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:
  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.  doi: 10.1017/S0004972708000282. Google Scholar  K.-C. Chang, "Infinite-Dimensional Morse Theory and Multiple Solution Problems,", Progress in Nonlinear Differential Equations and their Applications, 6 (1993). Google Scholar  S. Carl, V. K. Le and D. Motreanu, "Nonsmooth Variational Problems and Their Inequalities. Comparison Principles and Applications,", Springer Monographs in Mathematics, (2007). Google Scholar  S. Carl and D. Motreanu, Constant-sign and sign-changing solutions for nonlinear eigenvalue problems,, Nonlinear Anal., 68 (2008), 2668.  doi: 10.1016/j.na.2007.02.013. Google Scholar  X. Fan, Global $C^{1,\alpha}$ regularity for variable exponent elliptic equations in divergence form,, J. Differential Equations, 235 (2007), 397.  doi: 10.1016/j.jde.2007.01.008. Google Scholar  X. Fan, On the sub-supersolution method for $p(x)$-Laplacian equations,, J. Math. Anal. Appl., 330 (2007), 665.  doi: 10.1016/j.jmaa.2006.07.093. Google Scholar  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.  doi: 10.1006/jmaa.2000.7617. Google Scholar  X. Fan, J. Shen and D. Zhao, Sobolev embedding theorems for spaces $W^{k,p(x)}(\Omega)$,, J. Math. Anal. Appl., 262 (2001), 749.  doi: 10.1006/jmaa.2001.7618. Google Scholar  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. Google Scholar  O. Kováčik and J. Rákosník, On spaces $L^{p(x)}$ and $W^{k,p(x)}$,, Czechoslovak Math. J., 41(116) (1991), 592. Google Scholar  V. Maz'ja, "Sobolev Spaces,", Translated from the Russian by T. O. Shaposhnikova, (1985). Google Scholar  D. Motreanu, Three solutions with precise sign properties for systems of quasilinear elliptic equations,, Discrete Contin. Dyn. Syst. Ser. S, ().   Google Scholar  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. Google Scholar  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.   Google Scholar  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. Google Scholar  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.  doi: 10.1090/S0002-9939-2011-10884-0. Google Scholar  D. Motreanu and Z. Zhang, Constant sign and sign changing solutions for systems of quasilinear elliptic equations,, Set-Valued Anal., 19 (2011), 255.  doi: 10.1007/s11228-010-0142-z. Google Scholar  N. S. Papageorgiou and E. Rocha, A multiplicity theorem for a variable exponent Dirichlet problem,, Glasg. Math. J., 50 (2008), 335.  doi: 10.1017/S0017089508004242. Google Scholar

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##### References:
  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.  doi: 10.1017/S0004972708000282. Google Scholar  K.-C. Chang, "Infinite-Dimensional Morse Theory and Multiple Solution Problems,", Progress in Nonlinear Differential Equations and their Applications, 6 (1993). Google Scholar  S. Carl, V. K. Le and D. Motreanu, "Nonsmooth Variational Problems and Their Inequalities. Comparison Principles and Applications,", Springer Monographs in Mathematics, (2007). Google Scholar  S. Carl and D. Motreanu, Constant-sign and sign-changing solutions for nonlinear eigenvalue problems,, Nonlinear Anal., 68 (2008), 2668.  doi: 10.1016/j.na.2007.02.013. Google Scholar  X. Fan, Global $C^{1,\alpha}$ regularity for variable exponent elliptic equations in divergence form,, J. Differential Equations, 235 (2007), 397.  doi: 10.1016/j.jde.2007.01.008. Google Scholar  X. Fan, On the sub-supersolution method for $p(x)$-Laplacian equations,, J. Math. Anal. Appl., 330 (2007), 665.  doi: 10.1016/j.jmaa.2006.07.093. Google Scholar  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.  doi: 10.1006/jmaa.2000.7617. Google Scholar  X. Fan, J. Shen and D. Zhao, Sobolev embedding theorems for spaces $W^{k,p(x)}(\Omega)$,, J. Math. Anal. Appl., 262 (2001), 749.  doi: 10.1006/jmaa.2001.7618. Google Scholar  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. Google Scholar  O. Kováčik and J. Rákosník, On spaces $L^{p(x)}$ and $W^{k,p(x)}$,, Czechoslovak Math. J., 41(116) (1991), 592. Google Scholar  V. Maz'ja, "Sobolev Spaces,", Translated from the Russian by T. O. Shaposhnikova, (1985). Google Scholar  D. Motreanu, Three solutions with precise sign properties for systems of quasilinear elliptic equations,, Discrete Contin. Dyn. Syst. Ser. S, ().   Google Scholar  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. Google Scholar  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.   Google Scholar  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. Google Scholar  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.  doi: 10.1090/S0002-9939-2011-10884-0. Google Scholar  D. Motreanu and Z. Zhang, Constant sign and sign changing solutions for systems of quasilinear elliptic equations,, Set-Valued Anal., 19 (2011), 255.  doi: 10.1007/s11228-010-0142-z. Google Scholar  N. S. Papageorgiou and E. Rocha, A multiplicity theorem for a variable exponent Dirichlet problem,, Glasg. Math. J., 50 (2008), 335.  doi: 10.1017/S0017089508004242. Google Scholar
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