AI 中文总结
本文研究非规范不变非齐次非线性薛定谔方程，在强制性条件下证明低于Strauss型阈值时无散射态，并利用Lorentz空间中的非容许Strichartz估计证明阈值以上的小数据散射结果。
AI 中文摘要
我们考虑具有非规范不变非线性的非齐次非线性薛定谔方程解的渐近行为。非线性项中的空间系数在原点处可能具有不同阶的奇异性，并在无穷远处衰减。在涉及系数和非线性的强制性条件下，我们证明了在由无穷远处衰减确定的Strauss型阈值以下，方程不存在散射态。我们的类别包括具有主导非振荡分量的非线性。我们还证明了该阈值以上的互补小数据散射结果，使用适应奇异系数的Lorentz空间中的非容许Strichartz估计。
英文摘要
We consider the asymptotic behavior of solutions to inhomogeneous nonlinear Schrödinger equations with non-gauge-invariant nonlinearities. The spatial coefficient in the nonlinear term may have different orders of singularity at the origin and decay at infinity. Under a coercivity condition involving the coefficient and the nonlinearity, we show that no scattering states exist for the equation below a Strauss-type threshold determined by the decay at infinity. Our class includes nonlinearities with a dominant non-oscillatory component. We also prove a complementary small-data scattering result above this threshold, using non-admissible Strichartz estimates in Lorentz spaces adapted to the singular coefficient.
Comments28 pages
