Hostname: page-component-586b7cd67f-dlnhk Total loading time: 0 Render date: 2024-11-27T03:50:45.667Z Has data issue: false hasContentIssue false
  • English
  • Français

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

Published online by Cambridge University Press:  20 August 2013

FRANCISCO JULIO S. A. CORRÊA  and
AUGUSTO CÉSAR DOS REIS COSTA
FRANCISCO JULIO S. A. CORRÊA
Affiliation:
Centro de Ciências e Tecnologia, Unidade Acadêmica de Matemática e Estatística, Universidade Federal de Campina Grande, CEP:58.109-970, Campina Grande, Paraība, Brazil e-mail: [email protected]
AUGUSTO CÉSAR DOS REIS COSTA
Affiliation:
Instituto de Ciências Exatas e Naturais, Faculdade de Matemática, Universidade Federal do Pará, CEP:66075-110, Belém, Pará, Brazil e-mail: [email protected]
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

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.

Keywords

35J6035J7058E05
Type
Research Article
Information
Glasgow Mathematical Journal , Volume 56 , Issue 2 , May 2014 , pp. 317 - 333
Copyright
Copyright © Glasgow Mathematical Journal Trust 2013 

References

REFERENCES

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), 707–714; doi:10.3934/dcdss.2012.5.707.Google Scholar
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), 85–93.CrossRefGoogle Scholar
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), 927–953.Google Scholar
4.Bebernes, J. W. and Talaga, P., Nonlocal problems modelling shear banding, Comm. Appl. Nonlinear Anal. 3 (2) (1996), 79–103.Google Scholar
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).Google Scholar
6.Burns, T. J., Does a shear band result from a thermal explosion?, Mech. Mater. 17 (1994), 261–271.CrossRefGoogle Scholar
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), 501–525.Google Scholar
8.Carrillo, J. A., On a nonlocal elliptic equation with decreasing nonlinearity arising in plasma physics and heat conduction, Nonlinear Anal. 32 (1) (1998), 97–115.Google Scholar
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), 263–277.Google Scholar
10.Dolbeault, J., Stationary states in plasma physics: Maxwellian solutions of the Vlasov–Poisson system, Math. Models. Meth. Appl. Sci. 1 (1991), 183–148.CrossRefGoogle Scholar
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), 749–760.Google Scholar
12.Fan, X. L. and Zhang, Q. H., Existence of solutions for p(x)-Laplacian Dirichlet problems, Nonlinear Anal. 52 (2003), 1843–1852.Google Scholar
13.Fan, X. L. and Zhao, D., On the spaces Lp(x) and Wm,p(x), J. Math. Anal. Appl. 263 (2001), 424–446.Google Scholar
14.Fan, X., Zhao, Y. and Zhang, Q., A strong maximum principle for p(x)-Laplace equations, Chin. J. Contemp. Math. 21 (1) (2000), 1–7.Google Scholar
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), 137–153.CrossRefGoogle Scholar
16.Gomes, J. M. and Sanchez, L., On a variational approach to some non-local boundary value problems, Appl. Anal., 84 (9) (September 2005), 909–925.Google Scholar
17.Krzywick, A. and Nadzieja, T., Some results concerning the Poisson–Boltzmann equation, Zastosowania Math. Appl. Math., 21 (2) (1991), 265–272.Google Scholar
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), 2625–2641.Google Scholar
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), 2929–2937.CrossRefGoogle Scholar
20.Olmstead, W. E., Nemat-Nasser, S. and Ni, L., Shear bands as surfaces of discontinuity, J. Mech. Phys. Solids, 42 (1994), 697–709.Google Scholar
21.Perera, K. and Zhang, Z., Nontrivial solutions of Kirchhoff-type problems via Yang index, J. Diff. Equ. 221 (2006), 246–255.CrossRefGoogle Scholar
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), 456–463.Google Scholar
23.Ruzicka, M., Electrorheological fluids: modelling and mathematical theory (Springer-Verlag, Berlin, Germany, 2000).CrossRefGoogle Scholar
24.Zhang, Q., A strong maximum principle for differential equations with non-standardp(x)-growth conditions, J. Math. Anal. Appl. 312 (2005), 24–32.Google Scholar