Least energy sign-changing solutions for a nonlocal anisotropic Kirchhoff type equation
In this paper, we investigate the existence of sign-changing solutions for the following class of fractional Kirchhoff type equations with potential (1+b[u]α2)((-Δx)αu-Δyu)+V(x,y)u=f(x,y,u),(x,y)∈ℝN=ℝn×ℝm,\left( {1 + b\left[ u \right]_\alpha ^2} \right)\left( {{{\left( { - {\Delta _x}} \right)}^\alp...
| Main Authors: | , , , |
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| Format: | Article |
| Language: | English |
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Sciendo
2022-05-01
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| Series: | Moroccan Journal of Pure and Applied Analysis |
| Subjects: | |
| Online Access: | https://doi.org/10.2478/mjpaa-2022-0015 |
| _version_ | 1811253894735462400 |
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| author | Rahmani Mohammed Rahmani Mostafa Anane Aomar Massar Mohammed |
| author_facet | Rahmani Mohammed Rahmani Mostafa Anane Aomar Massar Mohammed |
| author_sort | Rahmani Mohammed |
| collection | DOAJ |
| description | In this paper, we investigate the existence of sign-changing solutions for the following class of fractional Kirchhoff type equations with potential
(1+b[u]α2)((-Δx)αu-Δyu)+V(x,y)u=f(x,y,u),(x,y)∈ℝN=ℝn×ℝm,\left( {1 + b\left[ u \right]_\alpha ^2} \right)\left( {{{\left( { - {\Delta _x}} \right)}^\alpha }u - {\Delta _y}u} \right) + V\left( {x,y} \right)u = f\left( {x,y,u} \right),\left( {x,y} \right) \in {\mathbb{R}^N} = {\mathbb{R}^n} \times {\mathbb{R}^m},
where [u]α=(∫ℝN(|(-Δx)α2u|2+|∇yu|2)dxdy)12{\left[ u \right]_\alpha } = {\left( {\int {_{{\mathbb{R}^N}}\left( {{{\left| {{{\left( { - {\Delta _x}} \right)}^{{\alpha \over 2}}}u} \right|}^2} + {{\left| {{\nabla _y}u} \right|}^2}} \right)dxdy} } \right)^{{1 \over 2}}}. Based on variational approach and a variant of the quantitative strain lemma, for each b > 0, we show the existence of a least energy nodal solution ub. In addition, a convergence property of ub as b ↘ 0 is established. |
| first_indexed | 2024-04-12T16:58:45Z |
| format | Article |
| id | doaj.art-b6aa5035f7ed4b01b1cac7845d8591d4 |
| institution | Directory Open Access Journal |
| issn | 2351-8227 |
| language | English |
| last_indexed | 2024-04-12T16:58:45Z |
| publishDate | 2022-05-01 |
| publisher | Sciendo |
| record_format | Article |
| series | Moroccan Journal of Pure and Applied Analysis |
| spelling | doaj.art-b6aa5035f7ed4b01b1cac7845d8591d42022-12-22T03:24:08ZengSciendoMoroccan Journal of Pure and Applied Analysis2351-82272022-05-018221222710.2478/mjpaa-2022-0015Least energy sign-changing solutions for a nonlocal anisotropic Kirchhoff type equationRahmani Mohammed0Rahmani Mostafa1Anane Aomar2Massar Mohammed3Department of Mathematics, Oriental Applied Mathematics Laboratory(LAMO), FSO,Mohamed first University, Morocco.Department of Mathematics, Oriental Applied Mathematics Laboratory(LAMO), FSO,Mohamed first University, Morocco.Department of Mathematics, Oriental Applied Mathematics Laboratory(LAMO), FSO,Mohamed first University, Morocco.Department of Mathematics, FTSH, Abdelmalek Essaadi University, Morocco.In this paper, we investigate the existence of sign-changing solutions for the following class of fractional Kirchhoff type equations with potential (1+b[u]α2)((-Δx)αu-Δyu)+V(x,y)u=f(x,y,u),(x,y)∈ℝN=ℝn×ℝm,\left( {1 + b\left[ u \right]_\alpha ^2} \right)\left( {{{\left( { - {\Delta _x}} \right)}^\alpha }u - {\Delta _y}u} \right) + V\left( {x,y} \right)u = f\left( {x,y,u} \right),\left( {x,y} \right) \in {\mathbb{R}^N} = {\mathbb{R}^n} \times {\mathbb{R}^m}, where [u]α=(∫ℝN(|(-Δx)α2u|2+|∇yu|2)dxdy)12{\left[ u \right]_\alpha } = {\left( {\int {_{{\mathbb{R}^N}}\left( {{{\left| {{{\left( { - {\Delta _x}} \right)}^{{\alpha \over 2}}}u} \right|}^2} + {{\left| {{\nabla _y}u} \right|}^2}} \right)dxdy} } \right)^{{1 \over 2}}}. Based on variational approach and a variant of the quantitative strain lemma, for each b > 0, we show the existence of a least energy nodal solution ub. In addition, a convergence property of ub as b ↘ 0 is established.https://doi.org/10.2478/mjpaa-2022-0015kirchhoff equationsbo-zk operatorsign-changing solutionsdeformation lemma35r1135d30 |
| spellingShingle | Rahmani Mohammed Rahmani Mostafa Anane Aomar Massar Mohammed Least energy sign-changing solutions for a nonlocal anisotropic Kirchhoff type equation Moroccan Journal of Pure and Applied Analysis kirchhoff equations bo-zk operator sign-changing solutions deformation lemma 35r11 35d30 |
| title | Least energy sign-changing solutions for a nonlocal anisotropic Kirchhoff type equation |
| title_full | Least energy sign-changing solutions for a nonlocal anisotropic Kirchhoff type equation |
| title_fullStr | Least energy sign-changing solutions for a nonlocal anisotropic Kirchhoff type equation |
| title_full_unstemmed | Least energy sign-changing solutions for a nonlocal anisotropic Kirchhoff type equation |
| title_short | Least energy sign-changing solutions for a nonlocal anisotropic Kirchhoff type equation |
| title_sort | least energy sign changing solutions for a nonlocal anisotropic kirchhoff type equation |
| topic | kirchhoff equations bo-zk operator sign-changing solutions deformation lemma 35r11 35d30 |
| url | https://doi.org/10.2478/mjpaa-2022-0015 |
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