Ground state energy threshold and blow-up for NLS with competing nonlinearities
Ground state energy threshold and blow-up for NLS with competing nonlinearities
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DOI：
10.2422/2036-2145.202005_044
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发表时间：
2020-12
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影响因子：
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通讯作者：
J. Bellazzini;Luigi Forcella;V. Georgiev
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文献类型：
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作者：
J. Bellazzini;Luigi Forcella;V. Georgiev
考虑一类非线性Schr\ odinger方程，其中首项为临界聚焦幂型非线性，扰动由幂型离焦非线性给出。我们完全回答了基态能量是否达到的问题，基态能量是全局存在和奇点形成之间的一个阈值。对于任何规定的质量，对于质量-超临界或质量-临界离焦扰动，基态能量是通过对相关平稳方程的径向对称和递减解来实现的。对于质量-亚临界摄动，我们证明了一个临界规定质量的存在，精确地说是相关椭圆方程唯一的、静态的、正解的质量，使得基态能量对于任何等于或小于临界质量的质量都可以实现。此外，当质量大于临界质量时，无法获得基态能量。作为基态能量变分特性的副产品，我们证明了在有限时间内，对于任何低于基态能量阈值的能量，爆破解的存在性。
We consider the nonlinear Schr\"odinger 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.