On Existence of Solution for a Class of Semilinear Elliptic Equations with Nonlinearities That Lies between Different Powers
We prove that the semilinear elliptic equation −Δu=f(u), in Ω, u=0, on ∂Ω has a positive solution when the nonlinearity f belongs to a class which satisfies μtq≤f(t)≤Ctp at infinity and behaves like tq near the origin, where 1<q<(N+2)/(N−2) if N≥3 and 1<q<+∞ if N=1,2. In our approach, we...
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| Format: | Article |
| Language: | English |
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Wiley
2008-01-01
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| Series: | Abstract and Applied Analysis |
| Online Access: | http://dx.doi.org/10.1155/2008/578417 |
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| author | Claudianor O. Alves Marco A. S. Souto |
| author_facet | Claudianor O. Alves Marco A. S. Souto |
| author_sort | Claudianor O. Alves |
| collection | DOAJ |
| description | We prove that the semilinear elliptic equation −Δu=f(u), in Ω, u=0, on ∂Ω has a positive solution when the nonlinearity f belongs to a class which satisfies μtq≤f(t)≤Ctp at infinity and behaves like tq near the origin, where 1<q<(N+2)/(N−2) if N≥3 and 1<q<+∞ if N=1,2. In our approach, we do not need the Ambrosetti-Rabinowitz condition, and the nonlinearity does not satisfy any hypotheses such those required by the blowup method. Furthermore, we do not impose any restriction on the growth of p. |
| format | Article |
| id | doaj-art-312c1ce3baeb4d87980067dcaac7576b |
| institution | Kabale University |
| issn | 1085-3375 1687-0409 |
| language | English |
| publishDate | 2008-01-01 |
| publisher | Wiley |
| record_format | Article |
| series | Abstract and Applied Analysis |
| spelling | doaj-art-312c1ce3baeb4d87980067dcaac7576b2025-02-03T01:29:16ZengWileyAbstract and Applied Analysis1085-33751687-04092008-01-01200810.1155/2008/578417578417On Existence of Solution for a Class of Semilinear Elliptic Equations with Nonlinearities That Lies between Different PowersClaudianor O. Alves0Marco A. S. Souto1Departamento de Matemática e Estatística, Universidade Federal de Campina Grande, CEP 58.109-970, Campina Grande - PB, BrazilDepartamento de Matemática e Estatística, Universidade Federal de Campina Grande, CEP 58.109-970, Campina Grande - PB, BrazilWe prove that the semilinear elliptic equation −Δu=f(u), in Ω, u=0, on ∂Ω has a positive solution when the nonlinearity f belongs to a class which satisfies μtq≤f(t)≤Ctp at infinity and behaves like tq near the origin, where 1<q<(N+2)/(N−2) if N≥3 and 1<q<+∞ if N=1,2. In our approach, we do not need the Ambrosetti-Rabinowitz condition, and the nonlinearity does not satisfy any hypotheses such those required by the blowup method. Furthermore, we do not impose any restriction on the growth of p.http://dx.doi.org/10.1155/2008/578417 |
| spellingShingle | Claudianor O. Alves Marco A. S. Souto On Existence of Solution for a Class of Semilinear Elliptic Equations with Nonlinearities That Lies between Different Powers Abstract and Applied Analysis |
| title | On Existence of Solution for a Class of Semilinear Elliptic Equations with Nonlinearities That Lies between Different Powers |
| title_full | On Existence of Solution for a Class of Semilinear Elliptic Equations with Nonlinearities That Lies between Different Powers |
| title_fullStr | On Existence of Solution for a Class of Semilinear Elliptic Equations with Nonlinearities That Lies between Different Powers |
| title_full_unstemmed | On Existence of Solution for a Class of Semilinear Elliptic Equations with Nonlinearities That Lies between Different Powers |
| title_short | On Existence of Solution for a Class of Semilinear Elliptic Equations with Nonlinearities That Lies between Different Powers |
| title_sort | on existence of solution for a class of semilinear elliptic equations with nonlinearities that lies between different powers |
| url | http://dx.doi.org/10.1155/2008/578417 |
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