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January  2018, 17(1): 39-52. doi: 10.3934/cpaa.2018003

## A nonlinear eigenvalue problem with $p(x)$-growth and generalized Robin boundary value condition

 1 Institute of Mathematics "Simion Stoilow" of the Romanian Academy, P.O. Box 1-764,014700 Bucharest, Romania, Department of Mathematics, University of Craiova, Street A.I. Cuza 13,200585 Craiova, Romania 2 School of Mathematics, Iran University of Science and Technology, Narmak, Tehran, Iran

Received  August 2016 Revised  July 2017 Published  September 2017

We are concerned with the study of the following nonlinear eigenvalue problem with Robin boundary condition
 $\begin{cases} -{\rm div}\,(a(x,\nabla u))=λ b(x,u)&\mbox{in} \ Ω\\\dfrac{\partial A}{\partial n}+β(x) c(x,u)=0&\mbox{on}\\partialΩ.\end{cases}$
The abstract setting involves Sobolev spaces with variable exponent. The main result of the present paper establishes a sufficient condition for the existence of an unbounded sequence of eigenvalues. Our arguments strongly rely on the Lusternik-Schnirelmann principle. Finally, we focus to the following particular case, which is a $p(x)$-Laplacian problem with several variable exponents:
 $\begin{cases} -{\rm div}\,(a_0(x) |\nabla u|^{p(x)-2}\nabla u)=λ b_0(x)|u|^{q(x)-2}u&\mbox{in} \ Ω\\|\nabla u|^{p(x)-2}\dfrac{\partial u}{\partial n}+β(x)|u|^{r(x)-2} u=0&\mbox{on}\\partialΩ.\end{cases}$
Combining variational arguments, we establish several properties of the eigenvalues family of this nonhomogeneous Robin problem.
Citation: VicenŢiu D. RǍdulescu, Somayeh Saiedinezhad. A nonlinear eigenvalue problem with $p(x)$-growth and generalized Robin boundary value condition. Communications on Pure and Applied Analysis, 2018, 17 (1) : 39-52. doi: 10.3934/cpaa.2018003
##### References:
 [1] R. Agarwal, M. B. Ghaemi and S. Saiedinezhad, The existence of weak solution for degenerate $\sum {{\Delta _{{p_i}(x)}}}$-equation, J. Comput. Anal. Appl., 13 (2011), 629-641. [2] C. Alves and Marco A. S. Souto, Existence of solutions for a class of problems in ${\mathbb R}^N$ involving the p(x)-Laplacian, in Contributions to nonlinear analysis, Birkhäuser Basel, (2005), 17-32. doi: 10.1007/3-7643-7401-2_2. [3] R. Aronson, Boundary conditions for diffusion of light, J. Opt. Soc. Am. A, 12 (1995), 2532-2539. [4] F. Browder, On the eigenfunctions and eigenvalues of the general linear elliptic differential operator, Proc. Nat. Acad. Sci. USA, 39 (1953), 433-439. [5] F. Browder, Lusternik-Schnirelmann category and nonlinear elliptic eigenvalue problems, Bull. Amer. Math. Soc., 71 (1965), 644-648.  doi: 10.1090/S0002-9904-1965-11378-7. [6] F. Browder, Variational methods for nonlinear elliptic eigenvalue problems, Bull. Amer. Math. Soc., 71 (1965), 176-183.  doi: 10.1090/S0002-9904-1965-11275-7. [7] F. Browder, Existence theorems for nonlinear partial differential equations, 1970 Global Analysis (Proc. Sympos. Pure Math., Vol. XVI, Berkeley, Calif., 1968), pp. 1-60, Amer. Math. Soc., Providence, R. I. [8] S.-G. Deng, Eigenvalues of the $p (x)$-Laplacian Steklov problem, J. Math. Anal. Appl., 339 (2008), 925-937.  doi: 10.1016/j.jmaa.2007.07.028. [9] X. Fan, Remarks on eigenvalue problems involving the $p (x)$-Laplacian, J. Math. Anal. Appl., 352 (2009), 85-98.  