SUT J. Math. 34 (2), 111-137, (June 1998) DOI: 10.55937/sut/991985326
Nakao Hayashi, Elena I. Kaikina
KEYWORDS: local existence, Nonlinear Schrôdinger equation, Local nonlinearity, 35Q55
In this paper we consider the local existence to the Cauchy problem for nonlinear Schrödinger equations with power nonlinearities
With . Classical energy method is useful to show local existence in time of solutions to when is pure imaginary (see, [10, 14-16]), and in this case it is known that there exists a unique solution if (see ), where . However, if is not pure imaginary, there are only a few results [2,12,13] that require higher order Sobolev spaces compared with [10, 14-16] because the classical energy method does not work for the problem. Our purpose in this paper is to show local existence in time of solutions to in the weighted Sobolev space without any size restriction on the data. Our function spaces are more natural than those used in [2,12,13].