August  2012, 5(4): 845-855. doi: 10.3934/dcdss.2012.5.845

Multiplicity of solutions for variable exponent Dirichlet problem with concave term

1. 

Ben Gurion University of the Negev, Department of Mathematics, Be'er Sheva 84105

Received  January 2011 Revised  April 2011 Published  November 2011

We consider a nonlinear Dirichlet boundary value problem involving the $p(x)$-Laplacian and a concave term. Our main result shows the existence of at least three nontrivial solutions. We use truncation techniques and the method of sub- and supersolutions.
Citation: 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
References:
[1]

D. Averna, S. A. Marano and D. Motreanu, Multiple solutions for a Dirichlet problem with $p$-Laplacian and set-valued nonlinearity, Bull. Aust. Math. Soc., 77 (2008), 285-303. doi: 10.1017/S0004972708000282.  Google Scholar

[2]

K.-C. Chang, "Infinite-Dimensional Morse Theory and Multiple Solution Problems," Progress in Nonlinear Differential Equations and their Applications, 6, Birkhäuser Boston, Inc., Boston, MA, 1993.  Google Scholar

[3]

S. Carl, V. K. Le and D. Motreanu, "Nonsmooth Variational Problems and Their Inequalities. Comparison Principles and Applications," Springer Monographs in Mathematics, Springer, New York, 2007.  Google Scholar

[4]

S. Carl and D. Motreanu, Constant-sign and sign-changing solutions for nonlinear eigenvalue problems, Nonlinear Anal., 68 (2008), 2668-2676. doi: 10.1016/j.na.2007.02.013.  Google Scholar

[5]

X. Fan, Global $C^{1,\alpha}$ regularity for variable exponent elliptic equations in divergence form, J. Differential Equations, 235 (2007), 397-417. doi: 10.1016/j.jde.2007.01.008.  Google Scholar

[6]

X. Fan, On the sub-supersolution method for $p(x)$-Laplacian equations, J. Math. Anal. Appl., 330 (2007), 665-682. doi: 10.1016/j.jmaa.2006.07.093.  Google Scholar

[7]

X. Fan and D. Zhao, On the spaces $L^{p(x)}(\Omega)$ and $W^{m,p(x)}(\Omega)$, J. Math. Anal. Appl., 263 (2001), 424-446. doi: 10.1006/jmaa.2000.7617.  Google Scholar

[8]

X. Fan, J. Shen and D. Zhao, Sobolev embedding theorems for spaces $W^{k,p(x)}(\Omega)$, J. Math. Anal. Appl., 262 (2001), 749-760. doi: 10.1006/jmaa.2001.7618.  Google Scholar

[9]

X. Fan, Y. Z. Zhao and Q. H. Zhang, A strong maximum principle for p(x)-Laplace equations, Chinese J. Contemp. Math., 24 (2003), 277-282.  Google Scholar

[10]

O. Kováčik and J. Rákosník, On spaces $L^{p(x)}$ and $W^{k,p(x)}$, Czechoslovak Math. J., 41(116) (1991), 592-618.  Google Scholar

[11]

V. Maz'ja, "Sobolev Spaces," Translated from the Russian by T. O. Shaposhnikova, Springer Series in Soviet Mathematics, Springer-Verlag, Berlin, 1985.  Google Scholar

[12]

D. Motreanu, Three solutions with precise sign properties for systems of quasilinear elliptic equations,, Discrete Contin. Dyn. Syst. Ser. S, ().   Google Scholar

[13]

D. Motreanu, V. V. Motreanu and N. S. Papageorgiou, A unified approach for multiple constant sign and nodal solutions, Adv. Differential Equations, 12 (2007), 1363-1392.  Google Scholar

[14]

D. Motreanu, V. V. Motreanu and N. S. Papageorgiou, Multiple constant sign and nodal solutions for nonlinear Neumann eigenvalue problems, Ann. Sc. Norm. Super. Pisa Cl. Sci., 10 (2011), 729-755. Google Scholar

[15]

D. Motreanu, V. V. Motreanu and N. S. Papageorgiou, On $p$-Laplace equations with concave terms and asymmetric perturbations, Proc. Roy. Soc. Edinburgh Sect. A, 141 (2011), 171-192.  Google Scholar

[16]

D. Motreanu and N. S. Papageorgiou, Multiple solutions for nonlinear Neumann problems driven by a nonhomogeneous differential operator, Proc. Amer. Math. Soc., 139 (2011), 3527-3535. doi: 10.1090/S0002-9939-2011-10884-0.  Google Scholar

[17]

D. Motreanu and Z. Zhang, Constant sign and sign changing solutions for systems of quasilinear elliptic equations, Set-Valued Anal., 19 (2011), 255-269. doi: 10.1007/s11228-010-0142-z.  Google Scholar

