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A VARIATIONAL APPROACH FOR A BI-NON-LOCAL ELLIPTIC PROBLEM INVOLVING THE p(x)-LAPLACIAN AND NON-LINEARITY WITH NON-STANDARD GROWTH

  • FRANCISCO JULIO S. A. CORRÊA (a1) and AUGUSTO CÉSAR DOS REIS COSTA (a2)

Abstract

In this paper we are concerned with a class of p(x)-Kirchhoff equation where the non-linearity has non-standard growth and contains a bi-non-local term. We prove, by using variational methods (Mountain Pass Theorem and Ekeland Variational Principle), several results on the existence of positive solutions.

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References

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1.Agarwal, R. P., Perera, K. and Zhang, Z., On some non-local eigenvalue problems, Discrete Contin. Dyn. Syst. Ser. S 5 (4) (August 2012), 707714; doi:10.3934/dcdss.2012.5.707.
2.Alves, C. O, Corrêa, F. J. S. A. and Ma, T. F., Positive solutions for a quasilinear elliptic equation of Kirchhoff type, Comput. Math. Appl. 49 (2005), 8593.
3.Bebernes, J. W. and Lacey, A., Global existence and finite-time blow-up for a class of nonlocal parabolic problems, Adv. Differ. Equ. 2 (6) (1997), 927953.
4.Bebernes, J. W. and Talaga, P., Nonlocal problems modelling shear banding, Comm. Appl. Nonlinear Anal. 3 (2) (1996), 79103.
5.Burns, T. J., On a combustion-like model for plastic strain localization, Chapter 2, in Shock induced transitions and phase structure in general media (Fosdick, R.et al., Editors) (Springer-Verlag, New York, 1992).
6.Burns, T. J., Does a shear band result from a thermal explosion?, Mech. Mater. 17 (1994), 261271.
7.Caglioti, E., Lions, P. L., Marchiori, C. and Pulvirenti, M., A special class of stationary flows for two - dimensional Euler equations: A statistical mechanics description, Comm. Math. Phys. 143 (1992), 501525.
8.Carrillo, J. A., On a nonlocal elliptic equation with decreasing nonlinearity arising in plasma physics and heat conduction, Nonlinear Anal. 32 (1) (1998), 97115.
9.Corrêa, F. J. S. A. and Figueiredo, G. M., On an elliptic equation of p-Kirchhof type via variational methods, Bull. Aust. Math. Soc. 74 (2006), 263277.
10.Dolbeault, J., Stationary states in plasma physics: Maxwellian solutions of the Vlasov–Poisson system, Math. Models. Meth. Appl. Sci. 1 (1991), 183–148.
11.Fan, X. L., Shen, J. S. and Zhao, D., Sobolev embedding theorems for spaces Wk,p(x)(Ω), J. Math. Anal. Appl. 262 (2001), 749760.
12.Fan, X. L. and Zhang, Q. H., Existence of solutions for p(x)-Laplacian Dirichlet problems, Nonlinear Anal. 52 (2003), 18431852.
13.Fan, X. L. and Zhao, D., On the spaces Lp(x) and Wm,p(x), J. Math. Anal. Appl. 263 (2001), 424446.
14.Fan, X., Zhao, Y. and Zhang, Q., A strong maximum principle for p(x)-Laplace equations, Chin. J. Contemp. Math. 21 (1) (2000), 17.
15.Gogny, D. and Lions, P. L., Sur les états déqulibre pour las densités életroniques das les plasmas, RAIRO Modél. Math. Anal. Numér. 23 (1998), 137153.
16.Gomes, J. M. and Sanchez, L., On a variational approach to some non-local boundary value problems, Appl. Anal., 84 (9) (September 2005), 909925.
17.Krzywick, A. and Nadzieja, T., Some results concerning the Poisson–Boltzmann equation, Zastosowania Math. Appl. Math., 21 (2) (1991), 265272.
18.Mihailescu, M. and Radulescu, V., A multiplicity result for a nonlinear degenerate problem arising in the theory of electrorheological fluids, Proc. R. Soc. A, 462 (2006), 26252641.
19.Mihailescu, M. and Radulescu, V., On a non-homogeneous quasilinear eigenvalue problem in Sobolev spaces with variable exponent, Proc. Amer. Math. Soc., 135 (9) (September 2007), 29292937.
20.Olmstead, W. E., Nemat-Nasser, S. and Ni, L., Shear bands as surfaces of discontinuity, J. Mech. Phys. Solids, 42 (1994), 697709.
21.Perera, K. and Zhang, Z., Nontrivial solutions of Kirchhoff-type problems via Yang index, J. Diff. Equ. 221 (2006), 246255.
22.Perera, K. and Zhang, Z., Sign changing solutions of Kirchhoff type problems via invariant sets of descent flows, J. Math. Anal. Appl. 317 (2006), 456463.
23.Ruzicka, M., Electrorheological fluids: modelling and mathematical theory (Springer-Verlag, Berlin, Germany, 2000).
24.Zhang, Q., A strong maximum principle for differential equations with non-standardp(x)-growth conditions, J. Math. Anal. Appl. 312 (2005), 2432.

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A VARIATIONAL APPROACH FOR A BI-NON-LOCAL ELLIPTIC PROBLEM INVOLVING THE p(x)-LAPLACIAN AND NON-LINEARITY WITH NON-STANDARD GROWTH

  • FRANCISCO JULIO S. A. CORRÊA (a1) and AUGUSTO CÉSAR DOS REIS COSTA (a2)

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