Exact Multiplicity of Positive Solutions for a Class of Semilinear Problems

Ouyang, Tiancheng; Shi, Junping

doi:10.1006/jdeq.1998.3414

Cited by 115 publications

(90 citation statements)

References 28 publications

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“…Now, we are ready to prove the uniqueness of stable solution. We use a similar argument to the one used in Theorem 3.7 of [21], see also [11], [12], [23], [24] and [30] where the idea of using the "turning direction" at all possible "turning points" is used.…”

Section: Taking Into Account (54) It Is Not Hard To Show That Sgn(1 mentioning

confidence: 99%

The Competition Between Incoming and Outgoing Fluxes in an Elliptic Problem

García-Melián

Rossi

Suárez

2007

Commun. Contemp. Math.

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In this work we consider existence and uniqueness of positive solutions to the elliptic equation −∆u = λu in Ω, with the nonlinear boundary conditionswhere Ω is a smooth bounded domain, ∂Ω = Γ 1 ∪ Γ 2 , Γ 1 ∩ Γ 2 = ∅, ν is the outward unit normal, p, q > 0 and λ is a real parameter. We obtain a complete picture of the bifurcation diagram of the problem, depending on the values of p, q and the parameter λ. Our proofs are based on different techniques: variational arguments, bifurcation techniques or comparison arguments, depending on the range of parameters considered.

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Section: Taking Into Account (54) It Is Not Hard To Show That Sgn(1 mentioning

confidence: 99%

The Competition Between Incoming and Outgoing Fluxes in an Elliptic Problem

García-Melián

Rossi

Suárez

2007

Commun. Contemp. Math.

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“…Observe that when δ = 0, w δ exists and is unique if and only if n ≤ 3. (Existence follows by a shooting method; see [8] and [11]. Radial symmetry is proved in [6] and uniqueness in [9].)…”

Section: Parameterized Ground Statesmentioning

confidence: 99%

On a cubic-quintic Ginzburg-Landau equation with global coupling

Wei

Winter

2004

Proc. Amer. Math. Soc.

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“…Rabinowitz [16] has studied the case p = 2 < q (for example, equations like λ∆u + u 2 (1 − u) = 0) by combining critical point theory and the Leray-Schauder degree theory, and proved there exists Λ > 0 such that if λ > Λ, then (P) λ has no solution and if λ < Λ, then (P) λ has at least two distinct solutions (see also Ambrosetti and Rabinowitz [1], and Rabinowitz [17]). Particularly, when Ω is a ball, Ouyang and Shi [14] have obtained a precise global bifurcation diagram and concluded that there exist exactly two solutions for small λ by using a bifurcation theorem of Crandall and Rabinowitz.…”

Section: ω|mentioning

confidence: 99%