## Abstract

In this paper we consider the local existence to the Cauchy problem for nonlinear Schrödinger equations with power nonlinearities(*){i∂ _{t}u+1/2Δu=N(u, ∇, u, ū, ∇ū) (t, x)∈R×R ^{n} u(O, X)=U _{O} (X), X∈R ^{n}, where n≥2 and N=N(u, w, ū, w̄)= ∑ _{l}0 _{≥|α|+|β|+|γ|≥L1} λ _{αβγ}U ^{α} _{1}U ^{α} _{2}Π _{j=1} ^{n}(W _{j}) ^{β} _{j}Π ^{n} _{k=1}(W _{K}) ^{γK}withw=(w _{j}) _{1<j<n},λ _{αβγ}∈Cl _{o}∈N,l _{1,} L _{0} > 2. Classical energy method is useful to show local existence in time of solutions to (*) when ∂ _{w}N is pure imaginary (see, [10, 14-16]), and in this case it is known that there exists a unique solution if U _{0}∈H ^{n/2 3,0}, (see [10]), whereH ^{m's}={f∈L ^{2};||f|| _{m,s}=||(1+|x| ^{2}) ^{s/2}(l -Δ) ^{m/2}f||L ^{2}<∞}. However,if ∂ _{w}N 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 H ^{[n/2]+6,0} H ^{[n/2]+3,2} without any size restriction on the data. Our function spaces are more natural than those used in [2,12,13].

Original language | English |
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Pages (from-to) | 11-137 |

Number of pages | 127 |

Journal | SUT Journal of Mathematics |

Volume | 34 |

Issue number | 2 |

Publication status | Published - 1998 |

## Keywords

- Local existence
- Local nonlinearity
- Nonlinear schrödinger equation