Explicit Formula Of The Critical Mass And The Energy Ground State Solution For The Mixed Local-Nonlocal Schrodinger Equation For The In-Between Critical Exponents Case
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arXiv
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| Auteurs principaux: | , |
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| Format: | Preprint |
| Publié: |
2026
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| _version_ | 1866918364079718400 |
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| author | Su, Yu Hajaiej, Hichem |
| author_facet | Su, Yu Hajaiej, Hichem |
| contents | Our first main contribution consists in establishing an explicit formula of the critical mass via the best constant of the Gagliardo-Nirenberg inequality for the mixed local-nonlocal Laplacian. We also prove the existence of an optimizer of the Gagliardo-Nirenberg inequality for the mixed local-nonlocal operator. We then show that the optimizer (after some suitable scaling) is an energy ground state solution (with critical mass ). This is a key ingredient to determine sufficient and necessary conditions of existence and non-existence of energy ground state solutions in the in-between critical exponents case. Finally, we show that the energy ground state solution uc0 is an optimizer of the Gagliardo-Nirenberg inequality for the mixed local-nonlocal operator. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2603_01138 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Explicit Formula Of The Critical Mass And The Energy Ground State Solution For The Mixed Local-Nonlocal Schrodinger Equation For The In-Between Critical Exponents Case Su, Yu Hajaiej, Hichem Analysis of PDEs Our first main contribution consists in establishing an explicit formula of the critical mass via the best constant of the Gagliardo-Nirenberg inequality for the mixed local-nonlocal Laplacian. We also prove the existence of an optimizer of the Gagliardo-Nirenberg inequality for the mixed local-nonlocal operator. We then show that the optimizer (after some suitable scaling) is an energy ground state solution (with critical mass ). This is a key ingredient to determine sufficient and necessary conditions of existence and non-existence of energy ground state solutions in the in-between critical exponents case. Finally, we show that the energy ground state solution uc0 is an optimizer of the Gagliardo-Nirenberg inequality for the mixed local-nonlocal operator. |
| title | Explicit Formula Of The Critical Mass And The Energy Ground State Solution For The Mixed Local-Nonlocal Schrodinger Equation For The In-Between Critical Exponents Case |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2603.01138 |