ON A CLASS OF p x -KIRHHOFF TYPE PROBLEMS WITH ROBIN BOUNDARY CONDITIONS AND INDEFINITE WEIGHTS

Year 2020, Volume: 10 Issue: 2, 0 - 2, 01.03.2020

N. T. Chung

Abstract

In this paper, we consider a class of p x -Kirhhoff type problems with Robin boundary conditions and indefinite weights. Under some suitable conditions on the nonlinearities, we establish the existence of at least one non-trivial weak solution for the problem by using the minimum principle and the Ekeland variational principle.

Keywords

p x -Kirhhoff type problems, Robin boundary condition, Singular weights, Variational methods

References

• Acerbi E. and Mingione G., (2002), Regularity results for stationary electro-rheological fluids, Arch. Ration. Mech. Anal., 164, (3), pp. 213-259.
• Allaoui M., (2017), Existence results for a class of p(x)-Kirchhoff problems, Studia Sci. Math. Hungar- ica, 54, (3), pp. 316-331.
• Allaoui M. and Ourraoui A., (2016), Existence results for a class of p(x)-Kirhhoff problem with a singular weight, Mediterr. J. Math., 13, (2), pp. 677-686.
• Amaziane B., Pankratov L. and A. Piatnitski, (2009), Nonlinear flow through double porosity media in variable exponent Sobolev spaces, Nonlinear Anal. Real World Appl., 10, (4), pp. 2521-2530.
• Ambrosetti A., Rabinowitz, P. H., (1973), Dual variational methods in critical points theory and applications, J. Funct. Anal., 14, pp. 349-381.
• Avci M., Cekic B. and Mashiyev R.A., (2011), Existence and multiplicity of the solutions of the p(x)- Kirchhoff type equation via genus theory, Math. Methods Appl. Sci., 34, (14), pp. 1751-1759.
• Avci M., (2013), Ni-Serrin type equations arising from capillarity phenomena with non-standard growth, Bound. Value Probl., 2013: 55.
• Bisci G. M., Radulescu V. D., (2015), Applications of local linking to nonlocal Neumann problems, Commun. Contemp. Math., 17, (1), 1450001.
• Blomgren P., Chan T. F., Mulet P. and Wong C. K., (1997), Total variation image restoration: nu- merical methods and extensions, in Proceedings of the International Conference on Image Processing, 1997, IEEE, 3, pp. 384-387
• Bouslimi, M. and Kefi, K., (2013), Existence of solution for an indefinite weight quasilinear problem with variable exponent, Complex Var. Elliptic Equa., 58, pp. 1655-1666.
• Cekic B., Kalinin A.V., Mashiyev R. A. and M. Avci, (2012), Lp(x)(Ω)-estimates of vector fields and some applications to magnetostatics problems, J. Math. Anal. Appl., 389, (2), pp. 838-851.
• Chung, N. T., (2013), Multiple solutions for a p(x)-Kirchhoff-type equation with sign-changing non- linearities, Complex Var. Elliptic Equa., 58(12), pp. 1637-1646.
• Chung, N. T., (2013), Multiple solutions for a class of p(x)-Kirchhoff type problems with Neumann boundary conditions, Adv. Pure Appl. Math., 4, (2), pp. 165-177.
• Chung, N. T., (2018), Some remarks on a class of p(x)-Laplacian Robin eigenvalue problems, Mediterr. J. Math., 15, (4): 147.
• Chipot, M, and Lovat, B., (1997), Some remarks on nonlocal elliptic and parabolic problems, Nonlinear Anal. (TMA), 30, (7), pp. 4619-4627.
• Colasuonno, F. and Pucci, P., (2011), Multiplicity of solutions for p(x)-polyharmonic Kirchhoff equa- tions, Nonlinear Anal. (TMA), 74, pp. 5962-5974.
• Correa, F. J. S. A. and Figueiredo, G. M., (2006), On an elliptic equation of p-Kirchhoff type via variational methods, Bull. Aust. Math. Soc., 74, pp. 263-277.
• Cruz-Uribe, D. V. and Fiorenza A., (2013), Variable Lebesgue Spaces: Foundations and Harmonic Analysis, Springer, Basel.
• Dai, G., (2013), Three solutions for a nonlocal Dirichlet boundary value problem involving the p(x)- Laplacian, Appl. Anal., 92(1), pp. 191-210.
• Dai, G. and Hao, R., (2009), Existence of solutions for a p(x)-Kirchhoff-type equation, J. Math. Anal. Appl., 359, pp. 275-284.
• Deng, S. G., (2009), Positive solutions for Robin problem involving the p(x)-Laplacian, J. Math. Anal. Appl., 360, pp. 548-560.
• Diening, L., Harjulehto, P., H¨ast¨o P. and Ru˘zi˘cka M., (2011), Lebesgue and Sobolev spaces with variable exponents, Lecture Notes in Mathematics, Vol. 2017, Springer-Verlag, Heidelberg.
• Ekeland I., (1974), On the variational principle, J. Math. Anal. Appl., 47, pp. 324-353.
• Ge, B. and Zhou, Q. M., (2017), Multiple solutions for a Robin-type differential inclusion problem involving the p(x)-Laplacian, Math. Meth. Appl. Sci., 40, (18), (2017), pp. 6229-6238.
• Kefi, K., (2018), On the Robin problem with indefinite weight in Sobolev spaces with variable expo- nents, Zeitschrift f¨ur Analysis und ihre Anwendugen (ZAA), 37, pp. 25-38.
• Kirchhoff, G., (1883), Mechanik, Teubner, Leipzig, Germany.
• Kov´aˇcik, O. and R´akosn´ık, J., (1991), On spaces Lp(x)and W1,p(x), Czechoslovak Math. J., 41, pp. 592-618.
• Ru˘zi˘cka, M., (2000), Electrorheological Fluids: Modeling and Mathematical Theory, Lecture Notes in Mathematics, Vol. 1748, Springer-Verlag, Berlin.
• Wang, L., Xie, K. and Zhang, B., (2018), Existence and multiplicity of solutions for critical Kirchhoff- type p-Laplacian problems, J. Math. Anal. Appl., 458, pp. 361-378.
• Zhikov. V. V., (1997), Meyer-type estimates for solving the nonlinear Stokes system, Differential Equa., 33, (1), pp. 108-115.