Schrödinger equation in dimension two with competing logarithmic self-interaction
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arXiv
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| Auteurs principaux: | , , |
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| Format: | Preprint |
| Publié: |
2024
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| _version_ | 1866913781878095872 |
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| author | Azzollini, Antonio d'Avenia, Pietro Pomponio, Alessio |
| author_facet | Azzollini, Antonio d'Avenia, Pietro Pomponio, Alessio |
| contents | In this paper we study the equation \[ -Δu +(\log |\cdot|*|u|^2)u=(\log|\cdot|*|u|^q)|u|^{q-2}u, \qquad \hbox{ in }\mathbb{R}^2, \] where $8/3 < q < 4$. By means of variational arguments, we find infinitely many radially symmetric classical solutions. The main difficulties rely on the competition between the two nonlocal terms and on the presence of logarithmic kernels, which have not a prescribed sign. In addition, in order to find finite energy solutions, a suitable functional setting analysis is required. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2404_03452 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Schrödinger equation in dimension two with competing logarithmic self-interaction Azzollini, Antonio d'Avenia, Pietro Pomponio, Alessio Analysis of PDEs 35J20, 35J60, 46E35 In this paper we study the equation \[ -Δu +(\log |\cdot|*|u|^2)u=(\log|\cdot|*|u|^q)|u|^{q-2}u, \qquad \hbox{ in }\mathbb{R}^2, \] where $8/3 < q < 4$. By means of variational arguments, we find infinitely many radially symmetric classical solutions. The main difficulties rely on the competition between the two nonlocal terms and on the presence of logarithmic kernels, which have not a prescribed sign. In addition, in order to find finite energy solutions, a suitable functional setting analysis is required. |
| title | Schrödinger equation in dimension two with competing logarithmic self-interaction |
| topic | Analysis of PDEs 35J20, 35J60, 46E35 |
| url | https://arxiv.org/abs/2404.03452 |