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Four Nontrivial Solutions for Kirchhoff Problems with Critical Potential, Critical Exponent and a Concave Term
Departement of Mathematics, College of Science, Qassim University, Buraidah, Kingdom of Saudi Arabia
- 1 Departement of Mathematics, College of Science, Qassim University, Buraidah, Kingdom of Saudi Arabia
Applied Mathematics·Volume 06 (2015)·Pages 2248–2256·Published 21 December 2015·DOI10.4236/am.2015.614198
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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.
KeywordsKirchhoff ProblemsCritical PotentialConcave termNehari ManifoldMountain Pass Theorem
- Kirchhoff, G.R. (1883) Vorlesungen über mathematische Physik—Mechanik. 3 Edition. Teubner, Leipzig.
- 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
- 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
- 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
- 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.
- 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
- 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
- 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
- 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
- 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
- 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
- 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