海森堡群上带参数和指数非线性的修正$Q$-拉普拉斯问题

中文摘要 AI 辅助

本文研究由$Q$-拉普拉斯驱动的如下修正拟线性方程：\[ \begin{cases} -\Delta_Q u - \Delta_Q\bigl(|u|^{2\alpha}\bigr)\,|u|^{2\alpha - 2} u = \lambda f(\xi,u) & \text{在 }\Omega 中, \\[2mm] u = 0 & \text{在 }\partial\Omega 上, \end{cases} \]其中$\Delta_Q(\cdot)$是海森堡群$\mathbb{H}^N$上的$Q$-拉普拉斯，$\Omega \subset \mathbb{H}^N$是有界光滑区域，$f$在莫泽 - 特鲁迪格意义下呈指数增长且$\alpha > \frac{1}{2}$。目的有二：一是建立非平凡正弱解的存在性，二是在$f$的次临界和临界指数增长假设下获得最小能量节点（变号）解。分析依赖于合适的变量变换将原拟线性结构化为半线性变分框架，以及临界点理论。所得结果即使在经典欧几里得情形下也是全新的，将拟线性薛定谔型方程的现有理论扩展到了海森堡群的次黎曼背景。

英文摘要

In this article, we investigate the following modified quasilinear equation with parameter driven by the $Q$-subLaplacian: \begin{align*} \begin{cases} -Δ_Q u - Δ_Q\bigl(|u|^{2α}\bigr)\,|u|^{2α-2} u = λf(ξ,u) & \text{in } Ω, \\[2mm] u = 0 & \text{on } \partialΩ, \end{cases} \end{align*} where $Δ_Q(\cdot):= \mathrm{div}_{\mathbb{H}}\bigl(|\nabla_{\mathbb{H}}(\cdot)|^{Q-2}\nabla_{\mathbb{H}}(\cdot)\bigr)$ denotes the $Q$-subLaplacian on the Heisenberg group $\mathbb{H}^N$, $Q=2N+2$ is the homogeneous dimension, $Ω\subset \mathbb{H}^N$ is a smooth bounded domain, $λ>0$, $α> \frac{1}{2}$, and $f$ has critical or subcritical exponential growth of order $\exp\bigl(β|t|^{2αQ/(Q-1)}\bigr)$. We prove three results: the existence of a nontrivial positive weak solution in the critical case for all large $λ$, and the existence of a least-energy nodal solution with exactly two nodal domains, under subcritical and critical exponential growth. A change of variables $u=g(v)$ reduces the problem to a quasilinear problem whose energy functional is of class $C^1$; the exponent $2αQ/(Q-1)$ arises from the growth of $g$. We handled the exponential growth using the sharp Moser-Trudinger inequality of Cohn and Lu. The positive solution is obtained by the mountain pass theorem and the nodal solutions by minimization on a nodal Nehari set.

发表机构

• Indian Institute of Technology (BHU), India（印度理工大学（贝拿勒斯 Hindu 大学））
• Netaji Subhash University of Technology, India（印度纳塔吉·苏巴什科技大学）

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