An elliptic problem with an indefinite nonlinearity and a parameter in the boundary condition

Chabrowski, J.; Tintarev, Cyril

doi:10.1007/s00030-013-0256-8

Cited by 9 publications

(3 citation statements)

References 15 publications

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“…In [10,Section 7] Chabrowski and Tintarev proved, by variational methods, that under (A.0) this problem has at least two nontrivial solutions w 1,α , w 2,α such that w 1,α < w 2,α on Ω for α > 0 small enough. Moreover, they also provided the following asymptotic profiles of w 1,α , w 2,α as α → 0 + :…”

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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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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“…To the best of our knowledge, prior to [12] the only work dealing with (P α ) for α > 0 is [7], where Chabrowski and Tintarev established a local multiplicity result for the related problem…”

Section: Introductionmentioning

confidence: 99%

“…with α > 0 small (note that (R α ) is equivalent to (P α ), after the change of variables w = α 1 1−q u). More precisely, by variational methods it was proved in [7,Propositions 7.4 and 7.7] that under (A.0), (R α ) has at least two nontrivial solutions w 1,α , w 2,α such that w 1,α < w 2,α on Ω for α > 0 small enough. Moreover, if…”

Section: Introductionmentioning

confidence: 99%

Nonnegative solutions of an indefinite sublinear Robin problem II: local and global exactness results

Kaufmann¹,

Quoirin²,

Umezu³

2020

Preprint

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We go further in the investigation of the Robin problem (Pα)on a bounded domain Ω ⊂ R N , with a sign-changing and 0 < q < 1.Assuming the existence of a positive solution for α = 0 (which holds if q is close enough to 1), we sharpen the description of the nontrivial solution set of (Pα) for α > 0. Moreover, strengthening the assumptions on a and q we provide a global (i.e. for every α > 0) exactness result on the number of solutions of (Pα) . Our approach also applies to the problem (Sα)−∆u = αu + a(x)u q in Ω, u ≥ 0 in Ω, ∂νu = 0 on ∂Ω.

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Semilinear elliptic problems with combined nonlinearities on the boundary

Furtado

Ruviaro

Silva

2017

Annali di Matematica

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An elliptic problem with an indefinite nonlinearity and a parameter in the boundary condition

Cited by 9 publications

References 15 publications

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

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

Nonnegative solutions of an indefinite sublinear Robin problem II: local and global exactness results

Semilinear elliptic problems with combined nonlinearities on the boundary

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