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| Main Authors: | , |
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| Format: | Preprint |
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2024
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2407.03717 |
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| _version_ | 1866917719036657664 |
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| author | Li, Chong Li, Xinyu |
| author_facet | Li, Chong Li, Xinyu |
| contents | In this paper, we establish the existence of one solution for a Schrödinger equation with jumping nonlinearities: $-Δu+V(x)u=f(x,u)$, $x\in \mathbb {R}^N$, and $u(x)\to 0$, $|x|\to +\infty$, where $V$ is a potential function on which we make hypotheses, and in particular allow $V$ which is unbounded below, and $f(x,u)=au^-+bu^++g(x,u)$. No restriction on $b$ is required, which implies that $f(x,s)s^{-1}$ may interfere with the essential spectrum of $ -Δ+V$ for $s\to +\infty$. Using the truncation method and the Morse theory, we can compute critical groups of the corresponding functional at zero and infinity, then prove the existence of one negative solution. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2407_03717 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | The existence of solutions for a Schrodinger equation with jumping nonlinearities crossing the essential spectrum Li, Chong Li, Xinyu Analysis of PDEs In this paper, we establish the existence of one solution for a Schrödinger equation with jumping nonlinearities: $-Δu+V(x)u=f(x,u)$, $x\in \mathbb {R}^N$, and $u(x)\to 0$, $|x|\to +\infty$, where $V$ is a potential function on which we make hypotheses, and in particular allow $V$ which is unbounded below, and $f(x,u)=au^-+bu^++g(x,u)$. No restriction on $b$ is required, which implies that $f(x,s)s^{-1}$ may interfere with the essential spectrum of $ -Δ+V$ for $s\to +\infty$. Using the truncation method and the Morse theory, we can compute critical groups of the corresponding functional at zero and infinity, then prove the existence of one negative solution. |
| title | The existence of solutions for a Schrodinger equation with jumping nonlinearities crossing the essential spectrum |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2407.03717 |