Four Nontrivial Solutions for Kirchhoff Problems with Critical Potential, Critical Exponent and a Concave Term

Mohammed El Mokhtar Ould El Mokhtar

Departement of Mathematics, College of Science, Qassim University, Buraidah, Kingdom of Saudi Arabia.

DOI: 10.4236/am.2015.614198                

Abstract

In this paper, we consider the existence of multiple solutions to the Kirchhoff problems with critical potential, critical exponent and a concave term. Our main tools are the Nehari manifold and mountain pass theorem.

Keywords

Kirchhoff Problems, Critical Potential, Concave term, Nehari Manifold, Mountain Pass Theorem

Conflicts of Interest

The authors declare no conflicts of interest.

References

[1]	Kirchhoff, G.R. (1883) Vorlesungen über mathematische Physik—Mechanik. 3 Edition. Teubner, Leipzig.
[2]	Alves, C.O., Correa, F.J.S.A. and Ma, T.F. (2005) Positive Solutions for a Quasilinear Elliptic Equation of Kirchhoff type. Computers & Mathematics with Applications, 49, 85-93. http://dx.doi.org/10.1016/j.camwa.2005.01.008
[3]	Cheng, C.T. and Wu, X. (2009) Existence Results of Positive Solutions of Kirchhoff Type Problems. Nonlinear Analysis, 71, 4883-4892. http://dx.doi.org/10.1016/j.na.2009.03.065
[4]	Ma, T.F. and Rivera, J.E.M. (2003) Positive Solutions for a Nonlinear Nonlocal Elliptic Transmission Problem. Applied Mathematics Letters, 16, 243-248. http://dx.doi.org/10.1016/S0893-9659(03)80038-1
[5]	Chen, C., Kuo, Y. and Wu, T. (2011) The Nehari Manifold for a Kirchhoff Type Problem Involving Sign Changing Weight Functions. Journal of Differential Equations, 250, 1876-1908.
[6]	Mao, A.M. and Zhang, Z.T. (2009) Sign-Changing and Multiple Solutions of Kirchhoff Type Problems without the P.S. Condition. Nonlinear Analysis, 70, 1275-1287. http://dx.doi.org/10.1016/j.na.2008.02.011
[7]	Mao, A.M. and Luan, S.X. (2011) Sign-Changing Solutions of a Class of Nonlocal Quasilinear Elliptic Boundary Value Problems. Journal of Mathematical Analysis and Applications, 383, 239-243. http://dx.doi.org/10.1016/j.jmaa.2011.05.021
[8]	Jin, J.H. and Wu, X. (2010) Infinitely Many Radial Solutions for Kirchhoff-Type Problems in RN. Journal of Mathematical Analysis and Applications, 369, 564-574. http://dx.doi.org/10.1016/j.jmaa.2010.03.059
[9]	Wei, L. and He, X.M. (2012) Multiplicity of High Energy Solutions for Superlinear Kirchho Equations. Journal of Applied Mathematics and Computing, 39, 473-487. http://dx.doi.org/10.1007/s12190-012-0536-1
[10]	He, X.M. and Zou, W.M. (2009) Infinitely Many Positive Solutions for Kirchhoff-Type Problems. Nonlinear Analysis, 70, 1407-1414. http://dx.doi.org/10.1016/j.na.2008.02.021
[11]	Brown, K.J. and Zhang, Y. (2003) The Nehari Manifold for a Semilinear Elliptic Equation with a Sign Changing Weight Function. Journal of Differential Equations, 2, 481-499. http://dx.doi.org/10.1016/S0022-0396(03)00121-9
[12]	Drabek, P., Kufner, A. and Nicolosi, F. (1997) Quasilinear Elliptic Equations with Degenerations and Singularities, Walter de Gruyter Series in Nonlinear Analysis and Applications Vol. 5. New York. http://dx.doi.org/10.1515/9783110804775