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

Sov. Math., Dokl. 29, 281-284 (1984); translation from Dokl. Akad. Nauk SSSR 275, 780-783 (1984).
For a bounded domain $$G\subset {\mathbb{R}}^ n$$ an initial-boundary-value problem for a nonlinear Schrödinger equation is considered: $\partial u/\partial t+i\Delta u+i\alpha | u|^ pu+\beta | u|^ qu=0,\quad u|_{\partial G\times [0,T]}=0,\quad u(x,0)=u_ 0(x).$ It is shown that for $$0\leq p<q$$, $$\beta >0$$ and any real $$\alpha$$ a global (weak) solution exists. For $$p=2k$$, $$k\in {\mathbb{N}}$$, $$q\geq 0$$ and $$\beta\geq 0$$ a uniqueness result is given.
Reviewer: N.Jacob

### MSC:

 35J10 Schrödinger operator, Schrödinger equation 35J65 Nonlinear boundary value problems for linear elliptic equations 35D05 Existence of generalized solutions of PDE (MSC2000) 49M15 Newton-type methods