Existence and multiplicity of solutions for Kirchhoff type equations

Abstract In this paper, the existence and multiplicity of weak solutions are obtained for a class of Kirchhoff type problems with Dirichlet boundary value conditions by using the mountain pass theorem, the local linking theorem, the fountain theorem and the symmetric mountain pass lemma in critical point theory.

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

[2]  M. Willem Minimax Theorems , 1997 .

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

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

[5]  M. Schechter A Variation of the Mountain Pass Lemma and Applications , 1991 .

[6]  Yang Yang,et al.  Nontrivial solutions of a class of nonlocal problems via local linking theory , 2010, Appl. Math. Lett..

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

[8]  Thomas Bartsch,et al.  Infinitely many solutions of a symmetric Dirichlet problem , 1993 .

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

[10]  Ryuji Kajikiya,et al.  A critical point theorem related to the symmetric mountain pass lemma and its applications to elliptic equations , 2005 .

[11]  Duchao Liu On a p-Kirchhoff equation via Fountain Theorem and Dual Fountain Theorem☆ , 2010 .

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

[13]  Bitao Cheng,et al.  Existence results of positive solutions of Kirchhoff type problems , 2009 .

[14]  Liu Shu Infinitely Many Solutions for a Superlinear Elliptic Equation , 2003 .

[15]  Louis Jeanjean,et al.  On the existence of bounded Palais–Smale sequences and application to a Landesman–Lazer-type problem set on ℝN , 1999, Proceedings of the Royal Society of Edinburgh: Section A Mathematics.

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

[17]  P. Rabinowitz Minimax methods in critical point theory with applications to differential equations , 1986 .

[18]  Anmin Mao,et al.  Periodic solutions for a class of non-autonomous Hamiltonian systems , 2005 .