In this paper, we study the Cauchy problem for the energy-critical inhomogeneous nonlinear Schrödinger equation \(i\partial _{t}u+\Delta u=\lambda |x|^{-\alpha }|u|^{\beta }u\) in \(H^1\). The well-posedness theory in \(H^1\) has been intensively studied in recent years, but the currently known approaches do not work for the critical case \(\beta =(4-2\alpha )/(n-2)\). It is still an open problem. The main contribution of this paper is to develop the theory in this case.

中文翻译：

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能量临界非齐次非线性薛定谔方程的柯西问题

在本文中，我们研究能量临界非齐次非线性薛定谔方程的柯西问题\(i\partial _{t}u+\Delta u=\lambda |x|^{-\alpha }|u|^{\beta }u\)在\(H^1\) 中。\(H^1\) 中的适定理论近年来得到了深入研究，但目前已知的方法不适用于临界情况\(\beta =(4-2\alpha )/(n-2 )\)。这仍然是一个悬而未决的问题。本文的主要贡献是在这种情况下发展理论。