Positive solutions of critical Hardy-Hénon equations with logarithmic term

TL;DR

This work studies positive solutions to the critical Hardy-Hénon equation with a logarithmic perturbation, -Δu=|x|^α|u|^{2^*_ ext{α}-2}u+μu log u^2+λu, on a unit ball domain with a radial function space framework. The authors develop a variational approach, employing Mountain Pass geometry, Nehari-type constraints, and bubble-approximation techniques via the extremals u_{ε,α}, to derive sharp energy estimates and concentration-compactness arguments. They establish the existence of a positive Mountain Pass solution (and a ground state) for N≥4 and μ>0, as well as the existence of at least a positive least energy solution and a Mountain Pass solution when μ<0 under suitable parameter regimes; they also prove nonexistence of positive solutions under a negative-μ inequality when α∈(-2,0]. The results reveal new phenomena induced by the logarithmic term, affecting compactness, radial symmetry, and the multiplicity of positive solutions in the critical Hardy-Hénon setting, with implications for related geometric and physical variational problems.

Abstract

We consider the existence, non-existence and multiplicity of positive solutions to the following critical Hardy-Hénon equation with logarithmic term \begin{equation*}\label{eq11}\left\{ \begin{array}{ll} -Δu =|x|^α|u|^{2^*_α-2}\cdot u+μu\log u^2+λu, &x\in Ω,\\ u=0, &x\in \partial Ω,\\ \end{array} \right.\end{equation*} where $ Ω=B$ for $α\geq 0$, $ Ω=B\setminus\{0\}$ for $α\in(-2,0)$, $B\subset\mathbb{R}^N$ is an unit ball, $λ, μ\in \mathbb{R}$, $N\geq 3, α>-2$, $2^*_α:=\frac{2(N+α)}{N-2}$ is the critical exponent for the embedding $H_{0,r}^{1}( Ω)\hookrightarrow L^p( Ω;|x|^α)$, and which can be seen as a Brézis-Nirenberg problem. When $N \geq 4$ and $μ>0$, we will show that the above problem has a positive Mountain pass solution, which is also a ground state solution. At the same time, when $μ<0$, under some assumptions on the $N$, $μ$, $λ$ and $α$, we will show that the above problem has at least a positive least energy solution and at least a positive Mountain pass solution, respectively. What's more, when certain inequality related to $N \geq 3$, $μ<0 $ and $α\in(-2,0]$ holds, we will demonstrate the non-existence of positive solutions to the above-mentioned problem. The presence of logarithmic term brings some new and interesting phenomena to this problem.

Theorems & Definitions (86)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Remark 1.4
• Remark 1.5
• Theorem 1.6
• Lemma 2.1
• Lemma 2.2
• proof
• Lemma 2.3