doi: 10.1016/j.jmaa.2008.05.086. [10] X. Fan, Q. Zhang and D. Zhao, Eigenvalues of $p (x)$-Laplacian Dirichlet problem, J. Math. Anal. Appl., 302 (2005), 306-317.  doi: 10.1016/j.jmaa.2003.11.020. [11] R. Filippucci, P. Pucci and V.D. Rădulescu, Existence and non-existence results for quasilinear elliptic exterior problems with nonlinear boundary conditions, Communications in Partial Differential Equations, 33 (2008), 706-717.  doi: 10.1080/03605300701518208. [12] Y. Fu and Y. Shan, On the removability of isolated singular points for elliptic equations involving variable exponent, Adv. Nonlinear Anal., 5 (2016), 121-132.  doi: 10.1515/anona-2015-0055. [13] O. Kovacik and J. Rakosnik, On spaces $L^{p (x)}$ and $W^{k, p (x)}$, Czechoslovak Mathematical Journal, 41 (1991), 592-618. [14] A. Le, Eigenvalue problems for the $p$-Laplacian, Nonlinear Analysis: Theory, Methods & Applications, 64 (2006), 1057-1099.  doi: 10.1016/j.na.2005.05.056. [15] L. A. Lusternik and L. G. Schnirelmann, Topological Methods in Variational Problems, Trudy Inst. Mat. Mech. Moscow State Univ. (1930), 1-68. [16] M. Mihailescu and V. Răadulescu, A multiplicity result for a nonlinear degenerate problem arising in the theory of electrorheological fluids, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 462 (2006), 2625-2641.  doi: 10.1098/rspa.2005.1633. [17] C. V. Pao, Nonlinear Parabolic and Elliptic Equations, Plenum Press, New York, 1992. [18] V. Răadulescu, Nonlinear elliptic equations with variable exponent: old and new, Nonlinear Anal., 121 (2015), 336-369.  doi: 10.1016/j.na.2014.11.007. [19] V. Răadulescu and D. Repovš, Partial Differential Equations with Variable Exponents: Variational Methods and Qualitative Analysis, CRC Press, Taylor & Francis Group, Boca Raton FL, 2015. doi: 10.1201/b18601. [20] D. Repovš, Stationary waves of Schrödinger-type equations with variable exponent, Anal. Appl. (Singap.), 13 (2015), 645-661.  doi: 10.1142/S0219530514500420. [21] M. Ruzicka, Electrorheological Fluids: Modeling and Mathematical Theory, Springer Science & Business Media, New York, 2000. [22] S. Samko, On a progress in the theory of Lebesgue spaces with variable exponent: maximal and singular operators, Integral Transforms and Special Functions, 16 (2005), 461-482.  doi: 10.1080/10652460412331320322. [23] O. Scherzer (Ed. ), Handbook of Mathematical Methods in Imaging, Springer, Berlin, 2011. [24] J. Simon, Régularité de la solution d'une équation non linéaire dans ${\mathbb R}^N$, Journées d'Analyse Non Linéaire (Proc. Conf., Besan¸con, 1977), pp. 205-227, Lecture Notes in Math., 665, Springer, Berlin, 1978. [25] Z. Yücedag, Solutions of nonlinear problems involving $p(x)$-Laplacian operator, Adv. Nonlinear Anal., 4 (2015), 285-293.  doi: 10.1515/anona-2015-0044. [26] E. Zeidler, Nonlinear Functional Analysis and Its Applications, Ⅲ. Variational Methods and Optimization, Springer Science & Business Media, New York, 2013. doi: 10.1007/978-1-4612-5020-3. [27] E. Zeidler, The Lusternik-Schnirelmann theory for indefinite and not necessarily odd nonlinear operators and its applications, Nonlinear Analysis: Theory, Methods & Applications, 4 (1980), 451-489.  doi: 10.1016/0362-546X(80)90085-1. [28] Q. Zhang, Existence of solutions for $p (x)$-Laplacian equations with singular coefficients in ${\mathbb R}^N$, J. Math. Anal. Appl., 348 (2008), 38-50.  doi: 10.1016/j.jmaa.2008.06.026.

