Multiplicity of Solutions for a Class of Elliptic Problem of p-Laplacian Type with a p-Gradient Term
We consider the following problem: -Δpu=c(x)|u|q-1u+μ|∇u|p+h(x) in Ω, u=0 on ∂Ω, where Ω is a bounded set in RN (N≥3) with a smooth boundary, 1<p<N, q>0, μ∈R⁎, and c and h belong to Lk(Ω) for some k>N/p. In this paper, we assume that c≩0 a.e. in Ω and h without sign condition and th...
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| Format: | Article |
| Language: | English |
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Wiley
2019-01-01
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| Series: | International Journal of Mathematics and Mathematical Sciences |
| Online Access: | http://dx.doi.org/10.1155/2019/6824502 |
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| author | Zakariya Chaouai Soufiane Maatouk |
| author_facet | Zakariya Chaouai Soufiane Maatouk |
| author_sort | Zakariya Chaouai |
| collection | DOAJ |
| description | We consider the following problem: -Δpu=c(x)|u|q-1u+μ|∇u|p+h(x) in Ω, u=0 on ∂Ω, where Ω is a bounded set in RN (N≥3) with a smooth boundary, 1<p<N, q>0, μ∈R⁎, and c and h belong to Lk(Ω) for some k>N/p. In this paper, we assume that c≩0 a.e. in Ω and h without sign condition and then we prove the existence of at least two bounded solutions under the condition that ck and hk are suitably small. For this purpose, we use the Mountain Pass theorem, on an equivalent problem to (P) with variational structure. Here, the main difficulty is that the nonlinearity term considered does not satisfy Ambrosetti and Rabinowitz condition. The key idea is to replace the former condition by the nonquadraticity condition at infinity. |
| format | Article |
| id | doaj-art-5cd4ed0e4653456784c6183598943b2f |
| institution | Kabale University |
| issn | 0161-1712 1687-0425 |
| language | English |
| publishDate | 2019-01-01 |
| publisher | Wiley |
| record_format | Article |
| series | International Journal of Mathematics and Mathematical Sciences |
| spelling | doaj-art-5cd4ed0e4653456784c6183598943b2f2025-02-03T01:31:28ZengWileyInternational Journal of Mathematics and Mathematical Sciences0161-17121687-04252019-01-01201910.1155/2019/68245026824502Multiplicity of Solutions for a Class of Elliptic Problem of p-Laplacian Type with a p-Gradient TermZakariya Chaouai0Soufiane Maatouk1Center of Mathematical Research and Applications of Rabat (CeReMAR), Laboratory of Mathematical Analysis and Applications (LAMA), Department of Mathematics, Faculty of Sciences, Mohammed V University, P.O. Box 1014, Rabat, MoroccoCenter of Mathematical Research and Applications of Rabat (CeReMAR), Laboratory of Mathematical Analysis and Applications (LAMA), Department of Mathematics, Faculty of Sciences, Mohammed V University, P.O. Box 1014, Rabat, MoroccoWe consider the following problem: -Δpu=c(x)|u|q-1u+μ|∇u|p+h(x) in Ω, u=0 on ∂Ω, where Ω is a bounded set in RN (N≥3) with a smooth boundary, 1<p<N, q>0, μ∈R⁎, and c and h belong to Lk(Ω) for some k>N/p. In this paper, we assume that c≩0 a.e. in Ω and h without sign condition and then we prove the existence of at least two bounded solutions under the condition that ck and hk are suitably small. For this purpose, we use the Mountain Pass theorem, on an equivalent problem to (P) with variational structure. Here, the main difficulty is that the nonlinearity term considered does not satisfy Ambrosetti and Rabinowitz condition. The key idea is to replace the former condition by the nonquadraticity condition at infinity.http://dx.doi.org/10.1155/2019/6824502 |
| spellingShingle | Zakariya Chaouai Soufiane Maatouk Multiplicity of Solutions for a Class of Elliptic Problem of p-Laplacian Type with a p-Gradient Term International Journal of Mathematics and Mathematical Sciences |
| title | Multiplicity of Solutions for a Class of Elliptic Problem of p-Laplacian Type with a p-Gradient Term |
| title_full | Multiplicity of Solutions for a Class of Elliptic Problem of p-Laplacian Type with a p-Gradient Term |
| title_fullStr | Multiplicity of Solutions for a Class of Elliptic Problem of p-Laplacian Type with a p-Gradient Term |
| title_full_unstemmed | Multiplicity of Solutions for a Class of Elliptic Problem of p-Laplacian Type with a p-Gradient Term |
| title_short | Multiplicity of Solutions for a Class of Elliptic Problem of p-Laplacian Type with a p-Gradient Term |
| title_sort | multiplicity of solutions for a class of elliptic problem of p laplacian type with a p gradient term |
| url | http://dx.doi.org/10.1155/2019/6824502 |
| work_keys_str_mv | AT zakariyachaouai multiplicityofsolutionsforaclassofellipticproblemofplaplaciantypewithapgradientterm AT soufianemaatouk multiplicityofsolutionsforaclassofellipticproblemofplaplaciantypewithapgradientterm |