Local uniqueness of multi-peak positive solutions to a class of fractional Kirchhoff equations
Fuente:
arXiv
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| Formato: | Preprint |
| Publicado: |
2022
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| _version_ | 1866914183148208128 |
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| author | Yang, Zhipeng |
| author_facet | Yang, Zhipeng |
| contents | This paper has two main purposes. In the first part, combining the nondegeneracy of the ground state with the Lyapunov--Schmidt reduction method, we prove the existence of multi-peak positive solutions to the singularly perturbed problem \[\Big(\varepsilon^{2s}a+\varepsilon^{4s-N} b\int_{\mathbb{R}^{N}}|(-Δ)^{\frac{s}{2}}u|^2\,dx\Big)(-Δ)^s u+V(x)u=u^p\quad \text{in }\mathbb{R}^{N},\] for all sufficiently small $\varepsilon> 0$, under the assumptions $2s<N<4s$, $1<p<2^*_s-1$, and some mild conditions on the potential $V$. The main difficulty comes from the interplay between the nonlocal operator $(-Δ)^s$ and the nonlocal Kirchhoff term, which makes the corresponding limiting problem a coupled system of partial differential equations rather than a single fractional Kirchhoff equation. In the second part, under additional assumptions on $V$, we establish the local uniqueness of positive multi-peak solutions by means of a local Pohozǎev identity. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2203_07468 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Local uniqueness of multi-peak positive solutions to a class of fractional Kirchhoff equations Yang, Zhipeng Analysis of PDEs 35R11, 35A15, 47G20 This paper has two main purposes. In the first part, combining the nondegeneracy of the ground state with the Lyapunov--Schmidt reduction method, we prove the existence of multi-peak positive solutions to the singularly perturbed problem \[\Big(\varepsilon^{2s}a+\varepsilon^{4s-N} b\int_{\mathbb{R}^{N}}|(-Δ)^{\frac{s}{2}}u|^2\,dx\Big)(-Δ)^s u+V(x)u=u^p\quad \text{in }\mathbb{R}^{N},\] for all sufficiently small $\varepsilon> 0$, under the assumptions $2s<N<4s$, $1<p<2^*_s-1$, and some mild conditions on the potential $V$. The main difficulty comes from the interplay between the nonlocal operator $(-Δ)^s$ and the nonlocal Kirchhoff term, which makes the corresponding limiting problem a coupled system of partial differential equations rather than a single fractional Kirchhoff equation. In the second part, under additional assumptions on $V$, we establish the local uniqueness of positive multi-peak solutions by means of a local Pohozǎev identity. |
| title | Local uniqueness of multi-peak positive solutions to a class of fractional Kirchhoff equations |
| topic | Analysis of PDEs 35R11, 35A15, 47G20 |
| url | https://arxiv.org/abs/2203.07468 |