J. Math. Anal. Appl. 367 (2010) 273–277 Contents lists available at ScienceDirect Journal of Mathematical Analysis and Applications www.elsevier.com/locate/jmaa Elliptic equations with indefinite concave nonlinearities near the origin Zuji Guo School of Mathematical Sciences, Capital Normal University, Beijing 100048, People’s Republic of China article info abstract Article history: Received 7 October 2009 Available online 21 January 2010 Submitted by J. Mawhin Keywords: Nonlinear boundary value problem Indefinite concave nonlinearity Infinitely many solutions In this note existence of infinitely many solutions is proved for an elliptic equation with indefinite concave nonlinearities.  2010 Elsevier Inc. All rights reserved. 1. Introduction This note concerns existence of infinitely many solutions to the equation  −u = b(x)|u| q−2 u + f (x, u) in Ω, u = 0 on ∂Ω, (1.1) where Ω ⊂ R N is a bounded domain and 1 < q < 2. Conditions on the weight function b and on the nonlinearity f will be formulated later. The weight function b will be possibly sign-changing and the nonlinearity f will be considered as a perturbation term. Equations of this type have been studied extensively in the literature, mainly in the case where b(x) is a positive con- stant; see for example [1–3,5,6]. In [1–3,5], infinitely many solutions were obtained provided that f satisfies some global assumptions for all x and u. A typical example of f satisfying those global assumptions is f (x, u) =|u| p−2 u with 2 < p  2 ∗ . These solutions are small solutions in the sense that the sequence of solutions converges to 0 in the L ∞ norm. It was first observed by Wang in [6] that existence of such a sequence of solutions relies only on local behavior of the equation and assumptions on f (x, u) only for small u are required. Among other results, it was proved in [6] that if b(x) = λ is a positive constant, 1 < q < 2, f (x, u) is odd in u for small |u|, and f (x, u) = o(|u| q−1 ) as u → 0 uniformly in x then (1.1) has a sequence of weak solutions (u n ) ⊂ H 1 0 (Ω) such that ‖u n ‖ L ∞ (Ω) = 0 as n →∞. Moreover, I (u n )  0 and I (u n ) → 0 as n →∞, where I (u) = 1 2  Ω |∇u| 2 dx − 1 q  Ω b(x)|u| q dx −  Ω F (x, u) dx with F (x, u) being the primitive of f (x, u). Note that positivity of b(x) = λ was used in [6] in an essential way. In this note, we consider (1.1) where b(x) is possibly sign-changing, a case which cannot be imbedded in the framework of [6]. The exact assumptions on b and f are as follows: E-mail address: guozuji@sina.com. 0022-247X/$ – see front matter  2010 Elsevier Inc. All rights reserved. doi:10.1016/j.jmaa.2010.01.012