ON A ROBIN PROBLEM IN ORLICZ-SOBOLEV SPACES

Yıl 2019, Cilt: 9 Sayı: 2, 246 - 256, 01.06.2019

Mustafa Avcı

Öz

In the present paper, we deal with the existence of solutions to a class of an elliptic equation with Robin boundary condition. The problem is settled in Orlicz-Sobolev spaces and the main tool used is Ekeland's variational principle.

Anahtar Kelimeler

Non-homogeneous Robin problem, Variational method, Ekeland variational principle, Orlicz-Sobolev spaces.

Kaynakça

• [1] Adams, R., (1975), Sobolev Spaces, Academic Press, New York.
• [2] Afrouzi, G.A., Mahdavi, S. and Naghizadeh, Z., (2007), The Nehari Manifold for p-Laplacian Equation with Dirichlet Boundary Condition, Nonlinear Analysis: Modelling and Control, 12 (2), pp. 143155.
• [3] Avci, M.,(2013), Ni-Serrin type equations arising from capillarity phenomena with non-standard growth, Boundary Value Problems, 55, pp. 1-13.
• [4] Avci, M. and Pankov, A., (2015), Nontrivial Solutions of Discrete Nonlinear Equations with Variable Exponent, J. Math. Anal. Appl., 431, pp. 22-33.
• [5] Avci, M., (2013), Existence and Multiplicity of Solutions for Dirichlet Problems Involving the p(x)- Laplace Operator, Elec. J. Dif. Eqn., 14, pp. 1-99.
• [6] Avci, M. and Pankov, A., (2018), Multivalued Elliptic Operators with Nonstandard Growth, Advances in Nonlinear Analysis, 7 (1), pp. 35-48.
• [7] Bonanno, G., Molica Bisci, G. and R˘adulescu, V., (2011), Existence of Three Solutions for a Nonhomogeneous Neumann Problem Through Orlicz-Sobolev Spaces, Nonlinear Anal. TMA, 18, pp. 4785- 4795.
• [8] Bonanno, G., Molica Bisci, G. and R˘adulescu, V., (2012), Arbitrarily Small Weak Solutions for a Nonlinear Eigenvalue Problem in Orlicz-Sobolev Spaces, Monatsh. Math., 165, pp. 305-318.
• [9] Bonanno, G., Molica Bisci, G. and R˘adulescu, V., (2011), Infinitely Many Solutions for a Class of Nonlinear Eigenvalue Problem in Orlicz-Sobolev Spaces, C. R. Acad. Sci. Paris. 349(I), pp. 263-268.
• [10] Bonanno, G., Molica Bisci, G. and R˘adulescu, V., (2012), Quasilinear Elliptic Non-homogeneous Dirichlet Problems Through Orlicz-Sobolev Spaces, Nonlinear Anal. TMA, 75, pp. 4441-4456.
• [11] Boureanu, M.M. and Udrea, D.N., (2013), Existence and Multiplicity Results for Elliptic Problems with p(·)-growth Conditions, Nonlinear Anal. Real World Applications, 14, pp. 1829-1844.
• [12] Boureanu, M.M. and Preda, F., (2012), Infinitely Many Solutions for Elliptic Problems with Variable Exponent and Nonlinear Boundary Conditions, Nonl. Diff. Eq. and Appl. (NoDEA), 19, pp. 235-251.
• [13] Cammaroto, F. and Vilasi, L., (2012), Multiple Solutions for a Non-homogeneous Dirichlet Problem in Orlicz-Sobolev Spaces, Appl. Math. Comput., 218, pp. 11518-11527.
• [14] Cekic, B., Kalinin, A.V., Mashiyev, R.A. and Avci, M., (2012), L p(x) (Ω)-estimates of vector fields and some applications to magnetostatics problems, J. Math. Anal. Appl. 389 (2), pp. 838-851.
• [15] Chipot, M., (2009), Elliptic Equations: An Introductory Course, Birkhuser Verlag AG, Basel.
• [16] Chung, N.T., (2014), Multiple Solutions for a Nonlocal Problem in Orlicz-Sobolev Space, Ricerche di Matematica, 63, pp. 169-182.