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##### References:
 [1] R. Agarwal, M. B. Ghaemi and S. Saiedinezhad, The existence of weak solution for degenerate $\sum {{\Delta _{{p_i}(x)}}}$-equation, J. Comput. Anal. Appl., 13 (2011), 629-641. [2] C. Alves and Marco A. S. Souto, Existence of solutions for a class of problems in ${\mathbb R}^N$ involving the p(x)-Laplacian, in Contributions to nonlinear analysis, Birkhäuser Basel, (2005), 17-32. doi: 10.1007/3-7643-7401-2_2. [3] R. Aronson, Boundary conditions for diffusion of light, J. Opt. Soc. Am. A, 12 (1995), 2532-2539. [4] F. Browder, On the eigenfunctions and eigenvalues of the general linear elliptic differential operator, Proc. Nat. Acad. Sci. USA, 39 (1953), 433-439. [5] F. Browder, Lusternik-Schnirelmann category and nonlinear elliptic eigenvalue problems, Bull. Amer. Math. Soc., 71 (1965), 644-648.  doi: 10.1090/S0002-9904-1965-11378-7. [6] F. Browder, Variational methods for nonlinear elliptic eigenvalue problems, Bull. Amer. Math. Soc., 71 (1965), 176-183.  doi: 10.1090/S0002-9904-1965-11275-7. [7] F. Browder, Existence theorems for nonlinear partial differential equations, 1970 Global Analysis (Proc. Sympos. Pure Math., Vol. XVI, Berkeley, Calif., 1968), pp. 1-60, Amer. Math. Soc., Providence, R. I. [8] S.-G. Deng, Eigenvalues of the $p (x)$-Laplacian Steklov problem, J. Math. Anal. Appl., 339 (2008), 925-937.  doi: 10.1016/j.jmaa.2007.07.028. [9] X. Fan, Remarks on eigenvalue problems involving the $p (x)$-Laplacian, J. Math. Anal. Appl., 352 (2009), 85-98.  doi: 10.1016/j.jmaa.2008.05.086. [10] X. Fan, Q. Zhang and D. Zhao, Eigenvalues of $p (x)$-Laplacian Dirichlet problem, J. Math. Anal. Appl., 302 (2005), 306-317.  doi: 10.1016/j.jmaa.2003.11.020. [11] R. Filippucci, P. Pucci and V.D. Rădulescu, Existence and non-existence results for quasilinear elliptic exterior problems with nonlinear boundary conditions, Communications in Partial Differential Equations, 33 (2008), 706-717.  doi: 10.1080/03605300701518208. [12] Y. Fu and Y. Shan, On the removability of isolated singular points for elliptic equations involving variable exponent, Adv. Nonlinear Anal., 5 (2016), 121-132.  doi: 10.1515/anona-2015-0055. [13] O. Kovacik and J. Rakosnik, On spaces $L^{p (x)}$ and $W^{k, p (x)}$, Czechoslovak Mathematical Journal, 41 (1991), 592-618. [14] A. Le, Eigenvalue problems for the $p$-Laplacian, Nonlinear Analysis: Theory, Methods & Applications, 64 (2006), 1057-1099.  doi: 10.1016/j.na.2005.05.056. [15] L. A. Lusternik and L. G. Schnirelmann, Topological Methods in Variational Problems, Trudy Inst. Mat. Mech. Moscow State Univ. (1930), 1-68. [16] M. Mihailescu and V. Răadulescu, A multiplicity result for a nonlinear degenerate problem arising in the theory of electrorheological fluids, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 462 (2006), 2625-2641.  doi: 10.1098/rspa.2005.1633. [17] C. V. Pao, Nonlinear Parabolic and Elliptic Equations, Plenum Press, New York, 1992. [18] V. Răadulescu, Nonlinear elliptic equations with variable exponent: old and new, Nonlinear Anal., 121 (2015), 336-369.  doi: 10.1016/j.na.2014.11.007. [19] V. Răadulescu and D. Repovš, Partial Differential Equations with Variable Exponents: Variational Methods and Qualitative Analysis, CRC Press, Taylor & Francis Group, Boca Raton FL, 2015. doi: 10.1201/b18601. [20] D. Repovš, Stationary waves of Schrödinger-type equations with variable exponent, Anal. Appl. (Singap.), 13 (2015), 645-661.  doi: 10.1142/S0219530514500420. [21] M. Ruzicka, Electrorheological Fluids: Modeling and Mathematical Theory, Springer Science & Business Media, New York, 2000. [22] S. Samko, On a progress in the theory of Lebesgue spaces with variable exponent: maximal and singular operators, Integral Transforms and Special Functions, 16 (2005), 461-482.  doi: 10.1080/10652460412331320322. [23] O. Scherzer (Ed. ), Handbook of Mathematical Methods in Imaging, Springer, Berlin, 2011. [24] J. Simon, Régularité de la solution d'une équation non linéaire dans ${\mathbb R}^N$, Journées d'Analyse Non Linéaire (Proc. Conf., Besan¸con, 1977), pp. 205-227, Lecture Notes in Math., 665, Springer, Berlin, 1978. [25] Z. Yücedag, Solutions of nonlinear problems involving $p(x)$-Laplacian operator, Adv. Nonlinear Anal., 4 (2015), 285-293.  doi: 10.1515/anona-2015-0044. [26] E. Zeidler, Nonlinear Functional Analysis and Its Applications, Ⅲ. Variational Methods and Optimization, Springer Science & Business Media, New York, 2013. doi: 10.1007/978-1-4612-5020-3. [27] E. Zeidler, The Lusternik-Schnirelmann theory for indefinite and not necessarily odd nonlinear operators and its applications, Nonlinear Analysis: Theory, Methods & Applications, 4 (1980), 451-489.  doi: 10.1016/0362-546X(80)90085-1. [28] Q. Zhang, Existence of solutions for $p (x)$-Laplacian equations with singular coefficients in ${\mathbb R}^N$, J. Math. Anal. Appl., 348 (2008), 38-50.  doi: 10.1016/j.jmaa.2008.06.026.
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