AI 中文总结
研究海森堡群上特定拟线性薛定谔方程驻波解的存在性，通过非线性变量变换转化方程，运用变分方法及集中紧致性论证，在福兰德 - 斯坦索伯列夫空间中建立了非平凡解的存在性。
AI 中文摘要
我们研究海森堡群上具有临界增长的如下拟线性薛定谔方程驻波解的存在性：\(-\Delta_{\mathbb{H}} u +V(\xi)u-\Delta_{\mathbb{H}} (\left|u\right|^{2\alpha})\left|u\right|^{2\alpha-2} u= \lambda \left|u\right|^{q-2}u + \left|u\right|^{p-2}u\)，其中\(\mathbb{H}^N\)是海森堡群，\(\Delta_{\mathbb{H}}\)是科恩拉普拉斯算子等。通过合适的非线性变量变换，将方程转化为半线性方程，利用变分方法及集中紧致性论证，建立了非平凡解的存在性。
英文摘要
We study the existence of standing wave solutions for the following quasilinear Schrödinger equations with critical growth on the Heisenberg group $$ -Δ_{\mathbb{H}} u +V(ξ)u-Δ_{\mathbb{H}} (\left|u\right|^{2α})\left|u\right|^{2α-2} u= λ\left|u\right|^{q-2}u + \left|u\right|^{p-2}u \text{ in }\mathbb{H}^N $$ where $\mathbb{H}^N$ is Heisenberg group, $Δ_{\mathbb{H}}$ is Kohn Laplacian operator, $4α<q<p \leq 2αQ^{*}$, \(Q^{*}= \frac{2Q}{Q-2}\) is the critical Folland--Stein exponent, $λ$ and $α$ are positive parameters, $α> \frac{1}{2}.$ By a suitable nonlinear change of variables, the quasilinear equation is transformed into a semilinear one, allowing the use of variational methods in the Folland--Stein Sobolev space $S^{1,2}(\mathbb{H}^N)$. Applying the mountain pass theorem together with a concentration--compactness argument adapted to the sub-Riemannian framework, we establish the existence of a nontrivial solution.
CommentsThis paper has been withdrawn by the authors due to a crucial error in the proof of regularity results
