On the Cauchy problem of 3-D energy-critical Schrödinger equations with subcritical perturbations

Abstract

We investigate the global well-posedness, scattering and blow up phenomena when the 3-D quintic nonlinear Schrödinger equation, which is energy-critical, is perturbed by a subcritical nonlinearity λ 1 |u| p u. We find when the quintic term is defocussing, then the solution is always global no matter what the sign of λ 1 is. Scattering will occur either when the perturbation is defocussing and 4 3 < p < 4 or when the mass of the solution is small enough and 4 3 p < 4. When the quintic term is focusing, we show the blow up for certain solutions.

The Cauchy problem of a focusing energy-critical nonlinear Schrödinger equation

“…Moreover, this result is discussed under the condition that the energy is positive. That result is quite different from the previous papers, such as [24]. On the other hand, combining with Theorem 1.1, we obtain sharp conditions of global existence and blowup of Eq.…”

“…In the double power case, Tao, Visan and Zhang [23] and Zhang [24] investigated the global well-posedness, scattering and blowup phenomena for Eq. (1.1).…”

The Nonlinear Schrödinger Equation with Combined Power-Type Nonlinearities

“…Hence, adding an energy-subcritical perturbation to (1.5), which destroys the scale invariance, is of particular interest. This particular problem was first pursued by the third author, [31], who considered the case n = 3. The perturbative approach used in [31] extends easily to dimensions n = 4, 5, 6.…”

“…This particular problem was first pursued by the third author, [31], who considered the case n = 3. The perturbative approach used in [31] extends easily to dimensions n = 4, 5, 6. However, in higher dimensions (n > 6) new difficulties arise, mainly related to the low power of the energy-critical nonlinearity.…”

Stability Solution of the Nonlinear Schr&#246;dinger Equation

“…Even though stability is a local question, it plays an important role in all existing treatments of the global well-posedness problem for nonlinear Schrödinger equation at critical case, for more see [7]. It has also proved useful in the treatment of local and global questions for more exotic nonlinearities [8,9]. In this section, we will only discus the stability theory for the mass-critical NLS.…”