We establish the existence of at least three solutions for elliptic problems driven by a p(x)-Laplacian. The existence of at least one nontrivial solution is also proved. The approaches are based on the variational methods and critical-point theory.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
S. N. Antontsev and J. F. Rodrigues, “On stationary thermo-rheological viscous flows,” Ann. Univ. Ferrara. Sez. VII, 52, 19–36 (2006).
S. N. Antontsev and S. I. Shmarev, “A model porous medium equation with variable exponent of nonlinearity: Existence, uniqueness, and localization properties of solutions,” Nonlin. Anal., 60, 515–545 (2005).
D. Averna and G. Bonanno, “A mountain pass theorem for a suitable class of functions,” Rocky Mountain J. Math., 39, 707–727 (2009).
D. Averna and G. Bonanno, “A three critical points theorem and its applications to the ordinary Dirichlet problem,” Top. Meth. Nonlin. Anal., 22, 93–103 (2003).
G. Bonanno, “A critical point theorem via the Ekeland variational principle,” Nonlinear Anal., 75, 2992–3007 (2012).
G. Bonanno and P. Candito, “Nondifferentiable functionals and applications to elliptic problems with discontinuous nonlinearities,” J. Different. Equat., 244, 3031–3059 (2008).
G. Bonanno and A. Chinnì, “Discontinuous elliptic problems involving the p(x)-Laplacian,” Math. Nachr., 284, 639–652 (2011).
G. Bonanno and A. Chinnì, “Multiple solutions for elliptic problems involving the p(x)-Laplacian,” Le Matematiche, 66, Fasc. I, 105–113 (2011).
G. Bonanno and A. Chinnì, “A Neumann boundary-value problem for the Sturm–Liouville equation,” Appl. Math. and Comput., 15, 318–327 (2009).
G. Bonanno, B. Di Bella, and D. O’Regan, “Nontrivial solutions for nonlinear fourth-order elastic beam equations,” Comput. and Math. Appl., 62, 1862–1869 (2011).
G. Bonanno, S. Heidarkhani, and D. O’Regan, “Multiple solutions for a class of Dirichlet quasilinear elliptic systems driven by a (p, q)-Laplacian operator,” Dynam. Syst. Appl., 20, 89–100 (2011).
G. Bonanno and S. A. Marano, “On the structure of the critical set of nondifferentiable functions with a weak compactness condition,” Appl. Anal., 89, 1–10 (2010).
G. Bonanno and P. F. Pizzimenti, “Neumann boundary-value problems with not coercive potential,” Mediterr. J. Math., DOI 10. 1007/s00009-011-0136-6.
G. Bonanno and A. Sciammentta, “An existence result of one nontrivial solution for two-point boundary-value problems,” Bull. Austral. Math. Soc., 84, 288–299 (2011).
H. Brézis, Analyse Functionelle-Théorie et Applications, Masson, Paris (1983).
Y. Chen, S. Levine, and M. Rao, “Variable exponent linear growth functional in image restoration,” SIAM J. Appl. Math., 66, No. 4, 1383–1406 (2006).
X. L. Fan and S. G. Deng, “Remarks on Ricceri’s variational principle and applications to the p(x)-Laplacian equations,” Nonlin. Anal., 67, 3064–3075 (2007).
X. L. Fan and X. Han, “Existence and multiplicity of solutions for p(x)-Laplacian equations in R N,” Nonlin. Anal., 59, 173–188 (2004).
X. L. Fan, J. Shen, and D. Zhao, “Sobolev embedding theorems for spaces W k,p(x),” J. Math. Anal. Appl., 262, 749–760 (2001).
X. L. Fan and Q. H. Zhang, “Existence of solutions for p(x)-Laplacian Dirichlet problem,” Nonlin. Anal., 52, 1843–1852 (2003).
X. L. Fan and Q. H. Zhang, “On the spaces L p(x)(Ω) and W m,p(x),” J. Math. Anal. Appl., 263, 424–446 (2001).
B. Ge and Q. M. Zhou, “Multiple solutions to a class of inclusion problem with the p(x)-Laplacian,” Appl. Anal., 91, 895–909 (2001).
B. Ge and Q. M. Zhou, “Three solutions for a differential inclusion problem involving the p(x)-Kirchhoff-type,” Appl. Anal., 1–12 (2011).
P. Harjulehto, P. Hästö, and V. Latvala, “Minimizers of the variable exponent, nonuniformly convex Dirichlet energy,” J. Math. Pures Appl., 89, 174–197 (2008).
S. Heidarkhani and J. Henderson, “Critical point approaches to quasilinear second-order differential equations depending on a parameter,” Top. Meth. Nonlin. Anal., 44, No. 1, 177–197 (2014).
C. Ji, “Remarks on the existence of three solutions for the p(x)-Laplacian equations,” Nonlin. Anal., 74, 2908–2915 (2011).
M. Kováčik and J. Rákosník, “On the spaces L p(x)(Ω) and W 1,p(x),” Czechoslovak Math., 41, 592–618 (1991).
B. Ricceri, “A general variational principle and some of its applications,” J. Comput. Appl. Math., 113, 401–410 (2000).
M. Ružička, Electrorheological Fluids: Modeling and Mathematical Theory, Springer-Verlag, Berlin (2000).
V. V. Zhikov, “Averaging of functionals in the variational calculus and elasticity theory,” Math. Izv. (USSR), 9, 33–66 (1987).
E. Zeidler, Nonlinear Functional Analysis and Its Applications, Springer, Berlin etc., Vol. 2 (1985).
Author information
Authors and Affiliations
Additional information
Published in Ukrains’kyi Matematychnyi Zhurnal, Vol. 66, No. 12, pp. 1676–1693, December, 2014.
Rights and permissions
About this article
Cite this article
Heidarkhani, S., Ge, B. Critical Points Approaches to Elliptic Problems Driven by a p(x)-Laplacian. Ukr Math J 66, 1883–1903 (2015). https://doi.org/10.1007/s11253-015-1057-5
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11253-015-1057-5