Least energy sign-changing solutions for a class of fractional $ (p, q) $-Laplacian problems with critical growth in $ \mathbb{R}^N $
This paper considers the following fractional $ (p, q) $-Laplacian equation: $ (-\Delta)_{p}^{s} u+(-\Delta)_{q}^{s} u+V(x)\left(|u|^{p-2} u+|u|^{q-2} u\right) = \lambda f(u)+|u|^{q^*_s-2}u \quad \text { in } \mathbb{R}^{N}, $ where $ s \in(0, 1), \lambda > 0, 2 < p <...
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| Format: | Article |
| Language: | English |
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AIMS Press
2023-04-01
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| Series: | AIMS Mathematics |
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| Online Access: | https://www.aimspress.com/article/doi/10.3934/math.2023675?viewType=HTML |
| _version_ | 1827964477113368576 |
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| author | Kun Cheng Shenghao Feng Li Wang Yuangen Zhan |
| author_facet | Kun Cheng Shenghao Feng Li Wang Yuangen Zhan |
| author_sort | Kun Cheng |
| collection | DOAJ |
| description | This paper considers the following fractional $ (p, q) $-Laplacian equation:
$ (-\Delta)_{p}^{s} u+(-\Delta)_{q}^{s} u+V(x)\left(|u|^{p-2} u+|u|^{q-2} u\right) = \lambda f(u)+|u|^{q^*_s-2}u \quad \text { in } \mathbb{R}^{N}, $
where $ s \in(0, 1), \lambda > 0, 2 < p < q < \frac{N}{s} $, $ (-\Delta)_{t}^{s} $ with $ t \in\{p, q\} $ is the fractional $ t $-Laplacian operator, and potential $ V $ is a continuous function. Using constrained variational methods, a quantitative Deformation Lemma and Brouwer degree theory, we prove that the above problem has a least energy sign-changing solution $ u_{\lambda} $ under suitable conditions on $ f $, $ V $ and $ \lambda $. Moreover, we show that the energy of $ u_{\lambda} $ is strictly larger than two times the ground state energy. |
| first_indexed | 2024-04-09T17:16:39Z |
| format | Article |
| id | doaj.art-d7e1c8b2526c4d739142ba47c6b101d6 |
| institution | Directory Open Access Journal |
| issn | 2473-6988 |
| language | English |
| last_indexed | 2024-04-09T17:16:39Z |
| publishDate | 2023-04-01 |
| publisher | AIMS Press |
| record_format | Article |
| series | AIMS Mathematics |
| spelling | doaj.art-d7e1c8b2526c4d739142ba47c6b101d62023-04-20T01:10:30ZengAIMS PressAIMS Mathematics2473-69882023-04-0186133251335010.3934/math.2023675Least energy sign-changing solutions for a class of fractional $ (p, q) $-Laplacian problems with critical growth in $ \mathbb{R}^N $Kun Cheng 0Shenghao Feng 1 Li Wang2Yuangen Zhan 31. School of Information Engineering, Jingdezhen Ceramic University, Jingdezhen 333403, China2. School of Mathematics and Computer Science, Nanchang University, Nanchang 330031, China3. College of Science, East China Jiaotong University, Nanchang 330013, China4. School of Information Engineering, Jingdezhen Ceramic University, Jingdezhen 333403, ChinaThis paper considers the following fractional $ (p, q) $-Laplacian equation: $ (-\Delta)_{p}^{s} u+(-\Delta)_{q}^{s} u+V(x)\left(|u|^{p-2} u+|u|^{q-2} u\right) = \lambda f(u)+|u|^{q^*_s-2}u \quad \text { in } \mathbb{R}^{N}, $ where $ s \in(0, 1), \lambda > 0, 2 < p < q < \frac{N}{s} $, $ (-\Delta)_{t}^{s} $ with $ t \in\{p, q\} $ is the fractional $ t $-Laplacian operator, and potential $ V $ is a continuous function. Using constrained variational methods, a quantitative Deformation Lemma and Brouwer degree theory, we prove that the above problem has a least energy sign-changing solution $ u_{\lambda} $ under suitable conditions on $ f $, $ V $ and $ \lambda $. Moreover, we show that the energy of $ u_{\lambda} $ is strictly larger than two times the ground state energy.https://www.aimspress.com/article/doi/10.3934/math.2023675?viewType=HTMLfractional $ (p， q) $-laplaciansign-changing solutionscritical problem |
| spellingShingle | Kun Cheng Shenghao Feng Li Wang Yuangen Zhan Least energy sign-changing solutions for a class of fractional $ (p, q) $-Laplacian problems with critical growth in $ \mathbb{R}^N $ AIMS Mathematics fractional $ (p， q) $-laplacian sign-changing solutions critical problem |
| title | Least energy sign-changing solutions for a class of fractional $ (p, q) $-Laplacian problems with critical growth in $ \mathbb{R}^N $ |
| title_full | Least energy sign-changing solutions for a class of fractional $ (p, q) $-Laplacian problems with critical growth in $ \mathbb{R}^N $ |
| title_fullStr | Least energy sign-changing solutions for a class of fractional $ (p, q) $-Laplacian problems with critical growth in $ \mathbb{R}^N $ |
| title_full_unstemmed | Least energy sign-changing solutions for a class of fractional $ (p, q) $-Laplacian problems with critical growth in $ \mathbb{R}^N $ |
| title_short | Least energy sign-changing solutions for a class of fractional $ (p, q) $-Laplacian problems with critical growth in $ \mathbb{R}^N $ |
| title_sort | least energy sign changing solutions for a class of fractional p q laplacian problems with critical growth in mathbb r n |
| topic | fractional $ (p， q) $-laplacian sign-changing solutions critical problem |
| url | https://www.aimspress.com/article/doi/10.3934/math.2023675?viewType=HTML |
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