Infinitely many solutions for a class of fractional Schrodinger equations coupled with neutral scalar field
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| Format: | Preprint |
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2024
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| _version_ | 1866917593256820736 |
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| author | Shen, Liejun Squassina, Marco Zeng, Xiaoyu |
| author_facet | Shen, Liejun Squassina, Marco Zeng, Xiaoyu |
| contents | We study the fractional Schrödinger equations coupled with a neutral scalar field
$$
(-Δ)^s u+V(x)u=K(x)ϕu +g(x)|u|^{q-2}u, \quad x\in \mathbb{R}^3,\qquad
(I-Δ)^t ϕ=K(x)u^2, \quad x\in \mathbb{R}^3,
$$ where $(-Δ)^s$ and $(I-Δ)^t$ denote the fractional Laplacian and Bessel operators with $\frac{3}{4} <s<1$ and $0<t<1$, respectively. Under some suitable assumptions for the external potentials $V$, $K$ and $g$, given $q\in(1,2)\cup(2,2_s^*)$ with $2_s^*:= \frac{6}{3-2s}$, with the help of an improved Fountain theorem dealing with a class of strongly indefinite variational problems approached by Gu-Zhou [Adv. Nonlinear Stud., {\bf 17} (2017), 727--738], we show that the system admits infinitely many nontrivial solutions. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2402_12006 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Infinitely many solutions for a class of fractional Schrodinger equations coupled with neutral scalar field Shen, Liejun Squassina, Marco Zeng, Xiaoyu Analysis of PDEs 35J60, 35Q55, 53C35 We study the fractional Schrödinger equations coupled with a neutral scalar field $$ (-Δ)^s u+V(x)u=K(x)ϕu +g(x)|u|^{q-2}u, \quad x\in \mathbb{R}^3,\qquad (I-Δ)^t ϕ=K(x)u^2, \quad x\in \mathbb{R}^3, $$ where $(-Δ)^s$ and $(I-Δ)^t$ denote the fractional Laplacian and Bessel operators with $\frac{3}{4} <s<1$ and $0<t<1$, respectively. Under some suitable assumptions for the external potentials $V$, $K$ and $g$, given $q\in(1,2)\cup(2,2_s^*)$ with $2_s^*:= \frac{6}{3-2s}$, with the help of an improved Fountain theorem dealing with a class of strongly indefinite variational problems approached by Gu-Zhou [Adv. Nonlinear Stud., {\bf 17} (2017), 727--738], we show that the system admits infinitely many nontrivial solutions. |
| title | Infinitely many solutions for a class of fractional Schrodinger equations coupled with neutral scalar field |
| topic | Analysis of PDEs 35J60, 35Q55, 53C35 |
| url | https://arxiv.org/abs/2402.12006 |