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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

VicenŢiu D. RǍdulescu 1, and  Somayeh Saiedinezhad 2,,

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.
Keywords: Nonlinear eigenvalue problem, Lusternik-Schnirelmann principle, variable exponent Sobolev space, p(x)-Laplacian operator.
Mathematics Subject Classification: Primary:34B09, 35P30;Secondary:35J20.
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

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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

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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

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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

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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

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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

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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

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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

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C. V. Pao, Nonlinear Parabolic and Elliptic Equations, Plenum Press, New York, 1992.  Google Scholar

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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

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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

show all references

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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