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

• Published in: Journal of mathematical analysis and applications Vol. 432; no. 2; pp. 965 - 982
• Main Author: Lu, Sheng-Sen
• Format: Journal Article
• Language: English
• Published: Elsevier Inc 15.12.2015
• Subjects: Kirchhoff-type equation; Signed and sign-changing solutions; Variational methods; Kirchhoff-type equation; Variational methods; Signed and sign-changing solutions
• ISSN: 0022-247X; 1096-0813
• DOI: 10.1016/j.jmaa.2015.07.033

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 RN with N≥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>0 under more restricted assumptions on M and f.