Classification and qualitative properties of positive solutions to double-power nonlinear stationary Schrödinger equations
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| Main Authors: | , , , , |
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| Format: | Preprint |
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2025
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| _version_ | 1866915553972584448 |
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| author | Akahori, Takafumi Ibrahim, Slim Kikuchi, Hiroaki Shibata, Masataka Wei, Juncheng |
| author_facet | Akahori, Takafumi Ibrahim, Slim Kikuchi, Hiroaki Shibata, Masataka Wei, Juncheng |
| contents | In this paper, we investigate positive radial solutions to double-power nonlinear stationary Schrodinger equations in three space dimensions. It is now known that the non-uniqueness of H^{1}-positive solutions can occur in three dimensions when the frequency is sufficiently small. Under suitable conditions, in addition to the ground state solution (whose L^{\infty} norm vanishes as the frequency tends to zero), there exists another positive solution that minimizes a different constrained variational problem, with an L^{\infty} norm diverging as the frequency tends to zero (see Theorem 1.4). We classify all positive solutions with small frequency into two categories: the ground state and the Aubin-Talenti type solution. As a consequence, we establish the multiplicity of positive solutions. Finally, we also examine the non-degeneracy and Morse index of each positive solution. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2510_12484 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Classification and qualitative properties of positive solutions to double-power nonlinear stationary Schrödinger equations Akahori, Takafumi Ibrahim, Slim Kikuchi, Hiroaki Shibata, Masataka Wei, Juncheng Analysis of PDEs 35J20, 35J61, 35B09, 34A34 In this paper, we investigate positive radial solutions to double-power nonlinear stationary Schrodinger equations in three space dimensions. It is now known that the non-uniqueness of H^{1}-positive solutions can occur in three dimensions when the frequency is sufficiently small. Under suitable conditions, in addition to the ground state solution (whose L^{\infty} norm vanishes as the frequency tends to zero), there exists another positive solution that minimizes a different constrained variational problem, with an L^{\infty} norm diverging as the frequency tends to zero (see Theorem 1.4). We classify all positive solutions with small frequency into two categories: the ground state and the Aubin-Talenti type solution. As a consequence, we establish the multiplicity of positive solutions. Finally, we also examine the non-degeneracy and Morse index of each positive solution. |
| title | Classification and qualitative properties of positive solutions to double-power nonlinear stationary Schrödinger equations |
| topic | Analysis of PDEs 35J20, 35J61, 35B09, 34A34 |
| url | https://arxiv.org/abs/2510.12484 |