Existence of positive solutions for fractional Schrödinger equation with general nonlinearities in exterior domains
Abstract In this paper, we investigate the existence of positive solutions of the following fractional Schrödinger equation with general nonlinearities: { ( − Δ ) s u + λ u = f ( u ) , in Ω , u = 0 , on R N ∖ Ω , $$\begin{aligned} \left \{ \textstyle\begin{array}{l@{\quad }l@{\quad }l} (-\Delta )^{s...
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2025-06-01
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| Series: | Boundary Value Problems |
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| Online Access: | https://doi.org/10.1186/s13661-025-02074-y |
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| author | Yalin Shen |
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| author_sort | Yalin Shen |
| collection | DOAJ |
| description | Abstract In this paper, we investigate the existence of positive solutions of the following fractional Schrödinger equation with general nonlinearities: { ( − Δ ) s u + λ u = f ( u ) , in Ω , u = 0 , on R N ∖ Ω , $$\begin{aligned} \left \{ \textstyle\begin{array}{l@{\quad }l@{\quad }l} (-\Delta )^{s} u+\lambda u=f(u), \quad &\text{in}\; \Omega , \\ u=0, \quad &\text{on}\; \mathbb{R}^{N} \backslash \Omega , \end{array}\displaystyle \right . \end{aligned}$$ where N ≥ 2 $N\ge 2$ , s ∈ ( 0 , 1 ) $s\in (0,1)$ , Ω ⊂ R N $\Omega \subset \mathbb{R}^{N}$ is an exterior domain, i.e., Ω is an unbounded domain in R N $\mathbb{R}^{N}$ with R N ∖ Ω $\mathbb{R}^{N} \backslash \Omega $ nonempty and bounded, λ > 0 $\lambda >0$ is a parameter, and f ∈ C 1 ( R , R ) $f \in C^{1}(\mathbb{R},\mathbb{R})$ satisfies some technical conditions. We use variational and topological methods to prove that there is a positive solution u ∈ H 0 s ( Ω ) $u\in H^{s}_{0}(\Omega )$ if R N ∖ Ω $\mathbb{R}^{N} \backslash \Omega $ is small enough. Furthermore, the result can be extended to the fractional Kirchhoff equation with general nonlinearities. |
| format | Article |
| id | doaj-art-95f6a34593134ceebca65ba4a93a41ba |
| institution | OA Journals |
| issn | 1687-2770 |
| language | English |
| publishDate | 2025-06-01 |
| publisher | SpringerOpen |
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| series | Boundary Value Problems |
| spelling | doaj-art-95f6a34593134ceebca65ba4a93a41ba2025-08-20T02:05:14ZengSpringerOpenBoundary Value Problems1687-27702025-06-012025111710.1186/s13661-025-02074-yExistence of positive solutions for fractional Schrödinger equation with general nonlinearities in exterior domainsYalin Shen0School of Mathematical Sciences, Jiangsu UniversityAbstract In this paper, we investigate the existence of positive solutions of the following fractional Schrödinger equation with general nonlinearities: { ( − Δ ) s u + λ u = f ( u ) , in Ω , u = 0 , on R N ∖ Ω , $$\begin{aligned} \left \{ \textstyle\begin{array}{l@{\quad }l@{\quad }l} (-\Delta )^{s} u+\lambda u=f(u), \quad &\text{in}\; \Omega , \\ u=0, \quad &\text{on}\; \mathbb{R}^{N} \backslash \Omega , \end{array}\displaystyle \right . \end{aligned}$$ where N ≥ 2 $N\ge 2$ , s ∈ ( 0 , 1 ) $s\in (0,1)$ , Ω ⊂ R N $\Omega \subset \mathbb{R}^{N}$ is an exterior domain, i.e., Ω is an unbounded domain in R N $\mathbb{R}^{N}$ with R N ∖ Ω $\mathbb{R}^{N} \backslash \Omega $ nonempty and bounded, λ > 0 $\lambda >0$ is a parameter, and f ∈ C 1 ( R , R ) $f \in C^{1}(\mathbb{R},\mathbb{R})$ satisfies some technical conditions. We use variational and topological methods to prove that there is a positive solution u ∈ H 0 s ( Ω ) $u\in H^{s}_{0}(\Omega )$ if R N ∖ Ω $\mathbb{R}^{N} \backslash \Omega $ is small enough. Furthermore, the result can be extended to the fractional Kirchhoff equation with general nonlinearities.https://doi.org/10.1186/s13661-025-02074-yFractional Schrödinger equationsExterior domainPositive solutions |
| spellingShingle | Yalin Shen Existence of positive solutions for fractional Schrödinger equation with general nonlinearities in exterior domains Boundary Value Problems Fractional Schrödinger equations Exterior domain Positive solutions |
| title | Existence of positive solutions for fractional Schrödinger equation with general nonlinearities in exterior domains |
| title_full | Existence of positive solutions for fractional Schrödinger equation with general nonlinearities in exterior domains |
| title_fullStr | Existence of positive solutions for fractional Schrödinger equation with general nonlinearities in exterior domains |
| title_full_unstemmed | Existence of positive solutions for fractional Schrödinger equation with general nonlinearities in exterior domains |
| title_short | Existence of positive solutions for fractional Schrödinger equation with general nonlinearities in exterior domains |
| title_sort | existence of positive solutions for fractional schrodinger equation with general nonlinearities in exterior domains |
| topic | Fractional Schrödinger equations Exterior domain Positive solutions |
| url | https://doi.org/10.1186/s13661-025-02074-y |
| work_keys_str_mv | AT yalinshen existenceofpositivesolutionsforfractionalschrodingerequationwithgeneralnonlinearitiesinexteriordomains |