Existence and multiplicity of solutions for nonlocal Schrödinger–Kirchhoff equations of convex–concave type with the external magnetic field
We are concerned with the following elliptic equations $ \begin{equation*} K(|z|^p_{s, {A}})(-\Delta)^s_{p, A}z+ V(x)|z|^{p-2}z = a(x)|z|^{r-2}z+\lambda f(x, |z|)z \quad {\rm{in}} \; \; \mathbb{R}^{N}, \end{equation*} $ where $ (-\Delta)^{s}_{p, A} $ is the fractional magnetic operator, $ K:\mat...
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| Format: | Article |
| Language: | English |
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AIMS Press
2022-01-01
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| Series: | AIMS Mathematics |
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| Online Access: | https://www.aimspress.com/article/doi/10.3934/math.2022367?viewType=HTML |
| _version_ | 1818330381054640128 |
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| author | Seol Vin Kim Yun-Ho Kim |
| author_facet | Seol Vin Kim Yun-Ho Kim |
| author_sort | Seol Vin Kim |
| collection | DOAJ |
| description | We are concerned with the following elliptic equations
$ \begin{equation*} K(|z|^p_{s, {A}})(-\Delta)^s_{p, A}z+ V(x)|z|^{p-2}z = a(x)|z|^{r-2}z+\lambda f(x, |z|)z \quad {\rm{in}} \; \; \mathbb{R}^{N}, \end{equation*} $
where $ (-\Delta)^{s}_{p, A} $ is the fractional magnetic operator, $ K:\mathbb{R}_0^+ \to\mathbb{R}^+_0 $ is a Kirchhoff function, $ A : \Bbb R^N \rightarrow \Bbb R^N $ is a magnetic potential and $ V:\Bbb R^{N}\to(0, \infty) $ is continuous potential. The main purpose is to show the existence of infinitely many large- or small- energy solutions to the problem above. The strategy of the proof for these results is to approach the problem variationally by employing the variational methods, namely, the fountain and the dual fountain theorem with Cerami condition. |
| first_indexed | 2024-12-13T13:03:03Z |
| format | Article |
| id | doaj.art-c0a080ee7ee943138376aaf91136606f |
| institution | Directory Open Access Journal |
| issn | 2473-6988 |
| language | English |
| last_indexed | 2024-12-13T13:03:03Z |
| publishDate | 2022-01-01 |
| publisher | AIMS Press |
| record_format | Article |
| series | AIMS Mathematics |
| spelling | doaj.art-c0a080ee7ee943138376aaf91136606f2022-12-21T23:44:55ZengAIMS PressAIMS Mathematics2473-69882022-01-01746583659910.3934/math.2022367Existence and multiplicity of solutions for nonlocal Schrödinger–Kirchhoff equations of convex–concave type with the external magnetic fieldSeol Vin Kim0Yun-Ho Kim1 Department of Mathematics Education, Sangmyung University, Seoul 03016, Republic of Korea Department of Mathematics Education, Sangmyung University, Seoul 03016, Republic of KoreaWe are concerned with the following elliptic equations $ \begin{equation*} K(|z|^p_{s, {A}})(-\Delta)^s_{p, A}z+ V(x)|z|^{p-2}z = a(x)|z|^{r-2}z+\lambda f(x, |z|)z \quad {\rm{in}} \; \; \mathbb{R}^{N}, \end{equation*} $ where $ (-\Delta)^{s}_{p, A} $ is the fractional magnetic operator, $ K:\mathbb{R}_0^+ \to\mathbb{R}^+_0 $ is a Kirchhoff function, $ A : \Bbb R^N \rightarrow \Bbb R^N $ is a magnetic potential and $ V:\Bbb R^{N}\to(0, \infty) $ is continuous potential. The main purpose is to show the existence of infinitely many large- or small- energy solutions to the problem above. The strategy of the proof for these results is to approach the problem variationally by employing the variational methods, namely, the fountain and the dual fountain theorem with Cerami condition.https://www.aimspress.com/article/doi/10.3934/math.2022367?viewType=HTMLschrödinger-kirchhoff equationfractional magnetic operatorsvariational methods |
| spellingShingle | Seol Vin Kim Yun-Ho Kim Existence and multiplicity of solutions for nonlocal Schrödinger–Kirchhoff equations of convex–concave type with the external magnetic field AIMS Mathematics schrödinger-kirchhoff equation fractional magnetic operators variational methods |
| title | Existence and multiplicity of solutions for nonlocal Schrödinger–Kirchhoff equations of convex–concave type with the external magnetic field |
| title_full | Existence and multiplicity of solutions for nonlocal Schrödinger–Kirchhoff equations of convex–concave type with the external magnetic field |
| title_fullStr | Existence and multiplicity of solutions for nonlocal Schrödinger–Kirchhoff equations of convex–concave type with the external magnetic field |
| title_full_unstemmed | Existence and multiplicity of solutions for nonlocal Schrödinger–Kirchhoff equations of convex–concave type with the external magnetic field |
| title_short | Existence and multiplicity of solutions for nonlocal Schrödinger–Kirchhoff equations of convex–concave type with the external magnetic field |
| title_sort | existence and multiplicity of solutions for nonlocal schrodinger kirchhoff equations of convex concave type with the external magnetic field |
| topic | schrödinger-kirchhoff equation fractional magnetic operators variational methods |
| url | https://www.aimspress.com/article/doi/10.3934/math.2022367?viewType=HTML |
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