Asymptotic profiles of ground state solutions for Choquard equations with a general local perturbation
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arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866916235551178752 |
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| author | Ma, Shiwang Moroz, Vitaly |
| author_facet | Ma, Shiwang Moroz, Vitaly |
| contents | In this paper, we study the asymptotic behavior of ground state solutions for the nonlinear Choquard equation with a general local perturbation $$ -Δu+\varepsilon u=(I_α\ast |u|^{p})|u|^{p-2}u+ g(u), \quad {\rm in} \ \mathbb R^N,
\eqno(P_\varepsilon)
$$
where $N\ge 3$ is an integer, $p=\frac{N+α}{N}$, or $\frac{N+α}{N-2}$, $I_α$ is the Riesz potential and $\varepsilon>0$ is a parameter. Under some mild conditions on $g(u)$, we show that as $\varepsilon\to \infty$, after {\em a suitable rescaling} the ground state solutions of $(P_\varepsilon)$ converge to a particular solution of some limit equations, and establish a sharp asymptotic characterisation of such a rescaling, which depend in a non-trivial way on the asymptotic behavior of the function $g(s)$ at infinity and the space dimension $N$. Based on this study, we also present some results on the existence and asymptotic behaviors of positive normalized solutions of $(P_\varepsilon)$ with the normalization constraint $\int_{\mathbb R^N}|u|^2=a^2$. Particularly, we obtain the asymptotic behavior of positive normalized solutions of such a problem as $a\to 0$ and $a\to \infty$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2405_02877 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Asymptotic profiles of ground state solutions for Choquard equations with a general local perturbation Ma, Shiwang Moroz, Vitaly Analysis of PDEs In this paper, we study the asymptotic behavior of ground state solutions for the nonlinear Choquard equation with a general local perturbation $$ -Δu+\varepsilon u=(I_α\ast |u|^{p})|u|^{p-2}u+ g(u), \quad {\rm in} \ \mathbb R^N, \eqno(P_\varepsilon) $$ where $N\ge 3$ is an integer, $p=\frac{N+α}{N}$, or $\frac{N+α}{N-2}$, $I_α$ is the Riesz potential and $\varepsilon>0$ is a parameter. Under some mild conditions on $g(u)$, we show that as $\varepsilon\to \infty$, after {\em a suitable rescaling} the ground state solutions of $(P_\varepsilon)$ converge to a particular solution of some limit equations, and establish a sharp asymptotic characterisation of such a rescaling, which depend in a non-trivial way on the asymptotic behavior of the function $g(s)$ at infinity and the space dimension $N$. Based on this study, we also present some results on the existence and asymptotic behaviors of positive normalized solutions of $(P_\varepsilon)$ with the normalization constraint $\int_{\mathbb R^N}|u|^2=a^2$. Particularly, we obtain the asymptotic behavior of positive normalized solutions of such a problem as $a\to 0$ and $a\to \infty$. |
| title | Asymptotic profiles of ground state solutions for Choquard equations with a general local perturbation |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2405.02877 |