论文标题
具有竞争非线性的NLS的基态能源阈值和爆炸
Ground state energy threshold and blow-up for NLS with competing nonlinearities
论文作者
论文摘要
我们考虑具有联合非线性的非线性schrödinger方程，其中领先术语是一个内部重点的功率类型的非线性，并且扰动由功率类型偏置的扰动给出。我们完全回答了基础状态能量的问题，这是全球存在与奇异之处形成之间的阈值。对于任何规定的质量，对于质量 - 质临界或质量批判性散落的扰动，基态能量是通过对相关固定方程的径向对称和减少的解决方案来实现的。对于质量临界扰动，我们显示了一个关键的规定质量的存在，正是相关椭圆方程的独特，静态，积极的质量的质量，因此，对于等于或小的质量，就可以实现基态能量。此外，对于大于关键的质量而言，基态能量没有实现。作为基态能量变异表征的副产品，我们证明在有限的时间内存在爆炸溶液的存在，对于基态能量阈值以下的任何能量。
We consider the nonlinear Schrödinger equation with combined nonlinearities, where the leading term is an intracritical focusing power-type nonlinearity, and the perturbation is given by a power-type defocusing one. We completely answer the question wether the ground state energy, which is a threshold between global existence and formation of singularities, is achieved. For any prescribed mass, for mass-supercritical or mass-critical defocusing perturbations, the ground state energy is achieved by a radially symmetric and decreasing solution to the associated stationary equation. For mass-subcritical perturbations, we show the existence of a critical prescribed mass, precisely the mass of the unique, static, positive solution to the associated elliptic equation, such that the ground state energy is achieved for any mass equal or smaller than the critical one. Moreover, the ground state energy is not achieved for mass larger than the critical one. As a byproduct of the variational characterization of the ground state energy, we prove the existence of blowing-up solutions in finite time, for any energy below the ground state energy threshold.