• [17] Cruz-Uribe, David V. and Fiorenza, A., (2013), Variable Lebesgue Spaces: Foundations and Harmonic Analysis, Springer, Basel.
• [18] Deng, S. G., (2009), Positive Solutions for Robin Problem Involving the p(x)-Laplacian, J. Math. Anal. Appl., 360, pp. 548-560.
• [19] Diening, L., Harjulehto, P., H¨ast¨o, P. and Ruzicka, M., (2011), Lebesgue and Sobolev Spaces with Variable Exponents, Lecture Notes in Mathematics, vol. 2017, Springer-Verlag, Heidelberg.
• [20] Dinca, G., Jebelean, J. and Mawhin, J., (2001), Variational and Topological Methods for Dirichlet Problems with p-Laplacian, Port. Math. (N.S.), 58, pp. 339378.
• [21] Ekeland, I., (1974), On the Variational Principle, J. Math. Anal. Appl., 47, pp. 324-353.
• [22] Fan, X.L., (2010), On Nonlocal p(x)-Laplacian Dirichlet Problems, Nonlinear Anal., 72, pp. 3314-3323.
• [23] Fang, F. and Tan, Z., (2012), Existence and Multiplicity of Solutions for a Class of Quasilinear Elliptic Equations: An Orlicz-Sobolev Setting, J. Math. Anal. Appl., 389, pp. 420-428.
• [24] Fan, X.L. and Zhao, D., (2001), On the Spaces L p(x) (Ω) and W m,p(x) (Ω), J. Math. Anal. Appl., 263, pp. 424446.
• [25] Fan, X.L., (2012), Differential equations of divergence form in MusielakSobolev spaces and a subsupersolution method, J. Math. Anal. Appl. 386 , 593-604.
• [26] Fukagai, N., Ito, M. and Narukawa, K., (2006), Positive Solutions of Quasilinear Elliptic Equations with Critical Orlicz-Sobolev Nonlinearity on R N , Funkcial. Ekvac., 49, pp. 235267.
• [27] Heidarkhani, S., Caristi, G. and Ferrara, M., (2016), Perturbed Kirchhoff-type Neumann Problems in Orlicz-Sobolev Spaces, Comput. Math. Appl., 71, pp. 2008-2019.
• [28] Heidarkhani, S., Afrouzi, G.A., Moradi, S.and Caristi, G., (2017), A Variational Approach for Solving p(x)-biharmonic Equations with Navier Boundary Conditions, Elec. J. Diff. Equations, 25, pp. 115.
• [29] Kufner, A., John, O. and Fuˇcik, S., (1977), Function Spaces, Noordhoff, Leyden.
• [30] Mih˘ailescu, M. and Repov˘s, D., (2011), Multiple Solutions for a Nonlinear and Non-homogeneous Problem in Orlicz-Sobolev Spaces, Appl. Math. Comput., 217, pp. 6624-6632.
• [31] Mih˘ailescu, M. and R˘adulescu, V., (2008), Neumann Problems Associated to Non-homogeneous Differential Operators in Orlicz-Sobolev Spaces, Ann. Inst. Fourier, 58, pp. 2087-2111.
• [32] Musielak, J., (1983), Modular Spaces and Orlicz Spaces, Lecture Notes in Math, vol.1034, SpringerVerlag, Berlin.
• [33] Perera, K. and Zhang, Z., (2005), Multiple Positive Solutions of Singular p-Laplacian Problems by Variational Methods, Bound. Value Probl. 3, pp. 377382.
• [34] R˘adulescu, V.D. and Repov˘s, D.D., (2015), Partial Differential Equations with Variable Equations: Variational Methods and Qualitative Analysis, CRC press, New York.
• [35] Rao, M.M. and Ren, Z.D., (1991), Theory of Orlicz Spaces, Marcel Dekker Inc., New York.
• [36] Yucedag, Z., (2015), Solutions of Nonlinear Problems Involving p(x)-Laplacian Operator, Adv. Nonlinear Anal., 4 , pp. 285-293.
• [37] Yucedag, Z., Avci, M. and Mashiyev, R., (2012), On an elliptic system of p (x)-Kirchhoff-type under Neumann boundary condition, Mathematical Modelling and Analysis, 17 (2), pp. 161-170.