On the number of positive solutions of elliptic systems

Donal O'Regan, Haiyan Wang

Research output: Contribution to journalArticle

10 Citations (Scopus)

Abstract

The paper deals with the existence, multiplicity and nonexistence of positive radial solutions for the elliptic system div(|∇u i|p-2∇ui)+λki(|x|) fi(u1, . . . , un) = 0, p > 1, R1 < |x| < R2, ui(x) = 0, on |x| = R1 and R2, i = 1, . . . , n, x ∈ ℝN, where k i and fi, i = 1, . . . , n, are continuous and nonnegative functions. Let u = (u1, . . . , un), φ(t) = |t| p-2t, f0 i = lim∥u∥→0o fi(u)/φ(∥u∥), f i = lim ∥u∥→∞ fi(u)/φ(∥u∥), i = 1, . . . , n, f = (f1 , . . . , fn), f0 = Σi=1 n f0 i and f = Σi=1 n f i. We prove that either f0 = 0 and f = ∞ (superlinear), or f0 = ∞ and f = 0 (sublinear), guarantee existence for all λ > 0. In addition, if f i(u) > 0 for ∥u∥ > 0, i = 1, . . . , n, then either f0 = f, = 0, or f0 = f = ∞, guarantee multiplicity for sufficiently large, or small λ, respectively. On The other hand, either f0 and f > 0, or f0 and f < ∞ imply nonexistence for sufficiently large, or small λ, respectively. Furthermore, all die results are valid for Dirichlet/Neumann boundary conditions. We shall use fixed point meorems in a cone.

Original languageEnglish (US)
Pages (from-to)1417-1430
Number of pages14
JournalMathematische Nachrichten
Volume280
Issue number12
DOIs
StatePublished - 2007

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Elliptic Systems
Nonexistence
Positive Solution
Multiplicity
Positive Radial Solutions
Neumann Boundary Conditions
Dirichlet Boundary Conditions
Cone
Die
Fixed point
Non-negative
Valid
Imply

Keywords

  • Annulus
  • Cone
  • Elliptic systems
  • Fixed point theorems
  • Positive radial solutions

ASJC Scopus subject areas

  • Mathematics(all)

Cite this

On the number of positive solutions of elliptic systems. / O'Regan, Donal; Wang, Haiyan.

In: Mathematische Nachrichten, Vol. 280, No. 12, 2007, p. 1417-1430.

Research output: Contribution to journalArticle

O'Regan, Donal ; Wang, Haiyan. / On the number of positive solutions of elliptic systems. In: Mathematische Nachrichten. 2007 ; Vol. 280, No. 12. pp. 1417-1430.
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abstract = "The paper deals with the existence, multiplicity and nonexistence of positive radial solutions for the elliptic system div(|∇u i|p-2∇ui)+λki(|x|) fi(u1, . . . , un) = 0, p > 1, R1 < |x| < R2, ui(x) = 0, on |x| = R1 and R2, i = 1, . . . , n, x ∈ ℝN, where k i and fi, i = 1, . . . , n, are continuous and nonnegative functions. Let u = (u1, . . . , un), φ(t) = |t| p-2t, f0 i = lim∥u∥→0o fi(u)/φ(∥u∥), f∞ i = lim ∥u∥→∞ fi(u)/φ(∥u∥), i = 1, . . . , n, f = (f1 , . . . , fn), f0 = Σi=1 n f0 i and f ∞ = Σi=1 n f∞ i. We prove that either f0 = 0 and f∞ = ∞ (superlinear), or f0 = ∞ and f∞ = 0 (sublinear), guarantee existence for all λ > 0. In addition, if f i(u) > 0 for ∥u∥ > 0, i = 1, . . . , n, then either f0 = f∞, = 0, or f0 = f∞ = ∞, guarantee multiplicity for sufficiently large, or small λ, respectively. On The other hand, either f0 and f∞ > 0, or f0 and f∞ < ∞ imply nonexistence for sufficiently large, or small λ, respectively. Furthermore, all die results are valid for Dirichlet/Neumann boundary conditions. We shall use fixed point meorems in a cone.",
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N2 - The paper deals with the existence, multiplicity and nonexistence of positive radial solutions for the elliptic system div(|∇u i|p-2∇ui)+λki(|x|) fi(u1, . . . , un) = 0, p > 1, R1 < |x| < R2, ui(x) = 0, on |x| = R1 and R2, i = 1, . . . , n, x ∈ ℝN, where k i and fi, i = 1, . . . , n, are continuous and nonnegative functions. Let u = (u1, . . . , un), φ(t) = |t| p-2t, f0 i = lim∥u∥→0o fi(u)/φ(∥u∥), f∞ i = lim ∥u∥→∞ fi(u)/φ(∥u∥), i = 1, . . . , n, f = (f1 , . . . , fn), f0 = Σi=1 n f0 i and f ∞ = Σi=1 n f∞ i. We prove that either f0 = 0 and f∞ = ∞ (superlinear), or f0 = ∞ and f∞ = 0 (sublinear), guarantee existence for all λ > 0. In addition, if f i(u) > 0 for ∥u∥ > 0, i = 1, . . . , n, then either f0 = f∞, = 0, or f0 = f∞ = ∞, guarantee multiplicity for sufficiently large, or small λ, respectively. On The other hand, either f0 and f∞ > 0, or f0 and f∞ < ∞ imply nonexistence for sufficiently large, or small λ, respectively. Furthermore, all die results are valid for Dirichlet/Neumann boundary conditions. We shall use fixed point meorems in a cone.

AB - The paper deals with the existence, multiplicity and nonexistence of positive radial solutions for the elliptic system div(|∇u i|p-2∇ui)+λki(|x|) fi(u1, . . . , un) = 0, p > 1, R1 < |x| < R2, ui(x) = 0, on |x| = R1 and R2, i = 1, . . . , n, x ∈ ℝN, where k i and fi, i = 1, . . . , n, are continuous and nonnegative functions. Let u = (u1, . . . , un), φ(t) = |t| p-2t, f0 i = lim∥u∥→0o fi(u)/φ(∥u∥), f∞ i = lim ∥u∥→∞ fi(u)/φ(∥u∥), i = 1, . . . , n, f = (f1 , . . . , fn), f0 = Σi=1 n f0 i and f ∞ = Σi=1 n f∞ i. We prove that either f0 = 0 and f∞ = ∞ (superlinear), or f0 = ∞ and f∞ = 0 (sublinear), guarantee existence for all λ > 0. In addition, if f i(u) > 0 for ∥u∥ > 0, i = 1, . . . , n, then either f0 = f∞, = 0, or f0 = f∞ = ∞, guarantee multiplicity for sufficiently large, or small λ, respectively. On The other hand, either f0 and f∞ > 0, or f0 and f∞ < ∞ imply nonexistence for sufficiently large, or small λ, respectively. Furthermore, all die results are valid for Dirichlet/Neumann boundary conditions. We shall use fixed point meorems in a cone.

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