Electronic Journal of Differential Equations, 
Vol. 2020 (2020), No. 44, pp. 1-15.

Title: Multiple positive solutions for biharmonic equation of Kirchhoff type
       involving concave-convex nonlinearities
       
Authors: Fengjuan Meng (Jiangsu Univ. of Technology, Changzhou, China)
         Fubao Zhang  (Southeast Univ., Nanjing, China)
         Yuanyuan Zhang (Jiangsu Univ. of Technology, Changzhou, China)

Abstract:
 In this article, we study the multiplicity of positive solutions for the
 biharmonic equation of Kirchhoff type involving concave-convex nonlinearities,
 $$
 \Delta^2u-\Big(a+b\int_{\mathbb{R}^N}|\nabla u|^2dx\Big)\Delta u+V(x)u
 =\lambda f_1(x)|u|^{q-2}u+f_2(x)|u|^{p-2}u.
 $$
 Using the Nehari manifold, Ekeland variational principle, and the theory
 of Lagrange multipliers, we prove that there are at least two positive solutions,
 one of which is a positive ground state solution.
 
Submitted March 11, 2019. Published May 19, 2020.
Math Subject Classifications: 35J35, 35J40, 35J91.
Key Words: Biharmonic  equation; ground state solution; Nehari manifold;
           concave-convex nonlinearity.