Nontrivial solutions of nonlocal fourth order elliptic equation of Kirchhoff type in R3

Abstract This paper is devoted to a class of important and general nonlocal fourth order elliptic problem Δ 2 u − ( 1 + λ ∫ R 3 | ∇ u | 2 d x ) Δ u + V ( x ) u = f ( x , u ) in R 3 , where Δ 2 = Δ ( Δ ) is the bi-harmonic operator, λ ≥ 0 is a constant. We focus on the case that f ( x , u ) involves a combination of convex and concave terms and the potential V ( x ) is allowed to be sign-changing. By new techniques, multiplicity results of two different type of solutions are established. Our results improves and generalizes that obtained in the literature.

[1]  Filippo Gazzola,et al.  A partially hinged rectangular plate as a model for suspension bridges , 2015 .

[2]  Yukun An,et al.  Existence of solutions for fourth order elliptic equations of Kirchhoff type , 2014 .

[3]  Zuji Guo,et al.  Ground states for Kirchhoff equations without compact condition , 2015 .

[4]  Xian Wu,et al.  Existence and multiplicity of nontrivial solutions for a class of modified nonlinear fourth-order elliptic equations on RN , 2014, Appl. Math. Comput..

[5]  Jianwen Zhou,et al.  Sign-changing solutions for some fourth-order nonlinear elliptic problems , 2008 .

[6]  C. O. Alves,et al.  Nodal ground state solution to a biharmonic equation via dual method , 2015, 1509.03086.

[7]  Yueh-Cheng Kuo,et al.  The Nehari manifold for a Kirchhoff type problem involving sign-changing weight functions , 2011 .

[8]  D. G. Figueiredo.  Positive solutions of semilinear elliptic problems , 1982 .

[10]  W. Shuai,et al.  Sign-changing solutions for a class of Kirchhoff-type problem in bounded domains☆ , 2015 .

[11]  Juntao Sun,et al.  Ground state solutions for an indefinite Kirchhoff type problem with steep potential well , 2014 .

[12]  I. Ekeland.  Convexity Methods In Hamiltonian Mechanics , 1990 .

[13]  Xiaoming He,et al.  Existence and concentration behavior of positive solutions for a Kirchhoff equation in R3 , 2012 .

[14]  Ó JoãoMarcosdo,et al.  On some fourth-order semilinear elliptic problems in R N , 2002 .

[15]  Fuyi Li,et al.  Existence of a positive solution to Kirchhoff type problems without compactness conditions , 2012 .

[16]  David G. Costa,et al.  An Invitation to Variational Methods in Differential Equations , 2007 .

[17]  Xianhua Tang,et al.  High energy solutions of modified quasilinear fourth-order elliptic equations with sign-changing potential , 2017, Comput. Math. Appl..

[18]  Jacques-Louis Lions,et al.  On Some Questions in Boundary Value Problems of Mathematical Physics , 1978 .

[19]  Fuyi Li,et al.  Positive solutions to Kirchhoff type equations with nonlinearity having prescribed asymptotic behavior , 2014 .

[20]  T. Ma.  Positive solutions for a nonlocal fourth order equation of Kirchhoff type , 2007 .