The Well-posedness Of A Class Of Nonlinear Schr(?)dinger Equations With Potential

Schr(o|¨)dinger equation is established by the Austrian physicist E. Schr(o|¨)dinger, and it is a basic equation and assume in the (non-relativity) quantum mechanics. Its status in quantum mechanics is equivalent to the Newton equations in classical mechanics. Schr(o|¨)dinger equation describes the laws for the state of micro-particle changing with time, and its form is as follows:Nonlinear Schr(o|¨)dinger equation is widely applied now, for example, in Nonlinear Optics, Condensed Matter Physics, Electromagnetism and so on. So, there is an important practical significance to study this kind of equations.In this paper, we mainly study the following Nonlinear Schr(o|¨)dinger equations with potential: With given initial condition:(?)_t=0 =u₀. We prove the local existence, uniqueness and continuous dependence of its initial data. Here V is potential, -λ₁ |(?)| ^p₁-1(?)-λ₂|(?)|^p₂-1(?)is the nonlinear perturbation. And potential V is bounded below and satisfies the condition that |D^αV| is bounded for all |α|≥2(e.g., V = |x|²).