MULTIPLE SOLUTIONS FOR A NONLINEAR ELLIPTIC SYSTEM SUBJECT TO NONAUTONOMOUS PERTURBATIONS
In this paper we consider the following Neumann problem { − Δ u = α ( x ) ( F u ( u , v ) − u ) + λ G u ( x , u , v ) in Ω − Δ v = α ( x ) ( F v ( u , v ) − v ) + λ G v ( x , u , v ) in Ω ∂ u ∂ ν = ∂ v ∂ ν = 0 on ∂ Ω In particular, by means of a multiplicity theorem obtained by Ricceri, we establish...
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| Published in | Taiwanese journal of mathematics Vol. 15; no. 3; pp. 1163 - 1169 |
|---|---|
| Main Author | |
| Format | Journal Article |
| Language | English |
| Published |
Mathematical Society of the Republic of China
01.06.2011
|
| Subjects | |
| Online Access | Get full text |
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| Summary: | In this paper we consider the following Neumann problem
{
−
Δ
u
=
α
(
x
)
(
F
u
(
u
,
v
)
−
u
)
+
λ
G
u
(
x
,
u
,
v
)
in
Ω
−
Δ
v
=
α
(
x
)
(
F
v
(
u
,
v
)
−
v
)
+
λ
G
v
(
x
,
u
,
v
)
in
Ω
∂
u
∂
ν
=
∂
v
∂
ν
=
0
on
∂
Ω
In particular, by means of a multiplicity theorem obtained by Ricceri, we establish that if the set of all global minima of the function
ℝ
2
∋
y
→
|
y
|
2
2
−
F
(
y
)
(whereF∈C
1(ℝ2) and it satisfies the conditionF(0, 0) = 0) has at leastk≥ 2 connected components, then the above Neumann problem admits at leastk+ 1 weak solutions,kof which are lying in a given set.
2000Mathematics Subject Classification: 35J20, 35J65.
Key words and phrases: Neumann problem, Multiplicity of solutions, Global minima, Connected components. |
|---|---|
| ISSN: | 1027-5487 2224-6851 |
| DOI: | 10.11650/twjm/1500406292 |