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A MULTIPLICITY THEOREM FOR A VARIABLE EXPONENT DIRICHLET PROBLEM

Published online by Cambridge University Press:  01 May 2008

NIKOLAOS S. PAPAGEORGIOU  and
EUGÉNIO M. ROCHA
NIKOLAOS S. PAPAGEORGIOU
Affiliation:
Dept. of Mathematics, National Technical University, Athens, Greece e-mail: [email protected]
EUGÉNIO M. ROCHA*
Affiliation:
Dept. of Mathematics, University of Aveiro, Portugal e-mail: [email protected]
*
*The corresponding author.
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Abstract

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We consider a nonlinear Dirichlet problem driven by the p(ċ)-Laplacian. Using variational methods based on the critical point theory, together with suitable truncation techniques and the use of upper-lower solutions and of critical groups, we show that the problem has at least three nontrivial solutions, two of which have constant sign (one positive, the other negative). The hypotheses on the nonlinearity incorporates in our framework of analysis, both coercive and noncoercive problems.

Keywords

35J6035J7058E05
Type
Research Article
Information
Glasgow Mathematical Journal , Volume 50 , Issue 2 , May 2008 , pp. 335 - 349
Copyright
Copyright © Glasgow Mathematical Journal Trust 2008

References

REFERENCES

1.Acerbi, E. and Mingione, G.Regularity results for stationary electro-rheological fluids, Arch. Rational Mech.Anal. 164 (2002), 213–259.CrossRefGoogle Scholar
2.Aizicovici, S., Papageorgiou, N. S., and Staicu, V., Degree theory for operators of monotone type and nonlinear elliptic equations with inequality constraints, Mem. Amer. Math. Soc. (in press).Google Scholar
3.Bartsch, T. and Liu, Z.On a superlinear elliptic p-Laplacian equation, J. Differential Equations 198 (2004), 149–175.CrossRefGoogle Scholar
4.Carl, S. and Perera, K.PereraSign-changing and multiplicity solutions for the p-Laplacian, Abstr. Appl. Anal. 7 (2003), 613–626.CrossRefGoogle Scholar
5.Chang, K.-C., Infinite dimensional Morse theory and multiple solution problems, (Birkhäuser, 1993).CrossRefGoogle Scholar
6.Fan, X.On the sub-supersolution method for p(x)-Laplacian equations, J. Math. Anal. Appl. 330 (2007), 665–682.CrossRefGoogle Scholar
7.Fan, X., Global C1,αregularity for variable exponent elliptic equations in divergence form, J. Differential Equations 235 (2007), 397–417.CrossRefGoogle Scholar
8.Fan, X. and Zhang, Q.Existence of solutions for p(x)-Laplacian Dirichelet problem, Nonlinear Anal. 52 (2003), 1843–1852.CrossRefGoogle Scholar
9.Fan, X. and Zhao, D.A class of De Giorgi type and Hölder continuity, Nonlinear Anal. 36 (1999), 295–318.CrossRefGoogle Scholar
10.Gasinski, L. and Papageorgiou, N. S., Nonlinear analysis, (Chapman and Hall/CRC, Boca Raton 2006).CrossRefGoogle Scholar
11.Guo, Y. and Liu, J.Solution of p-sublinear p-Laplacian equation via Morse theory, J. London Math. Soc. (2) 72 (2005), 632–644.CrossRefGoogle Scholar
12.Jiu, Q. and Su, J.Existence and multiplicity results for Dirichlet problems with p-Laplacian, J. Math. Anal. Appl. 281 (2003), 587–601.CrossRefGoogle Scholar
13.Kováčik, O. and Rákosník, J., On spaces Lp(x)(Ω) and Wk, p(x)(Ω), Czechosloval Math. J. 41 (1991), 592–618.CrossRefGoogle Scholar
14.Liu, S.Multiple solutions for coercive p-Laplacian equations, J. Math. Anal. Appl. 316 (2006), 229–236.CrossRefGoogle Scholar
15.Liu, J. and Liu, S.On the existence of multiple solutions to quasilinear elliptic equations, Bull. London Math. Soc. 37 (2005), 592–600.CrossRefGoogle Scholar
16.Mawhin, J. and Willem, M.Critical point theory and Hamilton systems (Springer-Verlag, 1989).CrossRefGoogle Scholar
17.Mihailescu, M.Existence and multiplicity of solutions for an elliptic equation with p(x)-growth conditions, Glasgow Math. J. 48 (2005), 411–418.CrossRefGoogle Scholar
18.Papageorgiou, E. and Papageorgiou, N.S.A multiplicity theorem for problems with the p-Laplacian, J. Funct. Anal. 244 (2007), 63–77.CrossRefGoogle Scholar
19.Ruzička, M., Electrorheological fluids modelling and mathematical theory (Springer-Verlag, 2000).CrossRefGoogle Scholar
20.Zang, A., p(x)-Laplacian equations satisfying Cerami condition, J.Math. Anal. Appl., to appear.Google Scholar
21.Zhang, Z., Chen, J., and Li, S.Construction of pseudogradient vector field and multiple solutions involving p-Laplacian, J. Differential Equations 201 (2004), 287–303.CrossRefGoogle Scholar
22.Zhang, Z. and Li, S.Sign-changing and multiple solutions of the p-Laplacian, J. Funct. Anal. 197 (2003), 447–468.CrossRefGoogle Scholar