Existence and Multiplicity of Positive Solutions for Dirichlet Problems in Unbounded Domains
We consider the elliptic problem −Δu+u=b(x)|u|p−2u+h(x) in Ω, u∈H01(Ω), where 2<p<(2N/(N−2)) (N≥3), 2<p<∞ (N=2), Ω is a smooth unbounded domain in ℝN, b(x)∈C(Ω), and h(x)∈H−1(Ω). We use the shape of domain Ω to prove that the above elliptic problem has a ground-state solution if the coef...
Saved in:
| Main Author: | |
|---|---|
| Format: | Article |
| Language: | English |
| Published: |
Wiley
2007-01-01
|
| Series: | Abstract and Applied Analysis |
| Online Access: | http://dx.doi.org/10.1155/2007/18187 |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1832562349894008832 |
|---|---|
| author | Tsung-Fang Wu |
| author_facet | Tsung-Fang Wu |
| author_sort | Tsung-Fang Wu |
| collection | DOAJ |
| description | We consider the elliptic problem −Δu+u=b(x)|u|p−2u+h(x) in Ω, u∈H01(Ω), where 2<p<(2N/(N−2)) (N≥3), 2<p<∞ (N=2), Ω is a smooth unbounded domain in ℝN, b(x)∈C(Ω), and h(x)∈H−1(Ω). We use the shape of domain Ω to prove that the above elliptic problem has a ground-state solution if the coefficient b(x) satisfies b(x)→b∞>0 as |x|→∞ and b(x)≥c for some suitable constants c∈(0,b∞), and h(x)≡0. Furthermore, we prove that the above elliptic problem has multiple positive solutions if the coefficient b(x) also satisfies the above conditions, h(x)≥0 and 0<‖h‖H−1<(p−2)(1/(p−1))(p−1)/(p−2)[bsupSp(Ω)]1/(2−p), where S(Ω) is the best Sobolev constant of subcritical operator in H01(Ω) and bsup=supx∈Ωb(x). |
| format | Article |
| id | doaj-art-2a6550479bd84c64a1d7e094c018f6df |
| institution | Kabale University |
| issn | 1085-3375 1687-0409 |
| language | English |
| publishDate | 2007-01-01 |
| publisher | Wiley |
| record_format | Article |
| series | Abstract and Applied Analysis |
| spelling | doaj-art-2a6550479bd84c64a1d7e094c018f6df2025-02-03T01:22:51ZengWileyAbstract and Applied Analysis1085-33751687-04092007-01-01200710.1155/2007/1818718187Existence and Multiplicity of Positive Solutions for Dirichlet Problems in Unbounded DomainsTsung-Fang Wu0Department of Applied Mathematics, National University of Kaohsiung, Kaohsiung 811, TaiwanWe consider the elliptic problem −Δu+u=b(x)|u|p−2u+h(x) in Ω, u∈H01(Ω), where 2<p<(2N/(N−2)) (N≥3), 2<p<∞ (N=2), Ω is a smooth unbounded domain in ℝN, b(x)∈C(Ω), and h(x)∈H−1(Ω). We use the shape of domain Ω to prove that the above elliptic problem has a ground-state solution if the coefficient b(x) satisfies b(x)→b∞>0 as |x|→∞ and b(x)≥c for some suitable constants c∈(0,b∞), and h(x)≡0. Furthermore, we prove that the above elliptic problem has multiple positive solutions if the coefficient b(x) also satisfies the above conditions, h(x)≥0 and 0<‖h‖H−1<(p−2)(1/(p−1))(p−1)/(p−2)[bsupSp(Ω)]1/(2−p), where S(Ω) is the best Sobolev constant of subcritical operator in H01(Ω) and bsup=supx∈Ωb(x).http://dx.doi.org/10.1155/2007/18187 |
| spellingShingle | Tsung-Fang Wu Existence and Multiplicity of Positive Solutions for Dirichlet Problems in Unbounded Domains Abstract and Applied Analysis |
| title | Existence and Multiplicity of Positive Solutions for Dirichlet Problems in Unbounded Domains |
| title_full | Existence and Multiplicity of Positive Solutions for Dirichlet Problems in Unbounded Domains |
| title_fullStr | Existence and Multiplicity of Positive Solutions for Dirichlet Problems in Unbounded Domains |
| title_full_unstemmed | Existence and Multiplicity of Positive Solutions for Dirichlet Problems in Unbounded Domains |
| title_short | Existence and Multiplicity of Positive Solutions for Dirichlet Problems in Unbounded Domains |
| title_sort | existence and multiplicity of positive solutions for dirichlet problems in unbounded domains |
| url | http://dx.doi.org/10.1155/2007/18187 |
| work_keys_str_mv | AT tsungfangwu existenceandmultiplicityofpositivesolutionsfordirichletproblemsinunboundeddomains |