Positivity results for indefinite sublinear elliptic problems via a continuity argument

Kaufmann, Uriel; Quoirin, Humberto Ramos; Umezu, Kenichiro

doi:10.1016/j.jde.2017.05.021

Cited by 20 publications

(39 citation statements)

References 20 publications

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“…It follows that Ω |∇u n | 2 → Ω |∇S(a)| 2 , and consequently u n → S(a) in H 1 0 (Ω). Using Remark 3.1 in [22], we get u n → S(a) in C 1 0 (Ω), as desired. Finally, if S(a) > 0 in Ω, then the last assertion of item (iii) follows from Lemma 2.1 (ii).…”

Section: Let Us First Deduce Thatmentioning

confidence: 61%

“…It follows that λ 1 (a) = 1 d ≤ 1, a contradiction. Hence {u n } is bounded, and arguing as in the proof of Theorem 1.3 in [22], we can show that, up to a subsequence, u n → u 0 in C 1 (Ω) and u 0 ≥ 0 solves (2.5). If u 0 ≡ 0 then, by the strong maximum principle, we have u 0 ∈ P • , so that λ 1 (a) = 1, and we reach a contradiction again.…”

Section: Let Us First Deduce Thatmentioning

confidence: 66%

“…More recently, we have proved in [22] that if Ω + has finitely many connected components and q is close enough to 1, then any nontrivial nonnegative solution of (P a,q ) belongs to P • , so that in this situation (P a,q ) has exactly one nontrivial nonnegative solution, which in addition belongs to P • (see [22,Corollary 1.5]).…”

Section: Let Us Setmentioning

confidence: 99%

See 2 more Smart Citations

A curve of positive solutions for an indefinite sublinear Dirichlet problem

Kaufmann¹,

Quoirin²,

Umezu³

2020

Discrete &Amp; Continuous Dynamical Systems - A

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We investigate the existence of a curve q → uq, with q ∈ (0, 1), of positive solutions for the problemwhere Ω is a bounded and smooth domain of R N and a : Ω → R is a sign-changing function (in which case the strong maximum principle does not hold). In addition, we analyze the asymptotic behavior of uq as q → 0 + and q → 1 − . We also show that in some cases uq is the ground state solution of (Pa,q). As a byproduct, we obtain existence results for a singular and indefinite Dirichlet problem. Our results are mainly based on bifurcation and sub-supersolutions methods.

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Section: Let Us First Deduce Thatmentioning

confidence: 61%

Section: Let Us First Deduce Thatmentioning

confidence: 66%

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A curve of positive solutions for an indefinite sublinear Dirichlet problem

Kaufmann¹,

Quoirin²,

Umezu³

2020

Discrete &Amp; Continuous Dynamical Systems - A

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“…Indeed, since (1.15) holds and b ≥ 0, we see that v n is a supersolution of (1.13) which is positive in Ω. So, condition (H ψ ) with ψ = a allows us to apply [15,Lemma 2.2], and deduce that there exist a ball B ⋐ Ω a + and a continuous function ψ on B such that v n ≥ ψ > 0 in B. Passing to the limit, we have that v 0 ≥ ψ in B, as desired.…”

Section: 2mentioning

confidence: 96%

Loop Type Subcontinua of Positive Solutions for Indefinite Concave-Convex Problems

Kaufmann

Humberto

Umezu

2018

Advanced Nonlinear Studies

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We establish the existence of loop type subcontinua of nonnegative solutions for a class of concave-convex type elliptic equations with indefinite weights, under Dirichlet and Neumann boundary conditions. Our approach depends on local and global bifurcation analysis from the zero solution in a non-regular setting, since the nonlinearities considered are not differentiable at zero, so that the standard bifurcation theory does not apply. To overcome this difficulty, we combine a regularization scheme with a priori bounds, and Whyburn's topological method. Furthermore, via a continuity argument we prove a positivity property for subcontinua of nonnegative solutions. These results are based on a positivity theorem for the associated concave problem proved in [15], and extend previous results established in the powerlike case.2010 Mathematics Subject Classification. 35J25, 35J61, 35B32.

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“…The investigation of (P α ) in the sublinear case has been carried out mostly for α = −∞ [5,9,12,14,15,16,17,19,27] and α = 0 [1,6,12,17,18]. To recall these results, we consider the conditions We also introduce the positivity set A α (a) := {q ∈ (0, 1) : any nontrivial solution of (P α ) lies in P • }.…”

Section: Introductionmentioning

confidence: 99%

Nonnegative solutions of an indefinite sublinear Robin problem I: positivity, exact multiplicity, and existence of a subcontinuum

Kaufmann

Quoirin

Umezu

2020

Annali di Matematica

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Let Ω ⊂ R N (N ≥ 1) be a smooth bounded domain, a ∈ C(Ω) a sign-changing function, and 0 ≤ q < 1. We investigate the Robin problemwhere α ∈ [−∞, ∞) and ν is the unit outward normal to ∂Ω. Due to the lack of strong maximum principle structure, this problem may have dead core solutions. However, for a large class of weights a we recover a positivity property when q is close to 1, which considerably simplifies the structure of the solution set. Such property, combined with a bifurcation analysis and a suitable change of variables, enables us to show the following exactness result for these values of q: (Pα) has exactly one nontrivial solution for α ≤ 0, exactly two nontrivial solutions for α > 0 small, and no such solution for α > 0 large. Assuming some further conditions on a, we show that these solutions lie on a subcontinuum. These results rely partially on (and extend) our previous work [17], where the cases α = −∞ (Dirichlet) and α = 0 (Neumann) have been considered. We also obtain some results for arbitrary q ∈ [0, 1). Our approach combines mainly bifurcation techniques, the sub-supersolutions method, and a priori lower and upper bounds. * 2010 Mathematics Subject Classification. 35J15, 35J25, 35J61. †

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Positivity results for indefinite sublinear elliptic problems via a continuity argument

Cited by 20 publications

References 20 publications

A curve of positive solutions for an indefinite sublinear Dirichlet problem

A curve of positive solutions for an indefinite sublinear Dirichlet problem

Loop Type Subcontinua of Positive Solutions for Indefinite Concave-Convex Problems

Nonnegative solutions of an indefinite sublinear Robin problem I: positivity, exact multiplicity, and existence of a subcontinuum

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