AI 中文总结
该研究针对具有Kato类势和$L^\u221e$有界非线性项的非线性薛定谔方程，运用Conley指标理论结合薛定谔半群性质，推导先验估计后得到非平凡驻波存在的条件，涵盖非共振与共振情形。
AI 中文摘要
我们建立了一类非线性薛定谔方程的驻波存在性，该方程具有属于Kato类的势和$L^\u221e$有界非线性项，其Lipschitz常数小于零到线性部分本质谱的距离。我们考虑了非共振和共振两种情形。我们的方法基于Conley指标理论，用于研究相关抛物半流的不变集。利用具有Kato类势的薛定谔半群的性质（这些性质源于其Feynman-Kac表示），我们推导了有界解在$L^\u221e$和$L^2$范数下的先验估计，以及在Sobolev空间中的正则性界。由此，我们得到了确保定态解之间连接轨道存在的条件，进而推导出非平凡驻波的存在性。
英文摘要
We establish the existence of standing waves for a nonlinear Schrodinger equation with potentials belonging to the Kato class and an $L^\infty$-bounded nonlinearity, whose Lipschitz constant is smaller than the distance from zero to the essential spectrum of the linear part. We consider both the nonresonant and resonant cases. Our approach is based on the Conley index theory applied to study invariant sets of the associated parabolic semiflow. Using properties of the Schrodinger semigroup with Kato-class potentials, which follow from its Feynman-Kac representation, we derive a priori estimates for bounded solutions in the $L^\infty$ and $L^2$ norms, as well as regularity bounds in Sobolev spaces. As a consequence, we obtain conditions ensuring the existence of connecting orbits between stationary solutions, which in turn yield the existence of nontrivial standing waves.