[18]

N. S. Papageorgiou and E. Rocha, A multiplicity theorem for a variable exponent Dirichlet problem, Glasg. Math. J., 50 (2008), 335-349. doi: 10.1017/S0017089508004242.  Google Scholar

show all references

References:
[1]

D. Averna, S. A. Marano and D. Motreanu, Multiple solutions for a Dirichlet problem with $p$-Laplacian and set-valued nonlinearity, Bull. Aust. Math. Soc., 77 (2008), 285-303. doi: 10.1017/S0004972708000282.  Google Scholar

[2]

K.-C. Chang, "Infinite-Dimensional Morse Theory and Multiple Solution Problems," Progress in Nonlinear Differential Equations and their Applications, 6, Birkhäuser Boston, Inc., Boston, MA, 1993.  Google Scholar

[3]

S. Carl, V. K. Le and D. Motreanu, "Nonsmooth Variational Problems and Their Inequalities. Comparison Principles and Applications," Springer Monographs in Mathematics, Springer, New York, 2007.  Google Scholar

[4]

S. Carl and D. Motreanu, Constant-sign and sign-changing solutions for nonlinear eigenvalue problems, Nonlinear Anal., 68 (2008), 2668-2676. doi: 10.1016/j.na.2007.02.013.  Google Scholar

[5]

X. Fan, Global $C^{1,\alpha}$ regularity for variable exponent elliptic equations in divergence form, J. Differential Equations, 235 (2007), 397-417. doi: 10.1016/j.jde.2007.01.008.  Google Scholar

[6]

X. Fan, On the sub-supersolution method for $p(x)$-Laplacian equations, J. Math. Anal. Appl., 330 (2007), 665-682. doi: 10.1016/j.jmaa.2006.07.093.  Google Scholar

[7]

X. Fan and D. Zhao, On the spaces $L^{p(x)}(\Omega)$ and $W^{m,p(x)}(\Omega)$, J. Math. Anal. Appl., 263 (2001), 424-446. doi: 10.1006/jmaa.2000.7617.  Google Scholar

[8]

X. Fan, J. Shen and D. Zhao, Sobolev embedding theorems for spaces $W^{k,p(x)}(\Omega)$, J. Math. Anal. Appl., 262 (2001), 749-760. doi: 10.1006/jmaa.2001.7618.  Google Scholar

[9]

X. Fan, Y. Z. Zhao and Q. H. Zhang, A strong maximum principle for p(x)-Laplace equations, Chinese J. Contemp. Math., 24 (2003), 277-282.  Google Scholar

[10]

O. Kováčik and J. Rákosník, On spaces $L^{p(x)}$ and $W^{k,p(x)}$, Czechoslovak Math. J., 41(116) (1991), 592-618.  Google Scholar

[11]

V. Maz'ja, "Sobolev Spaces," Translated from the Russian by T. O. Shaposhnikova, Springer Series in Soviet Mathematics, Springer-Verlag, Berlin, 1985.  Google Scholar

[12]

D. Motreanu, Three solutions with precise sign properties for systems of quasilinear elliptic equations,, Discrete Contin. Dyn. Syst. Ser. S, ().   Google Scholar

[13]

D. Motreanu, V. V. Motreanu and N. S. Papageorgiou, A unified approach for multiple constant sign and nodal solutions, Adv. Differential Equations, 12 (2007), 1363-1392.  Google Scholar

[14]

D. Motreanu, V. V. Motreanu and N. S. Papageorgiou, Multiple constant sign and nodal solutions for nonlinear Neumann eigenvalue problems, Ann. Sc. Norm. Super. Pisa Cl. Sci., 10 (2011), 729-755. Google Scholar

[15]

D. Motreanu, V. V. Motreanu and N. S. Papageorgiou, On $p$-Laplace equations with concave terms and asymmetric perturbations, Proc. Roy. Soc. Edinburgh Sect. A, 141 (2011), 171-192.  Google Scholar

[16]

D. Motreanu and N. S. Papageorgiou, Multiple solutions for nonlinear Neumann problems driven by a nonhomogeneous differential operator, Proc. Amer. Math. Soc., 139 (2011), 3527-3535. doi: 10.1090/S0002-9939-2011-10884-0.  Google Scholar

[17]

D. Motreanu and Z. Zhang, Constant sign and sign changing solutions for systems of quasilinear elliptic equations, Set-Valued Anal., 19 (2011), 255-269. doi: 10.1007/s11228-010-0142-z.  Google Scholar

[18]

N. S. Papageorgiou and E. Rocha, A multiplicity theorem for a variable exponent Dirichlet problem, Glasg. Math. J., 50 (2008), 335-349. doi: 10.1017/S0017089508004242.  Google Scholar

