Asymptotic profiles for Choquard equations with general critical nonlinearities
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arXiv
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| Autores principales: | , , |
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| Formato: | Preprint |
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2024
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| _version_ | 1866916243138674688 |
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| author | Liu, Xiaonan Ma, Shiwang Wang, Yachen |
| author_facet | Liu, Xiaonan Ma, Shiwang Wang, Yachen |
| contents | In this paper, we study asymptotic behavior of positive ground state solutions for the nonlinear Choquard equation: \begin{equation}\label{0.1} -Δu+\varepsilon u=\big(I_α\ast F(u)\big)F'(u),\quad u\in H^1(\mathbb R^N), \end{equation} where $F(u)=|u|^{\frac{N+α}{N-2}}+G(u)$, $N\geq3$ is an integer, $I_α$ is the Riesz potential of order $α\in(0,N)$, and $\varepsilon>0$ is a parameter. Under some mild subcritical growth assumptions on $G(u)$, we show that as $\varepsilon \to \infty$, the ground state solutions of \eqref{0.1}, after a suitable rescaling, converge to a particular solution of the critical Choquard equation $-Δu=\frac{N+α}{N-2}(I_α*|u|^{\frac{N+α}{N-2}})|u|^{\frac{N+α}{N-2}-2}u$. We establish a novel sharp asymptotic characterisation of such a rescaling, which depends in a non-trivial way on the asymptotic behavior of $G(u)$ at infinity and the space dimension $N=3$, $N=4$ or $N\geq5$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2405_07149 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Asymptotic profiles for Choquard equations with general critical nonlinearities Liu, Xiaonan Ma, Shiwang Wang, Yachen Analysis of PDEs In this paper, we study asymptotic behavior of positive ground state solutions for the nonlinear Choquard equation: \begin{equation}\label{0.1} -Δu+\varepsilon u=\big(I_α\ast F(u)\big)F'(u),\quad u\in H^1(\mathbb R^N), \end{equation} where $F(u)=|u|^{\frac{N+α}{N-2}}+G(u)$, $N\geq3$ is an integer, $I_α$ is the Riesz potential of order $α\in(0,N)$, and $\varepsilon>0$ is a parameter. Under some mild subcritical growth assumptions on $G(u)$, we show that as $\varepsilon \to \infty$, the ground state solutions of \eqref{0.1}, after a suitable rescaling, converge to a particular solution of the critical Choquard equation $-Δu=\frac{N+α}{N-2}(I_α*|u|^{\frac{N+α}{N-2}})|u|^{\frac{N+α}{N-2}-2}u$. We establish a novel sharp asymptotic characterisation of such a rescaling, which depends in a non-trivial way on the asymptotic behavior of $G(u)$ at infinity and the space dimension $N=3$, $N=4$ or $N\geq5$. |
| title | Asymptotic profiles for Choquard equations with general critical nonlinearities |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2405.07149 |