Existence, uniqueness, localization and minimization property of positive solutions for non-local problems involving discontinuous Kirchhoff functions
Let Ω⊂Rn\Omega \subset {{\bf{R}}}^{n} be a smooth bounded domain. In this article, we prove a result of which the following is a by-product: Let q∈]0,1[q\in ]0,1{[}, α∈L∞(Ω)\alpha \in {L}^{\infty }\left(\Omega ), with α>0\alpha \gt 0, and k∈Nk\in {\bf{N}}. Then, the problem −tan∫Ω∣∇u(x)∣2dxΔu=α(x...
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De Gruyter
2024-02-01
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| Series: | Advances in Nonlinear Analysis |
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| Online Access: | https://doi.org/10.1515/anona-2023-0104 |
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| author | Ricceri Biagio |
| author_facet | Ricceri Biagio |
| author_sort | Ricceri Biagio |
| collection | DOAJ |
| description | Let Ω⊂Rn\Omega \subset {{\bf{R}}}^{n} be a smooth bounded domain. In this article, we prove a result of which the following is a by-product: Let q∈]0,1[q\in ]0,1{[}, α∈L∞(Ω)\alpha \in {L}^{\infty }\left(\Omega ), with α>0\alpha \gt 0, and k∈Nk\in {\bf{N}}. Then, the problem −tan∫Ω∣∇u(x)∣2dxΔu=α(x)uqinΩu>0inΩu=0on∂Ω(k−1)π<∫Ω∣∇u(x)∣2dx<(k−1)π+π2\left\{\begin{array}{ll}-\tan \left(\mathop{\displaystyle \int }\limits_{\Omega }| \nabla u\left(x){| }^{2}{\rm{d}}x\right)\Delta u=\alpha \left(x){u}^{q}\hspace{1.0em}& \hspace{0.1em}\text{in}\hspace{0.1em}\hspace{0.33em}\Omega \\ u\gt 0\hspace{1.0em}& \hspace{0.1em}\text{in}\hspace{0.1em}\hspace{0.33em}\Omega \\ u=0\hspace{1.0em}& \hspace{0.1em}\text{on}\hspace{0.1em}\hspace{0.33em}\partial \Omega \\ \left(k-1)\pi \lt \mathop{\displaystyle \int }\limits_{\Omega }| \nabla u\left(x){| }^{2}{\rm{d}}x\lt \left(k-1)\pi +\frac{\pi }{2}\hspace{1.0em}\end{array}\right. has a unique weak solution u˜\tilde{u}, which is the unique global minimum in H01(Ω){H}_{0}^{1}\left(\Omega ) of the functional u→12tan∫Ω∣∇u˜(x)∣2dx∫Ω∣∇u(x)∣2dx−1q+1∫Ωα(x)∣u+(x)∣q+1dx,u\to \frac{1}{2}\tan \left(\mathop{\int }\limits_{\Omega }| \nabla \tilde{u}\left(x){| }^{2}{\rm{d}}x\right)\mathop{\int }\limits_{\Omega }| \nabla u\left(x){| }^{2}{\rm{d}}x-\frac{1}{q+1}\mathop{\int }\limits_{\Omega }\alpha \left(x)| {u}^{+}\left(x){| }^{q+1}{\rm{d}}x, where u+=max{0,u}{u}^{+}=\max \left\{0,u\right\}. |
| first_indexed | 2024-03-08T03:21:12Z |
| format | Article |
| id | doaj.art-0503aa7b5c7946e4a13ddb7eb9c100fa |
| institution | Directory Open Access Journal |
| issn | 2191-950X |
| language | English |
| last_indexed | 2024-03-08T03:21:12Z |
| publishDate | 2024-02-01 |
| publisher | De Gruyter |
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| series | Advances in Nonlinear Analysis |
