Existence results of positive solutions of Kirchhoff type problems

Abstract In the present paper, we use variational methods to prove two existence results of positive solutions of the following Kirchhoff type problems { − ( a + b ∫ Ω | ∇ u | 2 ) Δ u = f ( x , u ) , in  Ω ; u = 0 , on  ∂ Ω . One deals with the asymptotic behaviors of f near zero and infinity and the other deals with 4-superlinear of f at infinity.

[1]  Marcelo M. Cavalcanti,et al.  Global existence and uniform decay rates for the Kirchhoff-Carrier equation with nonlinear dissipation , 2001, Advances in Differential Equations.

[2]  Kanishka Perera,et al.  Sign changing solutions of Kirchhoff type problems via invariant sets of descent flow , 2006 .

[3]  Xiaoming He,et al.  Infinitely many positive solutions for Kirchhoff-type problems , 2009 .

[4]  Kanishka Perera,et al.  Nontrivial solutions of Kirchhoff-type problems via the Yang index , 2006 .

[5]  Jaime E. Muñoz Rivera,et al.  Positive solutions for a nonlinear nonlocal elliptic transmission problem , 2003, Appl. Math. Lett..

[6]  W. D. Evans,et al.  PARTIAL DIFFERENTIAL EQUATIONS , 1941 .

[7]  Olímpio H. Miyagaki,et al.  Nontrivial Solutions for Perturbations of the p-Laplacian on Unbounded Domains , 1995 .

[8]  Giovanna Cerami.  Un criterio di esistenza per i punti critici su varieta'illimitate , 1978 .

[9]  To Fu Ma,et al.  Positive solutions for a quasilinear elliptic equation of Kirchhoff type , 2005 .

[10]  Stefano Panizzi,et al.  On the Well-Posedness of the Kirchhoff String , 1996 .

[11]  S. Spagnolo,et al.  Global solvability for the degenerate Kirchhoff equation with real analytic data , 1992 .

[12]  Anmin Mao,et al.  Sign-changing and multiple solutions of Kirchhoff type problems without the P.S. condition , 2009 .

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

[14]  An Application of a Mountain Pass Theorem , 2002 .

[15]  Michel Chipot,et al.  Some remarks on non local elliptic and parabolic problems , 1997 .