关于Heisenberg群中零质量(p,Q)-Laplace次椭圆方程的尖锐奇异Moser-Trudinger型不等式及其应用
On sharp singular Moser-Trudinger type inequalities and applications to zero mass $(p,Q)$-Laplace subelliptic equations in the Heisenberg group
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中文总结 AI 辅助
本文在Heisenberg群中建立尖锐奇异Moser-Trudinger型不等式，分析其等价性，运用山路定理证明零质量(p,Q)-Laplace次椭圆方程正基态解的存在性。
中文摘要 AI 辅助
本文在新的函数空间中建立了一个尖锐奇异Moser-Trudinger型不等式，并受P.-L. Lions启发，在Heisenberg群中探讨了集中紧性原理。我们分析了尖锐临界与次临界奇异Moser-Trudinger型不等式的等价性，重点关注其渐近行为及上确界间的联系。此外，我们运用山路定理，证明了Heisenberg群$\boldsymbol{\text{H}}^n$中零质量$(p,Q)$-Laplace次椭圆方程正基态解的存在性，该方程为：$$ -\boldsymbol{\text{Δ}}_{\boldsymbol{\text{H}},p} u-\boldsymbol{\text{Δ}}_{\boldsymbol{\text{H}},Q} u=\frac{f(\boldsymbol{\text{ξ}},u)}{r(\boldsymbol{\text{ξ}})^\boldsymbol{\text{ϑ}}}\boldsymbol{\text{\textquad in}}\boldsymbol{\text{\textquad H}}^n, $$其中$1<p<Q$，$\boldsymbol{\text{ϑ}}\boldsymbol{\text{\text{∈}}}(0,Q)$，$Q=2n+2$，函数$r(\boldsymbol{\text{·}})$称为$\boldsymbol{\text{H}}^n$中的Korányi范数，非线性项$f:\boldsymbol{\text{H}}^n\times \boldsymbol{\text{R}}\to \boldsymbol{\text{R}}$是Carathéodory函数，当$|s|\to+\boldsymbol{\text{∞}}$时，其表现为$\boldsymbol{\text{exp}}(\boldsymbol{\text{α}}|s|^{\frac{Q}{Q-1}})$，其中$\boldsymbol{\text{α}}>0$。
英文摘要
This article establishes a sharp singular Moser-Trudinger type inequality in a new function space and explores a concentration-compactness principle in the Heisenberg group, inspired by P.\textcolor{blue}{-}L. Lions. We analyze the equivalence of sharp critical and subcritical singular Moser-Trudinger inequalities, focusing on their asymptotic behavior and connections between their suprema. Additionally, we apply the mountain pass theorem to demonstrate the existence of positive ground state solutions for zero mass $(p, Q)$-Laplace subelliptic equations with singular exponential nonlinearity for the following equation: $$ -Δ_{H,p} u-Δ_{H,Q} u=\frac{f(ξ,u)}{r(ξ)^\vartheta}\quad\text{in}\quad \mathbb{H}^n, $$ with $1<p<Q$, $\vartheta\in(0,Q)$, $Q=2n+2$, the function $r(\cdot)$ is called the Korányi norm in $\mathbb{H}^n$ and the nonlinearity $f:\mathbb{H}^n\times \mathbb{R}\to \mathbb{R}$ is a Carathéodory function, which behaves like $\exp{(α|s|^{\frac{Q}{Q-1}})}$ as $|s|\to~+\infty$ for some $α>0$.
发表机构
- Universidade de Brasília（巴西利亚大学）
- Indian Institute of Technology Jodhpur（印度理工学院乔德普尔分校）
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