KIRCHHOFF TYPE PROBLEMS INVOLVING <i>P</i>-BIHARMONIC OPERATORS AND CRITICAL EXPONENTS

Nguyen Thanh Chung; Pham Hong Minh

doi:10.11948/2017041

2017 Volume 7 Issue 2

Article Contents

Article navigation > Journal of Applied Analysis & Computation > 2017 > 7(2): 659-669

Next Article Previous Article

Nguyen Thanh Chung, Pham Hong Minh. KIRCHHOFF TYPE PROBLEMS INVOLVING P-BIHARMONIC OPERATORS AND CRITICAL EXPONENTS[J]. Journal of Applied Analysis & Computation, 2017, 7(2): 659-669. doi: 10.11948/2017041

Citation:

Nguyen Thanh Chung, Pham Hong Minh. KIRCHHOFF TYPE PROBLEMS INVOLVING P-BIHARMONIC OPERATORS AND CRITICAL EXPONENTS[J]. Journal of Applied Analysis & Computation, 2017, 7(2): 659-669. doi: 10.11948/2017041

KIRCHHOFF TYPE PROBLEMS INVOLVING P-BIHARMONIC OPERATORS AND CRITICAL EXPONENTS

Department of Mathematics, Quang Binh University, 312 Ly Thuong Kiet, Dong Hoi, Quang Binh, Viet Nam

Fund Project:

Abstract

Abstract

In this paper, we study the existence of solutions for a class of Kirchhoff type problems involving p-biharmonic operators and critical exponents. The proof is essentially based on the mountain pass theorem due to Ambrosetti and Rabinowitz[2] and the Concentration Compactness Principle due to Lions[18, 19].
- Kirchhoff type problems /
- p-biharmonic operators /
- critical exponents /
- mountain pass theorem
MSC: 47A75;35B38;35P30;34L05;34L30

FullText(HTML)

References

[1]	G. A. Afrouzi, M. Mirzapour and N. T. Chung, Existence and multiplicity of solutions for Kirchhoff type problems involving p(x)-biharmonic operators, Journal of Analysis and Its Applications (ZAA), 2014, (33), 289-303. Google Scholar
[2]	A. Ambrosetti and P. H. Rabinowitz, Dual variational methods in critical points theory and applications, J. Funct. Anal., 1973, 04, 349-381. Google Scholar
[3]	C. O. Alves, J. M. do Ó, and O. H. Miyagaki, On a class of singular biharmonic problems involving critical exponents, J. Math. Anal. Appl., 2003, 277, 12-26. Google Scholar
[4]	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., 2005, 49, 85-93. Google Scholar
[5]	C. O. Alves, F. J. S. A. Corrêa and G. M. Figueiredo, On a class of nonlocal elliptic problems with critical growth, Diff. Equa. Appl., 2010, 2, 409-417. Google Scholar
[6]	G. Autuori, F. Colasuonno and P. Pucci, On the existence of stationary solutions for higher-order p-Kirchhoff problems, Commun. Contemp. Math., 2014, 16(5), 43 pages. Google Scholar
[7]	G. Autuori, A. Fiscella and P. Pucci, Stationary Kirchhoff problems involving a fractional elliptic operator and a critical nonlinearity, Nonlinear Anal., 2015, 125, 699-714. Google Scholar
[8]	J. Benedikt and P. Drábek, Asymptotics for the principal eigenvalue of the pbiharmonic operator on the ball as p approaches 1, Nonlinear Anal., 2014, 95, 735-742. Google Scholar
[9]	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, 2011, 250, 1876-1908. Google Scholar
[10]	M. Chipot and B. Lovat, Some remarks on nonlocal elliptic and parabolic problems, Nonlinear Anal., 1997, 30(7), 4619-4627. Google Scholar
[11]	F. Colasuonno and P. Pucci, Multiplicity of solutions for p(x)-polyharmonic Kirchhoff equations, Nonlinear Anal., 2011, 74, 5962-5974. Google Scholar
[12]	P. Drábek and M. Otani, Global bifurcation result for the p-biharmonic operator, Electron. J. Differential Equations, 2011, 2011(48), 1-19. Google Scholar
[13]	G. M. Figueiredo, Existence of a positive solution for a Kirchhoff problem type with critical growth via truncation argument, J. Math. Anal. Appl., 2013, 401, 706-713. Google Scholar
[14]	G. M. Figueiredo and J. R. S. Junior, Multiplicity of solutions for a Kirchhoff equation with subcritical or critical growth, Diff. Int. Equa., 2012, 25(9-10), 853-868. Google Scholar
[15]	F. Gazzola, H. C. Grunau and M. Squassina, Existence and nonexistence results for critical growth biharmonic elliptic equations, Calc. Var., 2013, 18, 117-143. Google Scholar
[16]	E. M. Hssini, M. Massar and N. Tsouli, Solutions to Kirchhoff equations with critical exponent, Arab J. Math. Sci., 2016, 22, 138-149. Google Scholar
[17]	G. Kirchhoff, Mechanik, Teubner, Leipzig, Germany, 1883. Google Scholar
[18]	P. L. Lions, The concentration compactness principle in the calculus of variations, the limit case (I), Rev. Mat. Iberoamericana, 1995, 1(1), 145-201. Google Scholar
[19]	P. L. Lions, The concentration compactness principle in the calculus of variations, the limit case (Ⅱ), Rev. Mat. Iberoamericana, 1985, 1(2), 45-121. Google Scholar
[20]	V. F. Lubyshev, Multiple solutions of an even-order nonlinear problem with convex concave nonlinearity, Nonlinear Anal., 2011, 74(4), 1345-1354. Google Scholar
[21]	T. F. Ma, Remarks on an elliptic equation of Kirchhoff type, Nonlinear Anal., 2005, 63, 1967-1977. Google Scholar
[22]	D. Naimen, The critical problem of Kirchhoff type elliptic equations in dimension four, J. Differential Equations, 2014, 257, 1168-1193. Google Scholar
[23]	W. Wang and P. Zhao, Nonuniformly nonlinear elliptic equations of pbiharmonic type, J. Math. Anal. Appl., 2008, 348, 730-738. Google Scholar
[24]	Q. L. Xie, X. P. Wu and C. L. Tang, Existence and multiplicity of solutions for Kirchhoff type problem with critical exponent, Comm. Pure Appl. Anal., 2013, 12(6), 2773-2786. Google Scholar
[25]	L. Zhao and N. Zhang, Existence of solutions for a higher order Kirchhoff type problem with exponential critical growth, Nonlinear Anal., 2016, 132, 214-226. Google Scholar