Abstract
Let \(\Omega \subset \mathbb {R}^{N}\) (\(N\ge 1\)) be a bounded and smooth domain and \(a:\Omega \rightarrow \mathbb {R}\) be a sign-changing weight satisfying \(\int _{\Omega }a<0\). We prove the existence of a positive solution \(u_{q}\) for the problem
if \(q_{0}<q<1\), for some \(q_{0}=q_{0}(a)>0\). In doing so, we improve the existence result previously established in Kaufmann et al. (J Differ Equ 263:4481–4502, 2017). In addition, we provide the asymptotic behavior of \(u_{q}\) as \(q\rightarrow 1^{-}\). When \(\Omega \) is a ball and a is radial, we give some explicit conditions on q and a ensuring the existence of a positive solution of \((P_{a,q})\). We also obtain some properties of the set of q’s such that \((P_{a,q})\) admits a solution which is positive on \(\overline{\Omega }\). Finally, we present some results on nonnegative solutions having dead cores. Our approach combines bifurcation techniques, a priori bounds and the sub-supersolution method.
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Kaufmann, U., Ramos Quoirin, H. & Umezu, K. Positive solutions of an elliptic Neumann problem with a sublinear indefinite nonlinearity. Nonlinear Differ. Equ. Appl. 25, 12 (2018). https://doi.org/10.1007/s00030-018-0502-1
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DOI: https://doi.org/10.1007/s00030-018-0502-1