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A curve of positive solutions for an indefinite sublinear Dirichlet problem

  • * Corresponding author: Uriel Kaufmann

    * Corresponding author: Uriel Kaufmann 

U. Kaufmann was partially supported by Secyt-UNC 33620180100016CB; H. Ramos Quoirin was supported by Fondecyt grants 1161635, 1171532, 1171691, and 1181125; K. Umezu was supported by JSPS KAKENHI Grant Numbers JP15K04945 and JP18K03353

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  • We investigate the existence of a curve $ q\mapsto u_{q} $, with $ q\in(0, 1) $, of positive solutions for the problem

    $(P_{a, q}) \qquad \qquad \begin{cases} -\Delta u = a(x)u^{q} & \mbox{ in }\Omega, \\ u = 0 & \mbox{ on }\partial\Omega, \end{cases} $

    where $ \Omega $ is a bounded and smooth domain of $ \mathbb{R}^{N} $ and $ a:\Omega\rightarrow\mathbb{R} $ is a sign-changing function (in which case the strong maximum principle does not hold). In addition, we analyze the asymptotic behavior of $ u_{q} $ as $ q\rightarrow0^{+} $ and $ q\rightarrow1^{-} $. We also show that in some cases $ u_{q} $ is the ground state solution of $ (P_{a, 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.

    Mathematics Subject Classification: Primary: 35J25, 35J61; Secondary: 35B32.


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  • Figure 1.  The curve of positive solutions in the case $ \lambda_{1}(a) = 1 $

    Figure 2.  The curve of positive solutions emanating from $ (0, \mathcal{S}(a)) $: (ⅰ) The case $ \lambda_{1}(a)>1 $. (ⅱ) The case $ \lambda_{1}(a)<1 $

    Figure 3.  The situation of $ \underline{u} $ and $ u_{*} $

    Figure 4.  The curve of positive solutions in the case $ \lambda_{1}(a) = 1 $

    Figure 5.  The indefinite weight $ a $ in the case $ q = \frac{1}{3} $

    Figure 6.  The nontrivial nonnegative solutions $ u_{1}, u_{2} $ with dead cores and a possible solution $ u_{3} $ in $ \mathcal{P}^{\circ} $ (which is even, by uniqueness) in the case $ q = \frac{1}{3} $

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