Infinitely many positive solutions for Kirchhoff-type problems

Abstract

This paper is concerned with the existence of infinitely many positive solutions to a class of Kirchhoff-type problem − ( a + b ∫ Ω | ∇ u | 2 d x ) Δ u = λ f ( x , u ) in Ω and u = 0 on ∂ Ω , where Ω is a smooth bounded domain of R N , a , b > 0 , λ > 0 and f : Ω × R → R is a Carathéodory function satisfying some further conditions. We obtain a sequence of a.e. positive weak solutions to the above problem tending to zero in L ∞ ( Ω ) with f being more general than that of [K. Perera, Z. Zhang, Nontrivial solutions of Kirchhoff-type problems via the Yang index, J. Differential Equations 221 (2006) 246–255; Z. Zhang, K. Perera, Sign changing solutions of Kirchhoff type problems via invariant sets of descent flow, J. Math. Anal. Appl. 317 (2006) 456–463].