Infinitely many positive solutions for Kirchhoff-type problems

This paper is concerned with the existence of infinitely many positive solutions to a class of Kirchhoff-type problem −(a+b∫Ω|∇u|2dx)Δu=λf(x,u) in ΩΩ and u=0u=0 on ∂Ω∂Ω, where ΩΩ is a smooth bounded domain of RN,a,b>0,λ>0RN,a,b>0,λ>0 and f:Ω×R→Rf: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)L∞(Ω) with ff 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].