Existence of infinitely many solutions for degenerate and singular elliptic systems with indefinite concave nonlinearities
In this article, we consider degenerate and singular elliptic systems of the form $$displaylines{ - hbox{div}(h_1(x)abla u) = b_1(x)|u|^{r-2}u + F_u(x,u,v) quad hbox{in } Omega,cr - hbox{div}(h_2(x)abla v) = b_2(x)|v|^{r-2}v + F_v(x,u,v) quad hbox{in } Omega, }$$ where $Omega$ is a bounded d...
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| Format: | Article |
| Language: | English |
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Texas State University
2011-02-01
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| Series: | Electronic Journal of Differential Equations |
| Subjects: | |
| Online Access: | http://ejde.math.txstate.edu/Volumes/2011/30/abstr.html |
| _version_ | 1818758942347493376 |
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| author | Nguyen Thanh Chung |
| author_facet | Nguyen Thanh Chung |
| author_sort | Nguyen Thanh Chung |
| collection | DOAJ |
| description | In this article, we consider degenerate and singular elliptic systems of the form $$displaylines{ - hbox{div}(h_1(x)abla u) = b_1(x)|u|^{r-2}u + F_u(x,u,v) quad hbox{in } Omega,cr - hbox{div}(h_2(x)abla v) = b_2(x)|v|^{r-2}v + F_v(x,u,v) quad hbox{in } Omega, }$$ where $Omega$ is a bounded domain in $mathbb{R}^N$, $N geq 2$, with smooth boundary $partialOmega$; $h_i: Omega o [0, infty)$, $h_i in L^1_{loc}(Omega)$, and are allowed to have ``essential'' zeroes; $1 < r < 2$; the weight functions $b_i: Omega o mathbb{R}$, may be sign-changing; and $(F_u,F_v) = abla F$. Using variational techniques, a variant of the Caffarelli - Kohn - Nirenberg inequality, and a variational principle by Clark [9], we prove the rxistence of infinitely many solutions in a weighted Sobolev space. |
| first_indexed | 2024-12-18T06:34:51Z |
| format | Article |
| id | doaj.art-088a80baa69b4ac18067432f0b7982d0 |
| institution | Directory Open Access Journal |
| issn | 1072-6691 |
| language | English |
| last_indexed | 2024-12-18T06:34:51Z |
| publishDate | 2011-02-01 |
| publisher | Texas State University |
| record_format | Article |
| series | Electronic Journal of Differential Equations |
| spelling | doaj.art-088a80baa69b4ac18067432f0b7982d02022-12-21T21:17:48ZengTexas State UniversityElectronic Journal of Differential Equations1072-66912011-02-01201130,112Existence of infinitely many solutions for degenerate and singular elliptic systems with indefinite concave nonlinearitiesNguyen Thanh ChungIn this article, we consider degenerate and singular elliptic systems of the form $$displaylines{ - hbox{div}(h_1(x)abla u) = b_1(x)|u|^{r-2}u + F_u(x,u,v) quad hbox{in } Omega,cr - hbox{div}(h_2(x)abla v) = b_2(x)|v|^{r-2}v + F_v(x,u,v) quad hbox{in } Omega, }$$ where $Omega$ is a bounded domain in $mathbb{R}^N$, $N geq 2$, with smooth boundary $partialOmega$; $h_i: Omega o [0, infty)$, $h_i in L^1_{loc}(Omega)$, and are allowed to have ``essential'' zeroes; $1 < r < 2$; the weight functions $b_i: Omega o mathbb{R}$, may be sign-changing; and $(F_u,F_v) = abla F$. Using variational techniques, a variant of the Caffarelli - Kohn - Nirenberg inequality, and a variational principle by Clark [9], we prove the rxistence of infinitely many solutions in a weighted Sobolev space.http://ejde.math.txstate.edu/Volumes/2011/30/abstr.htmlDegenerate and singular Elliptic systemweight functionconcave nonlinearityinfinitely many solutions |
| spellingShingle | Nguyen Thanh Chung Existence of infinitely many solutions for degenerate and singular elliptic systems with indefinite concave nonlinearities Electronic Journal of Differential Equations Degenerate and singular Elliptic system weight function concave nonlinearity infinitely many solutions |
| title | Existence of infinitely many solutions for degenerate and singular elliptic systems with indefinite concave nonlinearities |
| title_full | Existence of infinitely many solutions for degenerate and singular elliptic systems with indefinite concave nonlinearities |
| title_fullStr | Existence of infinitely many solutions for degenerate and singular elliptic systems with indefinite concave nonlinearities |
| title_full_unstemmed | Existence of infinitely many solutions for degenerate and singular elliptic systems with indefinite concave nonlinearities |
| title_short | Existence of infinitely many solutions for degenerate and singular elliptic systems with indefinite concave nonlinearities |
| title_sort | existence of infinitely many solutions for degenerate and singular elliptic systems with indefinite concave nonlinearities |
| topic | Degenerate and singular Elliptic system weight function concave nonlinearity infinitely many solutions |
| url | http://ejde.math.txstate.edu/Volumes/2011/30/abstr.html |
| work_keys_str_mv | AT nguyenthanhchung existenceofinfinitelymanysolutionsfordegenerateandsingularellipticsystemswithindefiniteconcavenonlinearities |