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Existence of a ground state and blowup problem for a class of nonlinear Schrödinger equations involving mass and energy critical exponents

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Abstract

In this paper, we study the existence of the ground state and blowup problem for a class of nonlinear Schrödinger equations involving the mass and energy critical exponents. To show the existence of a ground state, we solve a minimization problem related to the virial identity, so that we need to compare the minimization value to the best constant of the Gagliardo–Nirenberg inequality because our nonlinearities contain the mass critical nonlinearity. Once we obtain the ground state, we can introduce a subset \({\mathcal {A}}_{\omega , -}\) of \(H^{1}({\mathbb {R}}^d)\) for each \(\omega > 0\) as in Berestycki and Cazenave (C R Acad Sci Paris Sér I Math 293:489–492, 1981). Then, it turn out that any radial solution starting from \({\mathcal {A}}_{\omega , -}\) blows up in a finite time.

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Notes

  1. We will define a so called “potential well”, which is a subset of \(H^{1}({\mathbb {R}}^d)\), and show that the solution starting from our potential well satisfies (1.5) (see (1.17) and Proposition 1.1 below).

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Acknowledgements

The authors would like to thank the anonymous referee for his/her comments. The work of H.K. was supported by JSPS KAKENHI Grant Number JP17K14223.

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Authors and Affiliations

  1. Department of Mathematics, Tsuda University, 2-1-1 Tsuda-machi, Kodaira-shi, Tokyo, 187-8577, Japan

    Hiroaki Kikuchi

  2. Graduate school of Mathematics, Tsuda University, 2-1-1 Tsuda-machi, Kodaira-shi, Tokyo, 187-8577, Japan

    Minami Watanabe

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  1. Hiroaki Kikuchi
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  2. Minami Watanabe
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Correspondence to Minami Watanabe.

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Kikuchi, H., Watanabe, M. Existence of a ground state and blowup problem for a class of nonlinear Schrödinger equations involving mass and energy critical exponents. Nonlinear Differ. Equ. Appl. 27, 56 (2020). https://doi.org/10.1007/s00030-020-00660-w

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