Existence of infinitely many solutions for a critical Hartree type equation with potential: local Pohožaev identities methods
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arXiv
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| Format: | Preprint |
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2024
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| author | Cassani, Daniele Yang, Minbo Zhang, Xinyun |
| author_facet | Cassani, Daniele Yang, Minbo Zhang, Xinyun |
| contents | This paper deals with the following equation $$-Δu =K(|x'|, x'')\Big(|x|^{-α}\ast (K(|x'|, x'')|u|^{2^{\ast}_α})\Big) |u|^{2^{\ast}_α-2}u\quad\mbox{in}\ \mathbb{R}^N,$$ where $N\geq5$, $α>5-\frac{6}{N-2}$, $2^{\ast}_α=\frac{2N-α}{N-2}$ is the so-called upper critical exponent in the Hardy-Littlewood-Sobolev inequality and $K(|x'|, x'')$, where $(x',x'')\in \mathbb{R}^2\times\mathbb{R}^{N-2}$, is bounded and nonnegative. Under proper assumptions on the potential function $K$, we obtain the existence of infinitely many solutions for the nonlocal critical equation by using a finite dimensional reduction argument and local Pohožaev identities. It is a remarkable fact that the order of the Riesz potential influences the existence/non-existence of solutions. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2402_04974 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Existence of infinitely many solutions for a critical Hartree type equation with potential: local Pohožaev identities methods Cassani, Daniele Yang, Minbo Zhang, Xinyun Analysis of PDEs This paper deals with the following equation $$-Δu =K(|x'|, x'')\Big(|x|^{-α}\ast (K(|x'|, x'')|u|^{2^{\ast}_α})\Big) |u|^{2^{\ast}_α-2}u\quad\mbox{in}\ \mathbb{R}^N,$$ where $N\geq5$, $α>5-\frac{6}{N-2}$, $2^{\ast}_α=\frac{2N-α}{N-2}$ is the so-called upper critical exponent in the Hardy-Littlewood-Sobolev inequality and $K(|x'|, x'')$, where $(x',x'')\in \mathbb{R}^2\times\mathbb{R}^{N-2}$, is bounded and nonnegative. Under proper assumptions on the potential function $K$, we obtain the existence of infinitely many solutions for the nonlocal critical equation by using a finite dimensional reduction argument and local Pohožaev identities. It is a remarkable fact that the order of the Riesz potential influences the existence/non-existence of solutions. |
| title | Existence of infinitely many solutions for a critical Hartree type equation with potential: local Pohožaev identities methods |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2402.04974 |