Asymptotic decay of solutions for sublinear fractional Choquard equations
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arXiv
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| Format: | Preprint |
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2023
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| _version_ | 1866909654720708608 |
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| author | Gallo, Marco |
| author_facet | Gallo, Marco |
| contents | Goal of this paper is to study the asymptotic behaviour of the solutions of the following doubly nonlocal equation $$(-Δ)^s u + μu = (I_α*F(u))f(u) \quad \hbox{on $\mathbb{R}^N$}$$ where $s \in (0,1)$, $N\geq 2$, $α\in (0,N)$, $μ>0$, $I_α$ denotes the Riesz potential and $F(t) = \int_0^t f(τ) d τ$ is a general nonlinearity with a sublinear growth in the origin. The found decay is of polynomial type, with a rate possibly slower than $\sim\frac{1}{|x|^{N+2s}}$. The result is new even for homogeneous functions $f(u)=|u|^{r-2}u$, $r\in [\frac{N+α}{N},2)$, and it complements the decays obtained in the linear and superlinear cases in [D'Avenia, Siciliano, Squassina (2015)] and [Cingolani, Gallo, Tanaka (2022)]. Differently from the local case $s=1$ in [Moroz, Van Schaftingen (2013)], new phenomena arise connected to a new "$s$-sublinear" threshold that we detect on the growth of $f$. To gain the result we in particular prove a Chain Rule type inequality in the fractional setting, suitable for concave powers. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2310_09251 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Asymptotic decay of solutions for sublinear fractional Choquard equations Gallo, Marco Analysis of PDEs 35B09, 35B40, 35D30, 35D40, 35Q40, 35Q55, 35R09, 35R11, 45M05, 45M20 Goal of this paper is to study the asymptotic behaviour of the solutions of the following doubly nonlocal equation $$(-Δ)^s u + μu = (I_α*F(u))f(u) \quad \hbox{on $\mathbb{R}^N$}$$ where $s \in (0,1)$, $N\geq 2$, $α\in (0,N)$, $μ>0$, $I_α$ denotes the Riesz potential and $F(t) = \int_0^t f(τ) d τ$ is a general nonlinearity with a sublinear growth in the origin. The found decay is of polynomial type, with a rate possibly slower than $\sim\frac{1}{|x|^{N+2s}}$. The result is new even for homogeneous functions $f(u)=|u|^{r-2}u$, $r\in [\frac{N+α}{N},2)$, and it complements the decays obtained in the linear and superlinear cases in [D'Avenia, Siciliano, Squassina (2015)] and [Cingolani, Gallo, Tanaka (2022)]. Differently from the local case $s=1$ in [Moroz, Van Schaftingen (2013)], new phenomena arise connected to a new "$s$-sublinear" threshold that we detect on the growth of $f$. To gain the result we in particular prove a Chain Rule type inequality in the fractional setting, suitable for concave powers. |
| title | Asymptotic decay of solutions for sublinear fractional Choquard equations |
| topic | Analysis of PDEs 35B09, 35B40, 35D30, 35D40, 35Q40, 35Q55, 35R09, 35R11, 45M05, 45M20 |
| url | https://arxiv.org/abs/2310.09251 |