Existence Results for Quasilinear Elliptic Equations with Indefinite Weight
We provide the existence of a solution for quasilinear elliptic equation −div(𝑎∞(𝑥)|∇𝑢|𝑝−2∇𝑢+̃𝑎(𝑥,|∇𝑢|)∇𝑢)=𝜆𝑚(𝑥)|𝑢|𝑝−2𝑢+𝑓(𝑥,𝑢)+ℎ(𝑥) in Ω under the Neumann boundary condition. Here, we consider the condition that ̃𝑎(𝑥,𝑡)=𝑜(𝑡𝑝−2) as 𝑡→+∞ and 𝑓(𝑥,𝑢)=𝑜(|𝑢|𝑝−1) as |𝑢|→∞. As a special case, our result imp...
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| Format: | Article |
| Language: | English |
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Wiley
2012-01-01
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| Series: | Abstract and Applied Analysis |
| Online Access: | http://dx.doi.org/10.1155/2012/568120 |
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| author | Mieko Tanaka |
| author_facet | Mieko Tanaka |
| author_sort | Mieko Tanaka |
| collection | DOAJ |
| description | We provide the existence of a solution for quasilinear elliptic equation −div(𝑎∞(𝑥)|∇𝑢|𝑝−2∇𝑢+̃𝑎(𝑥,|∇𝑢|)∇𝑢)=𝜆𝑚(𝑥)|𝑢|𝑝−2𝑢+𝑓(𝑥,𝑢)+ℎ(𝑥) in Ω under the Neumann boundary condition. Here, we consider the condition that ̃𝑎(𝑥,𝑡)=𝑜(𝑡𝑝−2) as 𝑡→+∞ and 𝑓(𝑥,𝑢)=𝑜(|𝑢|𝑝−1) as |𝑢|→∞. As a special case, our result implies that the following 𝑝-Laplace equation has at least one solution: −Δ𝑝𝑢=𝜆𝑚(𝑥)|𝑢|𝑝−2𝑢+𝜇|𝑢|𝑟−2𝑢+ℎ(𝑥) in Ω,𝜕𝑢/𝜕𝜈=0 on 𝜕Ω for every 1<𝑟<𝑝<∞, 𝜆∈ℝ, 𝜇≠0 and 𝑚,ℎ∈𝐿∞(Ω) with ∫Ω𝑚𝑑𝑥≠0. Moreover, in the nonresonant case, that is, 𝜆 is not an eigenvalue of the 𝑝-Laplacian with weight 𝑚, we present the existence of a solution of the above 𝑝-Laplace equation for every 1<𝑟<𝑝<∞, 𝜇∈ℝ and 𝑚,ℎ∈𝐿∞(Ω). |
| format | Article |
| id | doaj-art-aaea771d68364d7a8cc8223fa33a9ace |
| institution | Kabale University |
| issn | 1085-3375 1687-0409 |
| language | English |
| publishDate | 2012-01-01 |
| publisher | Wiley |
| record_format | Article |
| series | Abstract and Applied Analysis |
| spelling | doaj-art-aaea771d68364d7a8cc8223fa33a9ace2025-08-20T03:39:40ZengWileyAbstract and Applied Analysis1085-33751687-04092012-01-01201210.1155/2012/568120568120Existence Results for Quasilinear Elliptic Equations with Indefinite WeightMieko Tanaka0Department of Mathematics, Tokyo University of Science, Kagurazaka 1-3, Shinjuku-ku 162-8601, JapanWe provide the existence of a solution for quasilinear elliptic equation −div(𝑎∞(𝑥)|∇𝑢|𝑝−2∇𝑢+̃𝑎(𝑥,|∇𝑢|)∇𝑢)=𝜆𝑚(𝑥)|𝑢|𝑝−2𝑢+𝑓(𝑥,𝑢)+ℎ(𝑥) in Ω under the Neumann boundary condition. Here, we consider the condition that ̃𝑎(𝑥,𝑡)=𝑜(𝑡𝑝−2) as 𝑡→+∞ and 𝑓(𝑥,𝑢)=𝑜(|𝑢|𝑝−1) as |𝑢|→∞. As a special case, our result implies that the following 𝑝-Laplace equation has at least one solution: −Δ𝑝𝑢=𝜆𝑚(𝑥)|𝑢|𝑝−2𝑢+𝜇|𝑢|𝑟−2𝑢+ℎ(𝑥) in Ω,𝜕𝑢/𝜕𝜈=0 on 𝜕Ω for every 1<𝑟<𝑝<∞, 𝜆∈ℝ, 𝜇≠0 and 𝑚,ℎ∈𝐿∞(Ω) with ∫Ω𝑚𝑑𝑥≠0. Moreover, in the nonresonant case, that is, 𝜆 is not an eigenvalue of the 𝑝-Laplacian with weight 𝑚, we present the existence of a solution of the above 𝑝-Laplace equation for every 1<𝑟<𝑝<∞, 𝜇∈ℝ and 𝑚,ℎ∈𝐿∞(Ω).http://dx.doi.org/10.1155/2012/568120 |
| spellingShingle | Mieko Tanaka Existence Results for Quasilinear Elliptic Equations with Indefinite Weight Abstract and Applied Analysis |
| title | Existence Results for Quasilinear Elliptic Equations with Indefinite Weight |
| title_full | Existence Results for Quasilinear Elliptic Equations with Indefinite Weight |
| title_fullStr | Existence Results for Quasilinear Elliptic Equations with Indefinite Weight |
| title_full_unstemmed | Existence Results for Quasilinear Elliptic Equations with Indefinite Weight |
| title_short | Existence Results for Quasilinear Elliptic Equations with Indefinite Weight |
| title_sort | existence results for quasilinear elliptic equations with indefinite weight |
| url | http://dx.doi.org/10.1155/2012/568120 |
| work_keys_str_mv | AT miekotanaka existenceresultsforquasilinearellipticequationswithindefiniteweight |