Existence and Nonexistence of Positive Solutions for a Weighted Quasilinear Elliptic System
This paper deals with the existence and nonexistence of solutions for the following weighted quasilinear elliptic system, Sμ1,μ2q1,q2−divq1x∇up−2∇u=μ1up−2u+α+1uα−1uvβ+1in Ω−divq2x∇vp−2∇v=μ2vp−2v+β+1uα+1vβ−1vin Ωu>0, v>0in Ωu=v=0on ∂Ω , where Ω⊂ℜNN≥3,2≤p<N, q1, q2∈W1,pΩ∩CΩ¯, α,β≥0, μ1, μ2,...
| Main Authors: | , , |
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| Format: | Article |
| Language: | English |
| Published: |
Hindawi Limited
2023-01-01
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| Series: | Journal of Mathematics |
| Online Access: | http://dx.doi.org/10.1155/2023/8629590 |
| _version_ | 1797851486888132608 |
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| author | Yamina Hamzaoui Atika Matallah Mohammed El Mokhtar Ould El Mokhtar |
| author_facet | Yamina Hamzaoui Atika Matallah Mohammed El Mokhtar Ould El Mokhtar |
| author_sort | Yamina Hamzaoui |
| collection | DOAJ |
| description | This paper deals with the existence and nonexistence of solutions for the following weighted quasilinear elliptic system, Sμ1,μ2q1,q2−divq1x∇up−2∇u=μ1up−2u+α+1uα−1uvβ+1in Ω−divq2x∇vp−2∇v=μ2vp−2v+β+1uα+1vβ−1vin Ωu>0, v>0in Ωu=v=0on ∂Ω , where Ω⊂ℜNN≥3,2≤p<N, q1, q2∈W1,pΩ∩CΩ¯, α,β≥0, μ1, μ2, ≥0 and α,β>0 satisfy α+β=p∗−2 with p∗=pN/N−p is the critical Sobolev exponent. By means of variational methods we prove the existence of positive solutions which depends on the behavior of the weights q1, q2 near their minima and the dimension N. Moreover, we use the well known Pohozaev identity for prove the nonexistence result. |
| first_indexed | 2024-04-09T19:17:32Z |
| format | Article |
| id | doaj.art-12f504ba6cbe4eea88d81f526ed06959 |
| institution | Directory Open Access Journal |
| issn | 2314-4785 |
| language | English |
| last_indexed | 2024-04-09T19:17:32Z |
| publishDate | 2023-01-01 |
| publisher | Hindawi Limited |
| record_format | Article |
| series | Journal of Mathematics |
| spelling | doaj.art-12f504ba6cbe4eea88d81f526ed069592023-04-06T00:00:10ZengHindawi LimitedJournal of Mathematics2314-47852023-01-01202310.1155/2023/8629590Existence and Nonexistence of Positive Solutions for a Weighted Quasilinear Elliptic SystemYamina Hamzaoui0Atika Matallah1Mohammed El Mokhtar Ould El Mokhtar2Higher School of Management-TlemcenHigher School of Management-TlemcenQassim UniversityThis paper deals with the existence and nonexistence of solutions for the following weighted quasilinear elliptic system, Sμ1,μ2q1,q2−divq1x∇up−2∇u=μ1up−2u+α+1uα−1uvβ+1in Ω−divq2x∇vp−2∇v=μ2vp−2v+β+1uα+1vβ−1vin Ωu>0, v>0in Ωu=v=0on ∂Ω , where Ω⊂ℜNN≥3,2≤p<N, q1, q2∈W1,pΩ∩CΩ¯, α,β≥0, μ1, μ2, ≥0 and α,β>0 satisfy α+β=p∗−2 with p∗=pN/N−p is the critical Sobolev exponent. By means of variational methods we prove the existence of positive solutions which depends on the behavior of the weights q1, q2 near their minima and the dimension N. Moreover, we use the well known Pohozaev identity for prove the nonexistence result.http://dx.doi.org/10.1155/2023/8629590 |
| spellingShingle | Yamina Hamzaoui Atika Matallah Mohammed El Mokhtar Ould El Mokhtar Existence and Nonexistence of Positive Solutions for a Weighted Quasilinear Elliptic System Journal of Mathematics |
| title | Existence and Nonexistence of Positive Solutions for a Weighted Quasilinear Elliptic System |
| title_full | Existence and Nonexistence of Positive Solutions for a Weighted Quasilinear Elliptic System |
| title_fullStr | Existence and Nonexistence of Positive Solutions for a Weighted Quasilinear Elliptic System |
| title_full_unstemmed | Existence and Nonexistence of Positive Solutions for a Weighted Quasilinear Elliptic System |
| title_short | Existence and Nonexistence of Positive Solutions for a Weighted Quasilinear Elliptic System |
| title_sort | existence and nonexistence of positive solutions for a weighted quasilinear elliptic system |
| url | http://dx.doi.org/10.1155/2023/8629590 |
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