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On the solvability of a mixed problem for a nonlinear equation of Schrödinger type. (English) Zbl 0585.35019

For a bounded domain G n an initial-boundary-value problem for a nonlinear Schrödinger equation is considered:

u/t+iΔu+iα|u| p u+β|u| q u=0,u| G×[0,T] =0,u(x,0)=u 0 (x)·

It is shown that for 0p<q, β>0 and any real α a global (weak) solution exists. For p=2k, k, q0 and β0 a uniqueness result is given.

Reviewer: N.Jacob

MSC:
35J10Schrödinger operator
35J65Nonlinear boundary value problems for linear elliptic equations
35D05Existence of generalized solutions of PDE (MSC2000)
49M15Newton-type methods in calculus of variations