Electron. J. Differential Equations, Vol. 2016 (2016), No. 301, pp. 1-16.

Multiple positive solutions for Kirchhoff type problems involving concave and convex nonlinearities in R^3

Xiaofei Cao, Junxiang Xu, Jun Wang

Abstract:
In this article, we consider the multiplicity of positive solutions for a class of Kirchhoff type problems with concave and convex nonlinearities. Under appropriate assumptions, we prove that the problem has at least two positive solutions, moreover, one of which is a positive ground state solution. Our approach is mainly based on the Nehari manifold, Ekeland variational principle and the theory of Lagrange multipliers.

Submitted July 25, 2016. Published November 25, 2016.
Math Subject Classifications: 35A01, 35A15.
Key Words: Kirchhoff type problem; Ekeland variational principle; Nehari manifold.

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Xiaofei Cao
Department of Mathematics
Southeast University
Nanjing, Jiangsu 210096, China
email: caoxiaofei258@126.com
Junxiang Xu
Department of Mathematics
Southeast University
Nanjing, Jiangsu 210096, China
email: xujun@seu.edu.cn
Jun Wang
Faculty of Science
Jiangsu University
Zhenjiang, Jiangsu 212013, China
email: wangmath2011@126.com

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