Abstract
In this chapter we introduce results for semilinear Schrodinger models with power nonlinearity in the focusing and defocusing cases as well. First of all, we show how by a scaling argument a proposal for a critical exponent appears. This critical exponent heavily depends on the regularity of the data. The issue of L2 and H1 data is explained. As for the linear Schrodinger equation (see Sect. 11.2.3), some conserved quantities are given. Then, a global (in time) well-posedness result is proved for weak solutions in the subcritical L2 case. This result is valid for both cases focusing and defocusing, respectively. Finally, the subcritical H1 case is treated. Here the main concern is to show differences between both focusing and defocusing cases. A local (in time) well-posedness result is proved. This result contains, moreover, a blow up result in the focusing and a global (in time) well-posedness result in the defocusing case.
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