Existence of solutions for Kirchhoff-type systems with critical Sobolev exponents in $ \mathbb{R}^3 $
<p>In this paper, we study the following Kirchhoff-type system:</p> <p class="disp_formula">$ \begin{equation} \left\{ \begin{array}{ll} -(a_{1}+b_{1}\int_{\mathbb{R}^{3}}|\nabla u|^{2}dx)\Delta u = \frac{2\alpha}{\alpha+\beta}|u|^{\alpha-2}u|v|^{\beta}+\varepsilo...
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AIMS Press
2023-07-01
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| Online Access: | https://www.aimspress.com/article/doi/10.3934/era.2023269?viewType=HTML |
| _version_ | 1827791818983473152 |
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| author | Xing Yi Shuhou Ye |
| author_facet | Xing Yi Shuhou Ye |
| author_sort | Xing Yi |
| collection | DOAJ |
| description | <p>In this paper, we study the following Kirchhoff-type system:</p>
<p class="disp_formula">$ \begin{equation} \left\{ \begin{array}{ll} -(a_{1}+b_{1}\int_{\mathbb{R}^{3}}|\nabla u|^{2}dx)\Delta u = \frac{2\alpha}{\alpha+\beta}|u|^{\alpha-2}u|v|^{\beta}+\varepsilon f(x), \\ -(a_{2}+b_{2}\int_{\mathbb{R}^{3}}|\nabla v|^{2}dx)\Delta v = \frac{2\beta}{\alpha+\beta}|u|^{\alpha}|v|^{\beta-2}v+\varepsilon g(x), \\ (u, v)\in D^{1, 2}(\mathbb{R}^{3})\times D^{1, 2}(\mathbb{R}^{3}), \end{array} \right. \end{equation} $</p>
<p>where $ a_{1}, a_{2}\geq0, \; b_{1}, b_{2} > 0, \; \alpha, \beta > 1, \; \alpha+\beta = 6 $ and $ f(x), g(x)\geq0, \; f(x), g(x)\in L^{\frac{6}{5}}(\mathbb{R}^3). $ The aim of this paper is to demonstrate the existence of at least two solutions for system (0.1), utilizing the variational method. To achieve this, we construct an energy functional and analyze its critical points by applying the Ekeland variational principle, the mountain pass lemma and the concentration compactness principle.</p> |
| first_indexed | 2024-03-11T17:52:19Z |
| format | Article |
| id | doaj.art-c09f3d441d954197ab64769115939c2e |
| institution | Directory Open Access Journal |
| issn | 2688-1594 |
| language | English |
| last_indexed | 2024-03-11T17:52:19Z |
| publishDate | 2023-07-01 |
| publisher | AIMS Press |
| record_format | Article |
| series | Electronic Research Archive |
| spelling | doaj.art-c09f3d441d954197ab64769115939c2e2023-10-18T01:13:28ZengAIMS PressElectronic Research Archive2688-15942023-07-013195286531210.3934/era.2023269Existence of solutions for Kirchhoff-type systems with critical Sobolev exponents in $ \mathbb{R}^3 $Xing Yi0Shuhou Ye 1College of Mathematics and Statistics, Hunan Normal University, Hunan, ChinaCollege of Mathematics and Statistics, Hunan Normal University, Hunan, China<p>In this paper, we study the following Kirchhoff-type system:</p> <p class="disp_formula">$ \begin{equation} \left\{ \begin{array}{ll} -(a_{1}+b_{1}\int_{\mathbb{R}^{3}}|\nabla u|^{2}dx)\Delta u = \frac{2\alpha}{\alpha+\beta}|u|^{\alpha-2}u|v|^{\beta}+\varepsilon f(x), \\ -(a_{2}+b_{2}\int_{\mathbb{R}^{3}}|\nabla v|^{2}dx)\Delta v = \frac{2\beta}{\alpha+\beta}|u|^{\alpha}|v|^{\beta-2}v+\varepsilon g(x), \\ (u, v)\in D^{1, 2}(\mathbb{R}^{3})\times D^{1, 2}(\mathbb{R}^{3}), \end{array} \right. \end{equation} $</p> <p>where $ a_{1}, a_{2}\geq0, \; b_{1}, b_{2} > 0, \; \alpha, \beta > 1, \; \alpha+\beta = 6 $ and $ f(x), g(x)\geq0, \; f(x), g(x)\in L^{\frac{6}{5}}(\mathbb{R}^3). $ The aim of this paper is to demonstrate the existence of at least two solutions for system (0.1), utilizing the variational method. To achieve this, we construct an energy functional and analyze its critical points by applying the Ekeland variational principle, the mountain pass lemma and the concentration compactness principle.</p>https://www.aimspress.com/article/doi/10.3934/era.2023269?viewType=HTMLpositive solutionskirchhoff-type systemscritical sobolev exponentconcentration compactness principlemountain pass lemma |
| spellingShingle | Xing Yi Shuhou Ye Existence of solutions for Kirchhoff-type systems with critical Sobolev exponents in $ \mathbb{R}^3 $ Electronic Research Archive positive solutions kirchhoff-type systems critical sobolev exponent concentration compactness principle mountain pass lemma |
| title | Existence of solutions for Kirchhoff-type systems with critical Sobolev exponents in $ \mathbb{R}^3 $ |
| title_full | Existence of solutions for Kirchhoff-type systems with critical Sobolev exponents in $ \mathbb{R}^3 $ |
| title_fullStr | Existence of solutions for Kirchhoff-type systems with critical Sobolev exponents in $ \mathbb{R}^3 $ |
| title_full_unstemmed | Existence of solutions for Kirchhoff-type systems with critical Sobolev exponents in $ \mathbb{R}^3 $ |
| title_short | Existence of solutions for Kirchhoff-type systems with critical Sobolev exponents in $ \mathbb{R}^3 $ |
| title_sort | existence of solutions for kirchhoff type systems with critical sobolev exponents in mathbb r 3 |
| topic | positive solutions kirchhoff-type systems critical sobolev exponent concentration compactness principle mountain pass lemma |
| url | https://www.aimspress.com/article/doi/10.3934/era.2023269?viewType=HTML |
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