambda,\mu,\theta)\in M_3\cup M_4$, then problem 1.1 has a positive local minimum solution $u_0\in A$ and a positive ground state solution $u_1\in \eta$ such that $I_\nu(u_0)=c_{\rh

Theorems & Definitions (38)

  • Theorem 1.1
  • Theorem 1.2
  • Remark 1.3
  • Theorem 1.4
  • Remark 1.5
  • Remark 1.6
  • Remark 1.7
  • Remark 1.8
  • Lemma 2.1
  • proof
  • ...and 28 more