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(∫Ω|∇u|2)Δu=λf(u), where Ω is a bounded smooth domain in RNRN with N≥3N≥3 and λ is a positive parameter. Under certain assumptions on M and f, the existence results of signed and sign-changing solutions are established for λ large, and when λ 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 λ>0λ>0 under more restricted assumptions on M and f.