[1]

Liping Wang. Arbitrarily many solutions for an elliptic Neumann problem with sub- or supercritical nonlinearity. Communications on Pure & Applied Analysis, 2010, 9 (3) : 761-778. doi: 10.3934/cpaa.2010.9.761

[2]

Salvatore A. Marano, Nikolaos S. Papageorgiou. Positive solutions to a Dirichlet problem with $p$-Laplacian and concave-convex nonlinearity depending on a parameter. Communications on Pure & Applied Analysis, 2013, 12 (2) : 815-829. doi: 10.3934/cpaa.2013.12.815

[3]

Agnid Banerjee, Nicola Garofalo. On the Dirichlet boundary value problem for the normalized $p$-laplacian evolution. Communications on Pure & Applied Analysis, 2015, 14 (1) : 1-21. doi: 10.3934/cpaa.2015.14.1

[4]

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

[5]

Elhoussine Azroul, Abdelmoujib Benkirane, and Mohammed Shimi. On a nonlocal problem involving the fractional $ p(x,.) $-Laplacian satisfying Cerami condition. Discrete & Continuous Dynamical Systems - S, 2021, 14 (10) : 3479-3495. doi: 10.3934/dcdss.2020425

[6]

Said Taarabti. Positive solutions for the $ p(x)- $Laplacian : Application of the Nehari method. Discrete & Continuous Dynamical Systems - S, 2021  doi: 10.3934/dcdss.2021029

[7]

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

[8]

Zaizheng Li, Qidi Zhang. Sub-solutions and a point-wise Hopf's lemma for fractional $ p $-Laplacian. Communications on Pure & Applied Analysis, 2021, 20 (2) : 835-865. doi: 10.3934/cpaa.2020293

[9]

Michael E. Filippakis, Nikolaos S. Papageorgiou. Existence and multiplicity of positive solutions for nonlinear boundary value problems driven by the scalar $p$-Laplacian. Communications on Pure & Applied Analysis, 2004, 3 (4) : 729-756. doi: 10.3934/cpaa.2004.3.729

[10]

Jinguo Zhang, Dengyun Yang. Fractional $ p $-sub-Laplacian operator problem with concave-convex nonlinearities on homogeneous groups. Electronic Research Archive, , () : -. doi: 10.3934/era.2021036

[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]

Marek Galewski, Renata Wieteska. Multiple periodic solutions to a discrete $p^{(k)}$ - Laplacian problem. Discrete & Continuous Dynamical Systems - B, 2014, 19 (8) : 2535-2547. doi: 10.3934/dcdsb.2014.19.2535

[13]

Rossella Bartolo, Anna Maria Candela, Addolorata Salvatore. Infinitely many radial solutions of a non--homogeneous $p$--Laplacian problem. Conference Publications, 2013, 2013 (special) : 51-59. doi: 10.3934/proc.2013.2013.51

[14]

Everaldo S. de Medeiros, Jianfu Yang. Asymptotic behavior of solutions to a perturbed p-Laplacian problem with Neumann condition. Discrete & Continuous Dynamical Systems, 2005, 12 (4) : 595-606. doi: 10.3934/dcds.2005.12.595

[15]

Antonia Chinnì, Roberto Livrea. Multiple solutions for a Neumann-type differential inclusion problem involving the $p(\cdot)$-Laplacian. Discrete & Continuous Dynamical Systems - S, 2012, 5 (4) : 753-764. doi: 10.3934/dcdss.2012.5.753

[16]

K. D. Chu, D. D. Hai. Positive solutions for the one-dimensional singular superlinear $ p $-Laplacian problem. Communications on Pure & Applied Analysis, 2020, 19 (1) : 241-252. doi: 10.3934/cpaa.2020013

[17]

Alfonso Castro, Jorge Cossio, Sigifredo Herrón, Carlos Vélez. Infinitely many radial solutions for a $ p $-Laplacian problem with indefinite weight. Discrete & Continuous Dynamical Systems, 2021, 41 (10) : 4805-4821. doi: 10.3934/dcds.2021058

[18]

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

[19]

Francesca Colasuonno, Benedetta Noris. A p-Laplacian supercritical Neumann problem. Discrete & Continuous Dynamical Systems, 2017, 37 (6) : 3025-3057. doi: 10.3934/dcds.2017130

[20]

Francesca Colasuonno, Fausto Ferrari. The Soap Bubble Theorem and a $ p $-Laplacian overdetermined problem. Communications on Pure & Applied Analysis, 2020, 19 (2) : 983-1000. doi: 10.3934/cpaa.2020045

2020 Impact Factor: 2.425

Metrics

  • PDF downloads (75)
  • HTML views (0)
  • Cited by (11)

Other articles
by authors

[Back to Top]