Nonnegative solutions for indefinite Caffarelli–Kohn–Nirenberg-type problems with subcritical or critical growth
Abstract Using variational methods we prove the existence of nonnegative solutions for the following class of quasilinear problems given by: − div ( | x | − ϒ p | ∇ u | p − 2 ∇ u ) + | x | − b p ∗ | u | p − 2 u = λ | x | − b p ∗ a ( x ) g ( u ) + γ | x | − b p ∗ | u | p ∗ − 2 u in R N , for the subc...
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| Format: | Article |
| Language: | English |
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SpringerOpen
2024-11-01
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| Series: | Boundary Value Problems |
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| Online Access: | https://doi.org/10.1186/s13661-024-01969-6 |
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| author | Sami Baraket Anis Ben Ghorbal Giovany M. Figueiredo |
| author_facet | Sami Baraket Anis Ben Ghorbal Giovany M. Figueiredo |
| author_sort | Sami Baraket |
| collection | DOAJ |
| description | Abstract Using variational methods we prove the existence of nonnegative solutions for the following class of quasilinear problems given by: − div ( | x | − ϒ p | ∇ u | p − 2 ∇ u ) + | x | − b p ∗ | u | p − 2 u = λ | x | − b p ∗ a ( x ) g ( u ) + γ | x | − b p ∗ | u | p ∗ − 2 u in R N , for the subcritical case ( γ = 0 $\gamma =0$ ) and also for the critical case ( γ = 1 $\gamma =1$ ). The functions a : R N → R and g : R → R are continuous functions that satisfy some additional conditions, 1 < p < N $1 < p < N$ , 0 ≤ ϒ < N − p p $0 \leq \Upsilon < \frac{N-p}{p}$ , ϒ < b ≤ ϒ + 1 $\Upsilon < b \leq \Upsilon +1$ , p ∗ = p ∗ ( ϒ , b ) = p N N − d p $p^{*}=p^{*}(\Upsilon ,b)=\frac{pN}{N -d p}$ with d = 1 + ϒ − b $d = 1 + \Upsilon - b$ . |
| format | Article |
| id | doaj-art-3dfa823be3d44cbb8c43e8b40259dbc8 |
| institution | OA Journals |
| issn | 1687-2770 |
| language | English |
| publishDate | 2024-11-01 |
| publisher | SpringerOpen |
| record_format | Article |
| series | Boundary Value Problems |
| spelling | doaj-art-3dfa823be3d44cbb8c43e8b40259dbc82025-08-20T02:33:00ZengSpringerOpenBoundary Value Problems1687-27702024-11-012024111210.1186/s13661-024-01969-6Nonnegative solutions for indefinite Caffarelli–Kohn–Nirenberg-type problems with subcritical or critical growthSami Baraket0Anis Ben Ghorbal1Giovany M. Figueiredo2Department of Mathematics and Statistics, College of Science, Imam Mohammad Ibn Saud Islamic University (IMSIU)Department of Mathematics and Statistics, College of Science, Imam Mohammad Ibn Saud Islamic University (IMSIU)Departamento de Matemática CEP, Universidade de BrasíliaAbstract Using variational methods we prove the existence of nonnegative solutions for the following class of quasilinear problems given by: − div ( | x | − ϒ p | ∇ u | p − 2 ∇ u ) + | x | − b p ∗ | u | p − 2 u = λ | x | − b p ∗ a ( x ) g ( u ) + γ | x | − b p ∗ | u | p ∗ − 2 u in R N , for the subcritical case ( γ = 0 $\gamma =0$ ) and also for the critical case ( γ = 1 $\gamma =1$ ). The functions a : R N → R and g : R → R are continuous functions that satisfy some additional conditions, 1 < p < N $1 < p < N$ , 0 ≤ ϒ < N − p p $0 \leq \Upsilon < \frac{N-p}{p}$ , ϒ < b ≤ ϒ + 1 $\Upsilon < b \leq \Upsilon +1$ , p ∗ = p ∗ ( ϒ , b ) = p N N − d p $p^{*}=p^{*}(\Upsilon ,b)=\frac{pN}{N -d p}$ with d = 1 + ϒ − b $d = 1 + \Upsilon - b$ .https://doi.org/10.1186/s13661-024-01969-6Variational methodsCaffarelliKohnNirenberg problemNonnegative solutions |
| spellingShingle | Sami Baraket Anis Ben Ghorbal Giovany M. Figueiredo Nonnegative solutions for indefinite Caffarelli–Kohn–Nirenberg-type problems with subcritical or critical growth Boundary Value Problems Variational methods Caffarelli Kohn Nirenberg problem Nonnegative solutions |
| title | Nonnegative solutions for indefinite Caffarelli–Kohn–Nirenberg-type problems with subcritical or critical growth |
| title_full | Nonnegative solutions for indefinite Caffarelli–Kohn–Nirenberg-type problems with subcritical or critical growth |
| title_fullStr | Nonnegative solutions for indefinite Caffarelli–Kohn–Nirenberg-type problems with subcritical or critical growth |
| title_full_unstemmed | Nonnegative solutions for indefinite Caffarelli–Kohn–Nirenberg-type problems with subcritical or critical growth |
| title_short | Nonnegative solutions for indefinite Caffarelli–Kohn–Nirenberg-type problems with subcritical or critical growth |
| title_sort | nonnegative solutions for indefinite caffarelli kohn nirenberg type problems with subcritical or critical growth |
| topic | Variational methods Caffarelli Kohn Nirenberg problem Nonnegative solutions |
| url | https://doi.org/10.1186/s13661-024-01969-6 |
| work_keys_str_mv | AT samibaraket nonnegativesolutionsforindefinitecaffarellikohnnirenbergtypeproblemswithsubcriticalorcriticalgrowth AT anisbenghorbal nonnegativesolutionsforindefinitecaffarellikohnnirenbergtypeproblemswithsubcriticalorcriticalgrowth AT giovanymfigueiredo nonnegativesolutionsforindefinitecaffarellikohnnirenbergtypeproblemswithsubcriticalorcriticalgrowth |