Existence of a positive solution to Kirchhoff type problems without compactness conditions

Abstract The existence of a positive solution to a Kirchhoff type problem on R N is proved by using variational methods, and the new result does not require usual compactness conditions. A cut-off functional is utilized to obtain the bounded Palais–Smale sequences.

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

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

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

[4]  Chun-Lei Tang,et al.  Existence and multiplicity of solutions for Kirchhoff type equations , 2011 .

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

[6]  P. Lions.  On the Existence of Positive Solutions of Semilinear Elliptic Equations , 1982 .

[7]  Walter A. Strauss,et al.  Existence of solitary waves in higher dimensions , 1977 .

[8]  Xian Wu,et al.  Existence of nontrivial solutions and high energy solutions for Schrödinger–Kirchhoff-type equations in RN☆ , 2011 .

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

[10]  W. Rother,et al.  Nonlinear scalar field equations , 1992, Differential and Integral Equations.

[11]  Pierre-Louis Lions,et al.  Nonlinear scalar field equations, I existence of a ground state , 1983 .

[12]  Xiaoming He,et al.  Multiplicity of solutions for a class of Kirchhoff type problems , 2010 .

[13]  K Fan,et al.  Minimax Theorems. , 1953, Proceedings of the National Academy of Sciences of the United States of America.

[14]  W. Ni,et al.  On the existence of positive entire solutions of a semilinear elliptic equation , 1986 .

[15]  Zhi-Qiang Wang,et al.  On the Ambrosetti-Rabinowitz Superlinear Condition , 2004 .

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

[17]  Louis Jeanjean,et al.  Local conditions insuring bifurcation from the continuous spectrum , 1999 .

[18]  Stefan Le Coz,et al.  An existence and stability result for standing waves of nonlinear Schrödinger equations , 2006, Advances in Differential Equations.

[19]  Xian Wu,et al.  Infinitely many radial solutions for Kirchhoff-type problems in RN , 2010 .

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

[21]  Hiroaki Kikuchi,et al.  Existence and Stability of Standing Waves For Schrödinger-Poisson-Slater Equation , 2007 .

[22]  Michael Struwe,et al.  Variational methods: Applications to nonlinear partial differential equations and Hamiltonian systems , 1990 .

[23]  Jing Zeng,et al.  Ground states of nonlinear Schrödinger equations with potentials , 2006 .