AI 中文总结
该研究针对海森堡群上带临界指数的拟线性Schrödinger方程，在势函数满足不同条件时分别得到非平凡解的存在性与不存在性结果，还证明了弱解的有界性与指数衰减性。
AI 中文摘要
我们研究如下拟线性Schrödinger方程：\n\begin{align*}\n-Δ_{\mathbb{H}} u +V(ξ)u-Δ_{\mathbb{H}} (\left|u\right|^{2α})\left|u\right|^{2α-2} u= λ\left|u\right|^{q-2}u + \left|u\right|^{p-2}u \quad \text{ 在 } \mathbb{H}^N 中,\n\end{align*}\n其中$Δ_{\mathbb{H}}$是海森堡群$\mathbb{H}^N$上的Kohn拉普拉斯算子，$4α<q<p \leq 2αQ^*$，$α>\frac12$，$2αQ^{*}$为临界指数，齐次维数$Q=2N+2$，且$Q^{*}=\frac{2Q}{Q-2}$。当$p=2αQ^{*}$且$λ>0$时，在势函数有正的下界、且要么从上方渐近为常数要么在$\mathbb{H}^N$的一个离散子群下不变的假设下，我们得到了一个非平凡解。另一方面，我们证明了$Δ_{\mathbb{H}}$的Pohožaev恒等式，并将其与Nehari恒等式结合，得到一族恒等式，拟线性能量恰好在指数$2αQ^{*}$处消失。由此我们在势函数关于各向异性伸缩满足单调性条件的情况下得到了一个不存在性定理，表明当$λ\leq 0$时不存在非平凡解；次临界扰动的符号决定了可解性。在此过程中，我们证明了每个弱解都是有界的，且在Korányi规范下呈指数衰减。
英文摘要
We study the quasilinear Schrödinger equation \begin{align*} -Δ_{\mathbb{H}} u +V(ξ)u-Δ_{\mathbb{H}} (\left|u\right|^{2α})\left|u\right|^{2α-2} u= λ\left|u\right|^{q-2}u + \left|u\right|^{p-2}u \quad \text{ in } \mathbb{H}^N, \end{align*} where $Δ_{\mathbb{H}}$ is the Kohn Laplacian on the Heisenberg group $\mathbb{H}^N$, $4α<q<p \leq 2αQ^*$, $α>\frac12$, and $2αQ^{*}$ is the critical exponent, $Q=2N+2$ being the homogeneous dimension and $Q^{*}=\frac{2Q}{Q-2}$. For $p=2αQ^{*}$ and $λ>0$, we obtain a nontrivial solution, assuming that the potential is bounded below by a positive constant and is either asymptotically constant from above or invariant under a discrete subgroup of $\mathbb{H}^N$. In the opposite direction, we prove a Pohožaev identity for $Δ_{\mathbb{H}}$ and combine it with the Nehari identity, obtaining a family of identities from which the quasilinear energy disappears exactly at the exponent $2αQ^{*}$. This yields a nonexistence theorem under a monotonicity condition on the potential with respect to anisotropic dilations and shows that no nontrivial solution exists for $λ\leq 0$; the sign of the subcritical perturbation determines solvability. Along the way, we show that every weak solution is bounded and decays exponentially in the Korányi gauge.
