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Nonsmooth critical point theory and nonlinear elliptic equations at resonance

Part of: Variational problems in infinite-dimensional spaces Miscellaneous topics - Partial differential equations Elliptic equations and systems

Published online by Cambridge University Press:  09 April 2009

Nikolaos C. Kourogenis  and
Nikolaos S. Papageorgiou
Nikolaos S. Papageorgiou
Affiliation:
National Technical UniversityDepartment of Mathematics Zografou Campus Athens 157 80Greece e-mail: npapg@math.ntua.gr
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Abstract

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In this paper we complete two tasks. First we extend the nonsmooth critical point theory of Chang to the case where the energy functional satisfies only the weaker nonsmooth Cerami condition and we also relax the boundary conditions. Then we study semilinear and quasilinear equations (involving the p-Laplacian). Using a variational approach we establish the existence of one and of multiple solutions. In simple existence theorems, we allow the right hand side to be discontinuous. In that case in order to have an existence theory, we pass to a multivalued approximation of the original problem by, roughly speaking, filling in the gaps at the discontinuity points.

Keywords

Nonsmooth critical point theorylocally Lipschitz functionsubdifferentiallinkingnonsmooth Palais-Smale conditionnonsmooth Cerami conditionMountain pass theoremSaddle point theoremproblems at resonancep-Laplacianfirst eigenvalue

MSC classification

Secondary: 35J20: Variational methods for second-order elliptic equations 35R70: Partial differential equations with multivalued right-hand sides 58E05: Abstract critical point theory (Morse theory, Ljusternik-Schnirelman (Lyusternik-Shnirel'man) theory, etc.)
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 69 , Issue 2 , October 2000 , pp. 245 - 271
Copyright
Copyright © Australian Mathematical Society 2000

References

[1]Adams, R., Sobolev spaces (Academic Press, New York, 1975).Google Scholar
[2]Ahmad, S., ‘Multiple nontrivial solutions of resonant and nonresonant asymptotically linear problems’, Proc. Amer. Math. Soc. 96 (1986), 405–409.CrossRefGoogle Scholar
[3]Ahmad, S., Lazer, A. and Paul, J., ‘Elementary critical point theory and perturbations of elliptic boundary value problems at resonance’, Indiana Univ. Math. J. 25 (1976), 933–944.CrossRefGoogle Scholar
[4]Bartolo, P., Benci, V. and Fortunato, D., ‘Abstract critical point theorems and applications to some nonlinear problems with ‘strong resonance’ at infinity’, Nonlinear Anal. 9 (1983), 981–1012.CrossRefGoogle Scholar
[5]Brezis, H. and Nirenberg, L., ‘Remarks on finding critical points’, Comm. Pure Appl. Math. 64 (1991), 939–963.CrossRefGoogle Scholar
[6]Cerami, G., ‘Un criterio di esistenza per i punti critici su varieta illimitate’, Rend. Instit. Lombardo Sci. Lett. 112 (1978), 332–336.Google Scholar
[7]Chang, K.-C., ‘Variational methods for non-differentiable functionals and their applications to partial differential equations’, J. Math. Anal. Appl. 80 (1981), 102–129.CrossRefGoogle Scholar
[8]Clarke, F. H., Optimization and nonsmooth analysis (Wiley, New York, 1983).Google Scholar
[9]Ghoussoub, N., Duality and perturbation methods in critical point theory (Cambridge University Press, Cambridge, 1993).CrossRefGoogle Scholar
[10]Goncalves, J. and Miyagaki, O., ‘Multiple nontrivial solutions of semilinear strongly resonant elliptic equations’, Nonlinear Anal. 19 (1992), 43–52.CrossRefGoogle Scholar
[11]Goncalves, J. and Miyagaki, O., ‘Three solutions for a strongly resonant elliptic problems’, Nonlinear Anal. 24 (1995), 265–272.CrossRefGoogle Scholar
[12]Hu, S. and Papageorgiou, N. S., Handbook of multivalued analysis, volume I: theory (Kluwer, Dordrecht, 1997).CrossRefGoogle Scholar
[13]Kesavan, S., Topics in functional analysis and applications (Wiley, New York, 1989).Google Scholar
[14]Kourogenis, N. C. and Papageorgiou, N. S., ‘Discontinuous quasilinear elliptic problems at resonance’, Colloquium Math. 78 (1998), 213–223.CrossRefGoogle Scholar
[15]Kourogenis, N. C. and Papageorgiou, N. S., ‘Multiple solutions for nonlinear discontinuous elliptic equations near resonance’, Colloquium Math., to appear.Google Scholar
[16]Kourogenis, N. C. and Papageorgiou, N. S., ‘Multiple solutions for nonlinear discontinuous strongly resonant problems’, to appear.Google Scholar
[17]Kourogenis, N. C. and Papageorgiou, N. S., ‘Three nontrivial solutions for a quasilinear elliptic differential equation at resonance with discontinuous right hand side’, J. Math. Anal. Appl. 238 (1999), 477–490.CrossRefGoogle Scholar
[18]Landesman, E., Robinson, S. and Rumbos, A., ‘Multiple solutions of semilinear elliptic problems at resonance’, Nonlinear Anal. 24 (1995), 1045–1059.CrossRefGoogle Scholar
[19]Lieberman, G., ‘Boundary regularity for solutions of degenerate elliptic equations’, Nonlinear Anal. 12 (1988), 1203–1219.CrossRefGoogle Scholar
[20]Lindqvist, P., ‘On the equation div (‖ Dx ‖p-2Dx) + Λ|x|p-2x = 0’, Proc. Amer. Math. Soc. 109 (1990), 157–164.Google Scholar
[21]Rabinowitz, P., Minimax methods in critical point theory with applications to partial differential equations, CBMS 65 (Amer. Math. Soc., Providence, 1986).CrossRefGoogle Scholar
[22]Solimini, S., ‘On the solvability of some elliptic partial differential equations with linear part at resonance’, J. Math. Anal. Appl. 117 (1986), 138–152.CrossRefGoogle Scholar
[23]Thews, K., ‘Nontrivial solutions of elliptic equations at resonance’, Proc. Roy. Soc. Edinburgh 85A (1990), 119–129.Google Scholar
[24]Tolksdorf, P., ‘Regularity for a more general class of quasilinear elliptic equations’, J. Differential Equations 51 (1984), 126–150.CrossRefGoogle Scholar
[25]Ward, J., ‘Applications of critical point theory to weakly nonlinear boundary value problems at resonance’, Houston J. Math. 10 (1984), 291–305.Google Scholar
[26]Zhong, C.-K., ‘On Ekeland's variational principle and a minimax theorem’, J. Math. Anal. Appl. 205 (1997), 239–250.Google Scholar