# American Institute of Mathematical Sciences

• Previous Article
Infinitely many solutions for generalized quasilinear Schrödinger equations with sign-changing potential
• CPAA Home
• This Issue
• Next Article
Unilateral global interval bifurcation for Kirchhoff type problems and its applications
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 & 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

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
 [1] Yansheng Zhong, Yongqing Li. On a p-Laplacian eigenvalue problem with supercritical exponent. Communications on Pure & Applied Analysis, 2019, 18 (1) : 227-236. doi: 10.3934/cpaa.2019012 [2] Giuseppina Barletta, Roberto Livrea, Nikolaos S. Papageorgiou. A nonlinear eigenvalue problem for the periodic scalar $p$-Laplacian. Communications on Pure & Applied Analysis, 2014, 13 (3) : 1075-1086. doi: 10.3934/cpaa.2014.13.1075 [3] Lujuan Yu. The asymptotic behaviour of the $p(x)$-Laplacian Steklov eigenvalue problem. Discrete & Continuous Dynamical Systems - B, 2020, 25 (7) : 2621-2637. doi: 10.3934/dcdsb.2020025 [4] Anouar Bahrouni, VicenŢiu D. RĂdulescu. On a new fractional Sobolev space and applications to nonlocal variational problems with variable exponent. Discrete & Continuous Dynamical Systems - S, 2018, 11 (3) : 379-389. doi: 10.3934/dcdss.2018021 [5] Julián Fernández Bonder, Leandro M. Del Pezzo. An optimization problem for the first eigenvalue of the $p-$Laplacian plus a potential. Communications on Pure & Applied Analysis, 2006, 5 (4) : 675-690. doi: 10.3934/cpaa.2006.5.675 [6] Huiling Li, Xiaoliu Wang, Xueyan Lu. A nonlinear Stefan problem with variable exponent and different moving parameters. Discrete & Continuous Dynamical Systems - B, 2020, 25 (5) : 1671-1698. doi: 10.3934/dcdsb.2019246 [7] Futoshi Takahashi. An eigenvalue problem related to blowing-up solutions for a semilinear elliptic equation with the critical Sobolev exponent. Discrete & Continuous Dynamical Systems - S, 2011, 4 (4) : 907-922. doi: 10.3934/dcdss.2011.4.907 [8] Bernd Kawohl, Jiří Horák. On the geometry of the p-Laplacian operator. Discrete & Continuous Dynamical Systems - S, 2017, 10 (4) : 799-813. doi: 10.3934/dcdss.2017040 [9] M. Nakamura, Tohru Ozawa. The Cauchy problem for nonlinear wave equations in the Sobolev space of critical order. Discrete & Continuous Dynamical Systems - A, 1999, 5 (1) : 215-231. doi: 10.3934/dcds.1999.5.215 [10] V. V. Motreanu. Uniqueness results for a Dirichlet problem with variable exponent. Communications on Pure & Applied Analysis, 2010, 9 (5) : 1399-1410. doi: 10.3934/cpaa.2010.9.1399 [11] Ángel Arroyo, Joonas Heino, Mikko Parviainen. Tug-of-war games with varying probabilities and the normalized p(x)-laplacian. Communications on Pure & Applied Analysis, 2017, 16 (3) : 915-944. doi: 10.3934/cpaa.2017044 [12] Masahiro Ikeda, Takahisa Inui, Mamoru Okamoto, Yuta Wakasugi. $L^p$-$L^q$ estimates for the damped wave equation and the critical exponent for the nonlinear problem with slowly decaying data. Communications on Pure & Applied Analysis, 2019, 18 (4) : 1967-2008. doi: 10.3934/cpaa.2019090 [13] Bin Guo, Wenjie Gao. Finite-time blow-up and extinction rates of solutions to an initial Neumann problem involving the $p(x,t)-Laplace$ operator and a non-local term. Discrete & Continuous Dynamical Systems - A, 2016, 36 (2) : 715-730. doi: 10.3934/dcds.2016.36.715 [14] Kanishka Perera, Andrzej Szulkin. p-Laplacian problems where the nonlinearity crosses an eigenvalue. Discrete & Continuous Dynamical Systems - A, 2005, 13 (3) : 743-753. doi: 10.3934/dcds.2005.13.743 [15] Gabriele Bonanno, Giuseppina D'Aguì, Angela Sciammetta. One-dimensional nonlinear boundary value problems with variable exponent. Discrete & Continuous Dynamical Systems - S, 2018, 11 (2) : 179-191. doi: 10.3934/dcdss.2018011 [16] V. V. Motreanu. Multiplicity of solutions for variable exponent Dirichlet problem with concave term. Discrete & Continuous Dynamical Systems - S, 2012, 5 (4) : 845-855. doi: 10.3934/dcdss.2012.5.845 [17] Xing-Bin Pan. An eigenvalue variation problem of magnetic Schrödinger operator in three dimensions. Discrete & Continuous Dynamical Systems - A, 2009, 24 (3) : 933-978. doi: 10.3934/dcds.2009.24.933 [18] Goro Akagi, Kei Matsuura. Well-posedness and large-time behaviors of solutions for a parabolic equation involving $p(x)$-Laplacian. Conference Publications, 2011, 2011 (Special) : 22-31. doi: 10.3934/proc.2011.2011.22 [19] Xudong Shang, Jihui Zhang, Yang Yang. Positive solutions of nonhomogeneous fractional Laplacian problem with critical exponent. Communications on Pure & Applied Analysis, 2014, 13 (2) : 567-584. doi: 10.3934/cpaa.2014.13.567 [20] C. Fabry, Raul Manásevich. Equations with a $p$-Laplacian and an asymmetric nonlinear term. Discrete & Continuous Dynamical Systems - A, 2001, 7 (3) : 545-557. doi: 10.3934/dcds.2001.7.545

2018 Impact Factor: 0.925