Multiplicity of solutions for variable exponent Dirichlet problem with concave term

V. V. Motreanu

doi:10.3934/dcdss.2012.5.845

Article Contents

2012, Volume 5, Issue 4: 845-855. Doi: 10.3934/dcdss.2012.5.845

This issue Previous Article Three solutions with precise sign properties for systems of quasilinear elliptic equations Next Article Noncoercive elliptic equations with subcritical growth

Multiplicity of solutions for variable exponent Dirichlet problem with concave term

V. V. Motreanu^1,

1.
Ben Gurion University of the Negev, Department of Mathematics, Be'er Sheva 84105

Received: December 31, 2010

Revised: March 31, 2011

Published: July 2012

Abstract Related Papers Cited by

Abstract

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.

Keywords:

Mathematics Subject Classification: 35J20, 35J25, 35J70.

Citation:

Related Papers

Cited by

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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[10]	O. Kováčik and J. Rákosník, On spaces $L^{p(x)}$ and $W^{k,p(x)}$, Czechoslovak Math. J., 41(116) (1991), 592-618.
[11]	V. Maz'ja, "Sobolev Spaces," Translated from the Russian by T. O. Shaposhnikova, Springer Series in Soviet Mathematics, Springer-Verlag, Berlin, 1985.
[12]	D. Motreanu, Three solutions with precise sign properties for systems of quasilinear elliptic equations, Discrete Contin. Dyn. Syst. Ser. S, in print.
[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.
[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.
[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.
[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.
[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.
[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.