Multiplicity result for mixed local and nonlocal Kirchhoff problems involving critical growth
In this paper, we study the multiplicity of nonnegative solutions for the following nonlocal elliptic problem \[\begin{cases}M\Big(\ \int_{\mathbb{R}^N}|\nabla u|^2dx+\iint_{\mathbb{R}^{2N}}\frac{|u(x)-u(y)|^2}{|x-y|^{N+2s}}dxdy\Big)\mathcal{L}(u) \\ = \lambda {f(x)}|u|^{p-2}u+|u|^{2^*-2}u &\tex...
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AGH Univeristy of Science and Technology Press
2025-07-01
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| Series: | Opuscula Mathematica |
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| Online Access: | https://www.opuscula.agh.edu.pl/vol45/4/art/opuscula_math_4524.pdf |
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| author | Vinayak Mani Tripathi |
| author_facet | Vinayak Mani Tripathi |
| author_sort | Vinayak Mani Tripathi |
| collection | DOAJ |
| description | In this paper, we study the multiplicity of nonnegative solutions for the following nonlocal elliptic problem \[\begin{cases}M\Big(\ \int_{\mathbb{R}^N}|\nabla u|^2dx+\iint_{\mathbb{R}^{2N}}\frac{|u(x)-u(y)|^2}{|x-y|^{N+2s}}dxdy\Big)\mathcal{L}(u) \\ = \lambda {f(x)}|u|^{p-2}u+|u|^{2^*-2}u &\text{ in }\Omega, \\ u=0 &\text{ on }\mathbb R^N\setminus \Omega, \end{cases}\] where \(\Omega\subset\mathbb{R}^N\) is bounded domain with smooth boundary, \(1\lt p\lt 2\lt 2^*=\frac{2N}{N-2}\), \(N\geq 3\), \(\lambda\gt 0\), \(M\) is a Kirchhoff coefficient and \(\mathcal{L}\) denotes the mixed local and nonlocal operator. The weight function \(f\in L^{\frac{2^*}{2^*-p}}(\Omega)\) is allowed to change sign. By applying variational approach based on constrained minimization argument, we show the existence of at least two nonnegative solutions. |
| format | Article |
| id | doaj-art-fe1cf5553a1744839b199f876dce768c |
| institution | DOAJ |
| issn | 1232-9274 |
| language | English |
| publishDate | 2025-07-01 |
| publisher | AGH Univeristy of Science and Technology Press |
| record_format | Article |
| series | Opuscula Mathematica |
| spelling | doaj-art-fe1cf5553a1744839b199f876dce768c2025-08-20T02:40:29ZengAGH Univeristy of Science and Technology PressOpuscula Mathematica1232-92742025-07-01454523542https://doi.org/10.7494/OpMath.2025.45.4.5234524Multiplicity result for mixed local and nonlocal Kirchhoff problems involving critical growthVinayak Mani Tripathi0https://orcid.org/0009-0007-1710-3443Indian Institute of Technology Bhilai, Department of Mathematics, 491002, Durg, Chhattisgarh, IndiaIn this paper, we study the multiplicity of nonnegative solutions for the following nonlocal elliptic problem \[\begin{cases}M\Big(\ \int_{\mathbb{R}^N}|\nabla u|^2dx+\iint_{\mathbb{R}^{2N}}\frac{|u(x)-u(y)|^2}{|x-y|^{N+2s}}dxdy\Big)\mathcal{L}(u) \\ = \lambda {f(x)}|u|^{p-2}u+|u|^{2^*-2}u &\text{ in }\Omega, \\ u=0 &\text{ on }\mathbb R^N\setminus \Omega, \end{cases}\] where \(\Omega\subset\mathbb{R}^N\) is bounded domain with smooth boundary, \(1\lt p\lt 2\lt 2^*=\frac{2N}{N-2}\), \(N\geq 3\), \(\lambda\gt 0\), \(M\) is a Kirchhoff coefficient and \(\mathcal{L}\) denotes the mixed local and nonlocal operator. The weight function \(f\in L^{\frac{2^*}{2^*-p}}(\Omega)\) is allowed to change sign. By applying variational approach based on constrained minimization argument, we show the existence of at least two nonnegative solutions.https://www.opuscula.agh.edu.pl/vol45/4/art/opuscula_math_4524.pdfmixed local and nonlocal operatorskirchhoff type problemcritical nonlinearitynehari manifold |
| spellingShingle | Vinayak Mani Tripathi Multiplicity result for mixed local and nonlocal Kirchhoff problems involving critical growth Opuscula Mathematica mixed local and nonlocal operators kirchhoff type problem critical nonlinearity nehari manifold |
| title | Multiplicity result for mixed local and nonlocal Kirchhoff problems involving critical growth |
| title_full | Multiplicity result for mixed local and nonlocal Kirchhoff problems involving critical growth |
| title_fullStr | Multiplicity result for mixed local and nonlocal Kirchhoff problems involving critical growth |
| title_full_unstemmed | Multiplicity result for mixed local and nonlocal Kirchhoff problems involving critical growth |
| title_short | Multiplicity result for mixed local and nonlocal Kirchhoff problems involving critical growth |
| title_sort | multiplicity result for mixed local and nonlocal kirchhoff problems involving critical growth |
| topic | mixed local and nonlocal operators kirchhoff type problem critical nonlinearity nehari manifold |
| url | https://www.opuscula.agh.edu.pl/vol45/4/art/opuscula_math_4524.pdf |
| work_keys_str_mv | AT vinayakmanitripathi multiplicityresultformixedlocalandnonlocalkirchhoffproblemsinvolvingcriticalgrowth |