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ON A ROBIN PROBLEM IN ORLICZ-SOBOLEV SPACES

Year 2019, Volume: 9 Issue: 2, 246 - 256, 01.06.2019

Abstract

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.

References

  • [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.
Year 2019, Volume: 9 Issue: 2, 246 - 256, 01.06.2019

Abstract

References

  • [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.
There are 37 citations in total.

Details

Primary Language English
Journal Section Research Article
Authors

Mustafa Avcı This is me

Publication Date June 1, 2019
Published in Issue Year 2019 Volume: 9 Issue: 2

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