Signed and sign-changing solutions for a Kirchhoff-type equation in bounded domains

The main concern of this article is a Kirchhoff-type equation of the form -M (integral(Omega)vertical bar del u vertical bar(2)) Delta u = lambda f (u), where Omega is a bounded smooth domain in R-N with N >= 3 and lambda is a positive parameter. Under certain assumptions on M and f, the existence results of signed and sign-changing solutions are established for lambda large, and when lambda converges to infinity the asymptotic behavior of these solutions is also studied. The proofs are based on a careful study of the ground state and least energy nodal solutions of an auxiliary problem, which is constructed by making a refined truncation on M. Furthermore, we get the ground state and least energy nodal solutions, and prove the energy doubling property for all lambda > 0 under more restricted assumptions on M and f. (C) 2015 Elsevier Inc. All rights reserved.