Acta mathematica scientia, Series B >
EXISTENCE AND MULTIPLICITY OF SOLUTIONS FOR A CLASS OF p(x)-BIHARMONIC EQUATIONS
Received date: 2011-09-27
Revised date: 2012-06-06
Online published: 2013-01-20
Supported by
This work was supported by the National Natural Science Foundation of China (11071198), Scientific Research Fund of SUSE (2011KY03) and Scientific Reserch Fund of Sichuan Provincial Education Department (12ZB081).
In this paper, we study a class of p(x)-biharmonic equations with Navier boundary condition. Using the mountain pass theorem, fountain theorem, local linking theorem and symmetric mountain pass theorem, we establish the existence of at least one solution and infinitely many solutions of this problem, respectively.
LI Lin , TANG Chun-Lei . EXISTENCE AND MULTIPLICITY OF SOLUTIONS FOR A CLASS OF p(x)-BIHARMONIC EQUATIONS[J]. Acta mathematica scientia, Series B, 2013 , 33(1) : 155 -170 . DOI: 10.1016/S0252-9602(12)60202-1
[1] Ambrosetti A, Malchiodi A. Nonlinear Analysis and Semilinear Elliptic Problems. Cambridge Studies in Advanced Mathematics 104. Cambridge: Cambridge University Press, 2007
[2] Ambrosetti A, Rabinowitz P H. Dual variational methods in critical point theory and applications. J Funct Anal, 1973, 14: 349–381
[3] Bartolo P, Benci V, Fortunato D. Abstract critical point theorems and applications to some nonlinear problems with “strong” resonance at infinity. Nonlinear Anal, 1983, 7: 981–1012
[4] Bartsch T. Infinitely many solutions of a symmetric Dirichlet problem. Nonlinear Anal, 1993, 20: 1205–1216
[5] Boucheche Z, Yacoub R, Chtioui H. On a bi-harmonic equation involving critical exponent: existence and multiplicity results. Acta Math Sci, 2011, 31B: 1213–1244
[6] Cerami G. An existence criterion for the critical points on unbounded manifolds. Istit Lombardo Accad Sci Lett Rend A, 1978, 112: 332–336
[7] Costa D G, Magalhäes C A. Variational elliptic problems which are nonquadratic at infinity. Nonlinear Anal, 1994, 23: 1401–1412
[8] Deng Y, Li Y. Regularity of the solutions for nonlinear biharmonic equations in RN. Acta Math Sci, 2009, 29B: 1469–1480
[9] Drábek P, ˆOtani M. Global bifurcation result for the p-biharmonic operator. Electron. J Differ Equ, 2001, (48): 19 (electronic)
[10] Edmunds D E, Lang J, Nekvinda A. On Lp(x) norms. R Soc Lond Proc Ser A Math Phys Eng Sci, 1999, 455: 219–225
[11] Edmunds D E, R´akosn´?k J. Sobolev embeddings with variable exponent. Studia Math, 2000, 143: 267–293
[12] El Amrouss A, Moradi F, Moussaoui M. Existence of solutions for fourth-order PDEs with variable exponentsns. Electron J Differ Equ, 2009, (153): 13 (electronic)
[13] Fan X L, Shen J S, Zhao D. Sobolev embedding theorems for spaces Wk, p(x)(). J Math Anal Appl, 2001, 262: 749–760
[14] Fan X L, Zhao D. On the spaces Lp(x)() and Wm, p(x)(). J Math Anal Appl, 2001, 263: 424–446
[15] Harjulehto P, Häst¨o P, Lˆe ´U V, Nuortio M. Overview of differential equations with non-standard growth. Nonlinear Anal, 2010, 72: 4551–4574
[16] Jeanjean L. Local conditions insuring bifurcation from the continuous spectrum. Math Z, 1999, 232: 651–664
[17] Jeanjean L. On the existence of bounded Palais-Smale sequences and application to a Landesman-Lazertype problem set on RN. Proc Roy Soc Edinburgh Sect A, 1999, 129: 787–809
[18] Jeanjean L, Toland J F. Bounded Palais-Smale mountain-pass sequences. C R Acad Sci, Paris S´er I Math, 1998, 327: 23–28
[19] Kajikiya R. A critical point theorem related to the symmetric mountain pass lemma and its applications to elliptic equations. J Funct Anal, 2005, 225: 352–370
[20] Li G, Yang C. The existence of a nontrivial solution to a nonlinear elliptic boundary value problem of p-Laplacian type without the Ambrosetti-Rabinowitz condition. Nonlinear Anal, 2010, 72: 4602–4613
[21] Liu S, Squassina M. On the existence of solutions to a fourth-order quasilinear resonant problem. Abstr Appl Anal, 2002, 7: 125–133
[22] Liu Y, Wang Z. Biharmonic equations with asymptotically linear nonlinearities. Acta Math Sci, 2007, 27B: 549–560
[23] Liu Z L, Wang Z Q. On the Ambrosetti-Rabinowitz superlinear condition. Adv Nonlinear Stud, 2004, 4: 563–574
[24] Luan S, Mao A. Periodic solutions for a class of non-autonomous Hamiltonian systems. Nonlinear Anal, 2005, 61: 1413–1426
[25] Miyagaki O H, Souto M A S. Superlinear problems without Ambrosetti and Rabinowitz growth condition. J Differ Equ, 2008, 245: 3628–3638
[26] Ruzicka M. Electrorheological Fluids: Modeling and Mathematical Theory. Lecture Notes in Mathematics 1748. Berlin: Springer-Verlag, 2000
[27] Schechter M. A variation of the mountain pass lemma and applications. J London Math Soc. 1991, 44(2): 491–502
[28] Schechter M, Zou W M. Superlinear problems. Pacific J Math, 2004, 214: 145–160
[29] Willem M, Zou W M. On a Schr¨odinger equation with periodic potential and spectrum point zero. Indiana Univ Math J, 2003, 52: 109–132
[30] Zang A B, Fu Y. Interpolation inequalities for derivatives in variable exponent Lebesgue-Sobolev spaces. Nonlinear Anal, 2008, 69: 3629–3636
[31] Zhikov V V. Averaging of functionals of the calculus of variations and elasticity theory. Izv Akad Nauk SSSR Ser Mat, 1986, 50: 675–710
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