Abstract
This paper contains three parts. In the first part, we determine the best constant of an improved inequality of Gagliardo–Nirenberg interpolation (Chen, in Czechoslov Math J, in press). In the second part, we use this best constant to establish a sharp criterion for the global existence and blow-up of solutions of the inhomogeneous nonlinear Schrödinger equation with harmonic potential
in the critical nonlinearity p = 1 + (4 + 2b)/N. In the third part, we use this best constant to construct an unbounded subset \({\mathcal{S}}\) of Σ and prove that the solutions exist globally in time for \({\varphi_0\in \mathcal{S}}\) and p > 1 + (4 + 2b)/N.
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Chen, J., Guo, B. Sharp constant of an improved Gagliardo–Nirenberg inequality and its application. Annali di Matematica 190, 341–354 (2011). https://doi.org/10.1007/s10231-010-0152-3
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DOI: https://doi.org/10.1007/s10231-010-0152-3
Keywords
- Improved inequality of interpolation
- Sharp constants
- Global solutions
- Inhomogeneous Schrödinger equation