Nonnegative solutions of an indefinite sublinear Robin problem I: positivity, exact multiplicity, and existence of a sub
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Nonnegative solutions of an indefinite sublinear Robin problem I: positivity, exact multiplicity, and existence of a subcontinuum Uriel Kaufmann1 · Humberto Ramos Quoirin2,3
· Kenichiro Umezu4
Received: 23 October 2019 / Accepted: 27 January 2020 © Fondazione Annali di Matematica Pura ed Applicata and Springer-Verlag GmbH Germany, part of Springer Nature 2020
Abstract Let ⊂ R N (N ≥ 1) be a smooth bounded domain, a ∈ C() a sign-changing function, and 0 ≤ q < 1. We investigate the Robin problem ⎧ q ⎪ ⎨−u = a(x)u in , (Pα ) u≥0 in , ⎪ ⎩ on ∂, ∂ν u = αu where α ∈ [−∞, ∞) and ν is the unit outward normal to ∂. 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 in a subcontinuum. These results partially rely on (and extend) our previous work (Kaufmann et al. in J Differ Equ 263:4481–4502, 2017), 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. Keywords Elliptic problem · Indefinite · Sublinear · Positive solution · Robin boundary condition · Exact multiplicity Mathematics Subject Classification 35J15 · 35J25 · 35J61
Uriel Kaufmann: Partially supported by Secyt-UNC 33620180100016CB. Humberto Ramos Quoirin: Supported by Fondecyt Grants 1161635, 1171532, 1171691, 1181825. Kenichiro Umezu: Supported by JSPS KAKENHI Grant Numbers JP15K04945 and JP18K03353.
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Humberto Ramos Quoirin [email protected]
Extended author information available on the last page of the article
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1 Introduction This article is devoted to a class of indefinite elliptic pdes, whose prototype is the equation −u = a(x)u q in , where ⊂ R N (N ≥ 1) is a bounded and smooth domain and a ∈ C() is a sign-changing function. Over the past decades, many works have addressed basic issues on nonnegative solutions of this equation (under different boundary conditions) in the superlinear case q > 1 [2,4,7,8,22,26,32]. On the other hand, much less attention has been given to the sublinear problem, i.e., with 0 < q < 1, which will be considered here. In particular, we shall highlight the main contrasts between these two cases. We consider nonnegative solutions of the above equation under a Robin boundary condition, i.e., the problem: ⎧ q ⎪ ⎨−u = a(x)u in , (Pα ) u≥0 in , ⎪ ⎩ on ∂. ∂ν u = αu ∂ , and α ∈ [−∞, ∞). When α = −∞, the Here, ν is the unit
