AI 中文总结
研究分层李群上关于霍尔曼德 q - 次拉普拉斯算子的具有指数增长的拟线性方程，通过精心设计伽辽金方法求解，恢复并扩展了\(\mathbb{R}^n\)及海森堡群等已知结果，还讨论了其他特殊情况。
AI 中文摘要
我们证明了在任意分层李群上对于水平 q - 拉普拉斯算子（也称为这些群上的 q - 次拉普拉斯算子）的具有指数非线性的拟线性方程正弱解的存在性。该非线性项结合了一个凹项与指数增长，相对于分层李群上洛伦兹空间的特鲁迪格 - 莫泽不等式可能是次临界、临界或超临界的。我们在本文中证明了这个不等式的一个版本。由于该问题在自然能量空间中不是变分的，经典的极小化或临界点技术不能直接应用。通过精心设计的伽辽金方法应用于此设置得到了所得半线性方程的正解。我们的主要定理在完整范围\(q\in(1,\infty)\)内恢复了\(\mathbb{R}^n\)上先前已知的结果，以及\(q = 2n + 2\)时海森堡群\(\mathbb{H}^n\)上先前已知的结果，并在此扩展到完整范围\(q\in(1,\infty)\)。还讨论了包括恩格尔群在内的其他特殊和基本情况。
英文摘要
We prove the existence of positive weak solutions for a quasilinear equation with exponential nonlinearity on arbitrary stratified Lie groups for \emph{horizontal $q$-Laplacians,} also called $q$-sub-Laplacians on these groups. The nonlinearity combines a concave term with an exponential growth that can be subcritical, critical, or supercritical with respect to the Trudinger--Moser inequality for Lorentz spaces on stratified Lie groups. We prove a version of this inequality in this paper. Since the problem is not variational in the natural energy space, classical minimisation or critical-point techniques do not apply directly. Positive solutions to the resulting semilinear equation are then obtained via a carefully designed Galerkin method applied to this setting. Our main theorem recovers previous known results on $\mathbb{R}^n$ in the complete range $q\in (1,\infty),$ and the previous known result on the Heisenberg group $\mathbb{H}^n$ for $q=2n+2,$ and is extended here in the full range $q\in (1,\infty)$. Other particular and fundamental cases, including the Engel group, are discussed.
Comments27 pages
