# American Institute of Mathematical Sciences

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

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##### 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
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