Infinitely many solutions for a bi-nonlocal‎ ‎equation with sign-changing weight functions

Document Type: Research Paper


Department of Mathematics‎, ‎Razi University‎, ‎Kermanshah‎, ‎Iran.


In this paper, we investigate the existence of infinitely many solutions for a bi-nonlocal equation with sign-changing weight functions. We use some natural constraints and the Ljusternik-Schnirelman critical point theory on C1-manifolds, to prove our main results.


Main Subjects

C. O. Alves and F. J. S. A. Corrêa, On existence of solutions for a class of problem involving a nonlinear operator, Comm. Appl. Nonlinear Anal. 8 (2001), no. 2, 43--56.

C. O. Alves, F. J. S. A. Corrêa and T. F. Ma, Positive solutions for a quasilinear elliptic equation of Kirchhoff type, Comput. Math. Appl. 49 (2005), no. 1, 85--93.

G. Anello, A uniqueness result for a nonlocal equation of Kirchhoff type and some related open problem, J. Math. Anal. Appl. 373 (2011), no. 1, 248--251.

G. Anello, On a perturbed Dirichlet problem for a nonlocal differential equation of Kirchhoff type, Bound. Value Probl. 2011 (2011), Article ID 891430, 10 pages.

K. J. Brown and Y. Zhang, The Nehari manifold for a semilinear elliptic equation with a sign-changing weight function, J. Differential Equations 193 (2003), no. 2, 481--499.

A. Bonnet, A deformation lemma on a C1 manifold, Manuscripta Math. 81 (1993), no. 3-4, 339--359.

C. Y. Chen, Y. C. Kuo and T. F. Wu, The Nehari manifold for a Kirchhoff type problem involving sign-changing weight functions, J. Differential Equations 250 (2011), no, 4, 1876--1908.

S. J. Chen and L. Li, Multiple solutions for the nonhomogeneous Kirchhoff equation on RN, Nonlinear Anal. Real World Appl. 14 (2013), no. 3, 1477--1486.

B. Cheng, New existence and multiplicity of nontrivial solutions for nonlocal elliptic Kirchhoff type problems, J. Math. Anal. Appl. 394 (2012), no. 2, 488--495.

M. Chipot and J. F. Rodriguse, On a class of nonlocal nonlinear elliptic problems, RAIRO Model. Math. Anal. Numer. 29 (1992), no. 3, 447--468. F. J. S. A. Corra, M. Delgado and A. Suarez, Some non-local population models with

non-linear diffusion, Rev. Integr. Temas Mat. 28 (2010), no. 1, 37--49.

F. J. S. A. Corrêa and S. D. B. Menezes, Existence of solutions to nonlocal and singular elliptic problems via Galerkin method, Electron. J. Differential Equations 2004 (2004), no. 19, 1--10.

F. J. S. A. Corrêa and G. M. Figueiredo, Existence and multiplicity of nontrivial solutions for a bi-nonlocal equation, Adv. Differential Equations 18 (2013), no. 5-6, 587--608.

P. Drabek and S. I. Pohozaev, Positive solutions for the p-Laplacian: application of the fibrering method, Proc. Roy. Soc. Edinburgh Sect. A 127 (1997), no. 4, 703--726.

G. M. Figueiredo, Existence of a positive solution for a Kirchhoff problem type with critical growth via truncation argument, J. Math. Anal. Appl. 401 (2013), no. 2, 706--713.

J. Furter and M. Grinfeld, Local vs. nonlocal interactions in population dynamics, J. Math. Biol. 27 (1989), no. 1, 65--80.

X. He and W. Zou, Existence and concentration behavior of positive solutions for a Kirchhoff equation in R3, J. Differential Equations 252 (2012), no. 2, 1813--1834.

G. Kirchhoff, Mechanik, Teubner, Leipzig, 1883.

P. H. Rabinowitz, Minimax Methods in Critical Point Theory with Applications to Differential Equations, CBMS Reg. Conf. Series Math., 65, Amer. Math. Soc., Providence, 1986.

M. Struwe, Variational Methods, Springer-Verlag, Berlin, 2008.

A. Szulkin, Ljusternik-Schnirelmann theory on C1-manifolds, Ann. Inst. H. Poincare Anal. Non Lineaire 5 (1988) 119--139.

L. Wang, On a quasilinear Schrodinger-Kirchhoff-type equation with radial potentials, Nonlinear Anal. 83 (2013) 58--68.

M. Willem, Minimax Theorems, BirkhäuserBoston, Inc., Boston, 1996.