Positive solutions for sublinear periodic parabolic problems

Godoy, T.; Kaufmann, Uriel; Paczka, S.

doi:10.1016/s0362-546x(03)00212-8

Cited by 3 publications

(4 citation statements)

References 3 publications

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“…[11], Remark 2.1 (e)). The following lemma compiles some necessary facts about some semilinear periodic parabolic problems.…”

Section: Preliminaries and Auxiliary Resultsmentioning

confidence: 91%

“…Next, we note that ξ → (λaξ p + bξ) /ξ is decreasing, and thus the uniqueness assertion can be proved as in Theorem 3.5 in [10] (it can also be proved in a different way as in [11], Theorem 3.3). Observe also that, due to the homogeneity, (2.5) follows immediately from (2.3) taking V as the positive solution of (2.3) with λ = 1.…”

Section: Moreover In Both Cases the Positive Solution Is Unique Andmentioning

confidence: 93%

“…For applications we refer to [13], [3]. In [8] and [11], bifurcation of positive solutions for (1.2) was proved assuming that ξ → g (x, t, ξ) /ξ is nonincreasing in (0, ∞) and that g ξ (x, t, 0) belongs to L r T for some r > (N + 2) /2. On the other hand, in [9], existence results of positive solutions for (1.2) were given without monotonicity conditions on g and allowing g ξ (x, t, 0) = +∞, but assuming that sup 0<ξ≤σ…”

Section: Introductionmentioning

confidence: 99%

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Periodic parabolic problems with concave and convex nonlinearities

Godoy

Kaufmann

2007

Nonlinear differ. equ. appl.

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Abstract. Let Ω ⊂ RN be a smooth bounded domain, let a, b be two functions that are possibly discontinuous and unbounded with a ≥ 0 in Ω×R and b > 0 in a set of positive measure and let 0 < p < 1 < q. We prove that there exists some 0 < Λ < ∞ such that the nonlinear Dirichlet periodic parabolic problem Lu = λa (x, t) u p + b (x, t) u q in Ω × R has a positive solution for all 0 < λ < Λ and that there is no positive solution if λ > Λ. In some cases we also show the existence of a minimal solution for all 0 < λ < Λ and that the solution u λ can be chosen such that λ → u λ is differentiable and increasing. We also give some upper and lower estimates for such a Λ. All results remain true for the analogous elliptic problems.2000 Mathematics Subject Classification: 35K20, 35P05, 35B10

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“…[11], Remark 2.1 (e)). The following lemma compiles some necessary facts about some semilinear periodic parabolic problems.…”

Section: Preliminaries and Auxiliary Resultsmentioning

confidence: 91%

Section: Moreover In Both Cases the Positive Solution Is Unique Andmentioning

confidence: 93%

Section: Introductionmentioning

confidence: 99%

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Periodic parabolic problems with concave and convex nonlinearities

Godoy

Kaufmann

2007

Nonlinear differ. equ. appl.

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“…For applications we refer to [15], [5]. In [11] and [13], bifurcation of positive solutions for (1.2) was proved assuming that ξ → g (x, t, ξ) /ξ is nonincreasing in (0, ∞) and that g ξ (x, t, 0) belongs to L r T for some r > (N + 2) /2. On the other hand, in [12], existence results of positive solutions for (1.2) were given without monotonicity conditions on g and allowing g ξ (x, t, 0) = +∞, but assuming that inf 0<σ≤ξ (g (x, t, σ) /σ) belongs to L r T for some r > (N + 2) /2.…”

Section: 1mentioning

confidence: 99%

Periodic parabolic problems with nonlinearities indefinite in sign

Godoy

Kaufmann²

2007

Publicacions Matemàtiques

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Let Ω ⊂ R N be a smooth bounded domain. We give sufficient conditions (which are also necessary in many cases) on two nonnegative functions a, b that are possibly discontinuous and unbounded for the existence of nonnegative solutions for semilinear Dirichlet periodic parabolic problems of the form Lu = λa (x, t) u p − b (x, t) u q in Ω × R, where 0 < p, q < 1 and λ > 0. In some cases we also show the existence of solutions u λ in the interior of the positive cone and that u λ can be chosen such that λ → u λ is differentiable and increasing. A uniqueness theorem is also given in the case p ≤ q. All results remain valid for the corresponding elliptic problems.

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