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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

* Corresponding author: Somayeh Saiedinezhad

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 & 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. Google Scholar [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. Google Scholar [3] R. Aronson, Boundary conditions for diffusion of light, J. Opt. Soc. Am. A, 12 (1995), 2532-2539. Google Scholar [4] F. Browder, On the eigenfunctions and eigenvalues of the general linear elliptic differential operator, Proc. Nat. Acad. Sci. USA, 39 (1953), 433-439. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [13] O. Kovacik and J. Rakosnik, On spaces $L^{p (x)}$ and $W^{k, p (x)}$, Czechoslovak Mathematical Journal, 41 (1991), 592-618. Google Scholar [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. Google Scholar [15] L. A. Lusternik and L. G. Schnirelmann, Topological Methods in Variational Problems, Trudy Inst. Mat. Mech. Moscow State Univ. (1930), 1-68.Google Scholar [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. Google Scholar [17] C. V. Pao, Nonlinear Parabolic and Elliptic Equations, Plenum Press, New York, 1992. Google Scholar [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. Google Scholar [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. Google Scholar [20] D. Repovš, Stationary waves of Schrödinger-type equations with variable exponent, Anal. Appl. (Singap.), 13 (2015), 645-661. doi: 10.1142/S0219530514500420. Google Scholar [21] M. Ruzicka, Electrorheological Fluids: Modeling and Mathematical Theory, Springer Science & Business Media, New York, 2000. Google Scholar [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. Google Scholar [23] O. Scherzer (Ed. ), Handbook of Mathematical Methods in Imaging, Springer, Berlin, 2011.Google Scholar [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.Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar

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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. Google Scholar [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. Google Scholar [3] R. Aronson, Boundary conditions for diffusion of light, J. Opt. Soc. Am. A, 12 (1995), 2532-2539. Google Scholar [4] F. Browder, On the eigenfunctions and eigenvalues of the general linear elliptic differential operator, Proc. Nat. Acad. Sci. USA, 39 (1953), 433-439. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [13] O. Kovacik and J. Rakosnik, On spaces $L^{p (x)}$ and $W^{k, p (x)}$, Czechoslovak Mathematical Journal, 41 (1991), 592-618. Google Scholar [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. Google Scholar [15] L. A. Lusternik and L. G. Schnirelmann, Topological Methods in Variational Problems, Trudy Inst. Mat. Mech. Moscow State Univ. (1930), 1-68.Google Scholar [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. Google Scholar [17] C. V. Pao, Nonlinear Parabolic and Elliptic Equations, Plenum Press, New York, 1992. Google Scholar [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. Google Scholar [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. Google Scholar [20] D. Repovš, Stationary waves of Schrödinger-type equations with variable exponent, Anal. Appl. (Singap.), 13 (2015), 645-661. doi: 10.1142/S0219530514500420. Google Scholar [21] M. Ruzicka, Electrorheological Fluids: Modeling and Mathematical Theory, Springer Science & Business Media, New York, 2000. Google Scholar [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. Google Scholar [23] O. Scherzer (Ed. ), Handbook of Mathematical Methods in Imaging, Springer, Berlin, 2011.Google Scholar [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.Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar
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