| spelling | doaj.art-0503aa7b5c7946e4a13ddb7eb9c100fa2024-02-12T09:11:36ZengDe GruyterAdvances in Nonlinear Analysis2191-950X2024-02-01131435610.1515/anona-2023-0104Existence, uniqueness, localization and minimization property of positive solutions for non-local problems involving discontinuous Kirchhoff functionsRicceri Biagio0Department of Mathematics and Informatics, University of Catania, Viale A. Doria 6, 95125 Catania, ItalyLet Ω⊂Rn\Omega \subset {{\bf{R}}}^{n} be a smooth bounded domain. In this article, we prove a result of which the following is a by-product: Let q∈]0,1[q\in ]0,1{[}, α∈L∞(Ω)\alpha \in {L}^{\infty }\left(\Omega ), with α>0\alpha \gt 0, and k∈Nk\in {\bf{N}}. Then, the problem −tan∫Ω∣∇u(x)∣2dxΔu=α(x)uqinΩu>0inΩu=0on∂Ω(k−1)π<∫Ω∣∇u(x)∣2dx<(k−1)π+π2\left\{\begin{array}{ll}-\tan \left(\mathop{\displaystyle \int }\limits_{\Omega }| \nabla u\left(x){| }^{2}{\rm{d}}x\right)\Delta u=\alpha \left(x){u}^{q}\hspace{1.0em}& \hspace{0.1em}\text{in}\hspace{0.1em}\hspace{0.33em}\Omega \\ u\gt 0\hspace{1.0em}& \hspace{0.1em}\text{in}\hspace{0.1em}\hspace{0.33em}\Omega \\ u=0\hspace{1.0em}& \hspace{0.1em}\text{on}\hspace{0.1em}\hspace{0.33em}\partial \Omega \\ \left(k-1)\pi \lt \mathop{\displaystyle \int }\limits_{\Omega }| \nabla u\left(x){| }^{2}{\rm{d}}x\lt \left(k-1)\pi +\frac{\pi }{2}\hspace{1.0em}\end{array}\right. has a unique weak solution u˜\tilde{u}, which is the unique global minimum in H01(Ω){H}_{0}^{1}\left(\Omega ) of the functional u→12tan∫Ω∣∇u˜(x)∣2dx∫Ω∣∇u(x)∣2dx−1q+1∫Ωα(x)∣u+(x)∣q+1dx,u\to \frac{1}{2}\tan \left(\mathop{\int }\limits_{\Omega }| \nabla \tilde{u}\left(x){| }^{2}{\rm{d}}x\right)\mathop{\int }\limits_{\Omega }| \nabla u\left(x){| }^{2}{\rm{d}}x-\frac{1}{q+1}\mathop{\int }\limits_{\Omega }\alpha \left(x)| {u}^{+}\left(x){| }^{q+1}{\rm{d}}x, where u+=max{0,u}{u}^{+}=\max \left\{0,u\right\}.https://doi.org/10.1515/anona-2023-0104discontinuous kirchhoff functionexistenceuniquenesslocalizationminimization property35j1535j2535j6149j35 |
| spellingShingle | Ricceri Biagio Existence, uniqueness, localization and minimization property of positive solutions for non-local problems involving discontinuous Kirchhoff functions Advances in Nonlinear Analysis discontinuous kirchhoff function existence uniqueness localization minimization property 35j15 35j25 35j61 49j35 |
| title | Existence, uniqueness, localization and minimization property of positive solutions for non-local problems involving discontinuous Kirchhoff functions |
| title_full | Existence, uniqueness, localization and minimization property of positive solutions for non-local problems involving discontinuous Kirchhoff functions |
| title_fullStr | Existence, uniqueness, localization and minimization property of positive solutions for non-local problems involving discontinuous Kirchhoff functions |
| title_full_unstemmed | Existence, uniqueness, localization and minimization property of positive solutions for non-local problems involving discontinuous Kirchhoff functions |
| title_short | Existence, uniqueness, localization and minimization property of positive solutions for non-local problems involving discontinuous Kirchhoff functions |
| title_sort | existence uniqueness localization and minimization property of positive solutions for non local problems involving discontinuous kirchhoff functions |
| topic | discontinuous kirchhoff function existence uniqueness localization minimization property 35j15 35j25 35j61 49j35 |
| url | https://doi.org/10.1515/anona-2023-0104 |
| work_keys_str_mv | AT ricceribiagio existenceuniquenesslocalizationandminimizationpropertyofpositivesolutionsfornonlocalproblemsinvolvingdiscontinuouskirchhofffunctions |