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Nontrivial solutions for a multivalued problem with strong resonance

Published online by Cambridge University Press:  18 May 2009

Vicenţiu D. Rădulescu
Vicenţiu D. Rădulescu
Affiliation:
Department of Mathematics, University of Craiova, 1100 Craiova, Romania
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The Mountain-Pass Theorem of Ambrosetti and Rabinowitz (see [1]) and the Saddle Point Theorem of Rabinowitz (see [21]) are very important tools in the critical point theory of C1-functional. That is why it is natural to ask us what happens if the functional fails to be differentiable. The first who considered such a case were Aubin and Clarke (see [6]) and Chang (see [12]),who gave suitable variants of the Mountain-Pass Theorem for locally Lipschitz functionals which are denned on reflexive Banach spaces. For this aim they replaced the usual gradient with a generalized one, which was firstly defined by Clarke (see [13], [14]).As observed by Brezis (see [12, p. 114]), these abstract critical point theorems remain valid in non-reflexive Banach spaces.

Type
Research Article
Information
Glasgow Mathematical Journal , Volume 38 , Issue 1 , January 1996 , pp. 53 - 59
Copyright
Copyright © Glasgow Mathematical Journal Trust 1996

References

REFERENCES

1.Ambrosetti, A. and Rabinowitz, P. H., Dual variational methods in critical point theory and applications, J. Functional Analysis 14 (1973), 349–381.CrossRefGoogle Scholar
2.Arcoya, D., Periodic solutions of Hamiltonian systems with strong resonance at infinity, Differential Integral Equations 3 (1990), 909–921.CrossRefGoogle Scholar
3.Arcoya, D. and Cañada, A., Critical point theorems and applications to nonlinear boundary value problems, Nonlinear Anal. 14 (1990), 393–411.CrossRefGoogle Scholar
4.Arcoya, D. and Cañada, A., The dual variational principle and discontinuous elliptic problems with strong resonance at infinity, Nonlinear Anal. 15(1990), 1145–1154.CrossRefGoogle Scholar
5.Arcoya, D. and Costa, D. G., Nontrivial solutions for a strongly resonant problem, Differential Integral Equations, to appear.Google Scholar
6.Aubin, J. P. and Clarke, F. H., Shadow prices and duality for a class of optimal control problems, SIAM J. Control Optim. 17 (1979), 567–586.CrossRefGoogle Scholar
7.Aubin, T., Nonlinear analysis on manifolds. Monge-Ampère equations (Springer, 1982).CrossRefGoogle Scholar
8.Bartolo, P., Benci, V. and Fortunato, D., Abstract critical point theorems and applications to some nonlinear problems with “strong” resonance at infinity, Nonlinear Anal. 7 (1983), 981–1012.CrossRefGoogle Scholar
9.Brézis, H., Coron, J. M. and Nirenberg, L., Free vibrations for a nonlinear wave equation and a theorem of P. Rabinowitz, Comm. Pure Appl. Math. 33 (1980), 667–684.CrossRefGoogle Scholar
10.Capozzi, A., Lupo, D. and Solimini, S., Double resonance in semilinear elliptic problems, Comm. Partial Differential Equations 6 (1991), 91–120.Google Scholar
11.Choulli, M., Deville, R. and Rhandi, A., A general mountain pass principle for nondifferentiable functions, Rev. Mat. Apl. 13 (1992), 45–58.Google Scholar
12.Chang, K. C., Variational methods for nondifferentiable functionals and applications to partial differential equations, J. Math. Anal. Appl 80 (1981), 102–129.CrossRefGoogle Scholar
13.Clarke, F. H., Generalized gradients and applications, Trans. Amer. Math. Soc. 205 (1975), 247–262.CrossRefGoogle Scholar
14.Clarke, F. H., Generalized gradients of Lipschitz functionals, Adv. in Math. 40 (1981), 52–67.CrossRefGoogle Scholar
15.Costa, D. G. and Silva, E. A., The Palais-Smale condition versus coercivity, Nonlinear Anal. 16 (1991), 371–381.CrossRefGoogle Scholar
16.Ekeland, I., On the variational principle, J. Math. Anal. Appl. 47 (1974), 324–353.CrossRefGoogle Scholar
17.Hess, P., Nonlinear perturbations of linear elliptic and parabolic problems at resonance, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 5 (1978), 527–537.Google Scholar
18.Landesman, E. A. and Lazer, A. C., Nonlinear perturbations of linear elliptic boundary value problems at resonance, J. Math. Mech. 19 (1969/1970), 609–623.Google Scholar
19.Lupo, D. and Solimini, S., A note on a resonance problem, Proc. Roy. Soc. Edinburgh Sect. A 102 (1986), 1–7.CrossRefGoogle Scholar
20.Mironescu, P. and Rădulescu, V., A multiplicity theorem for locally Lipschitz periodic functionals, J. Math. Anal. Appl.,to appear.Google Scholar
21.Rabinowitz, P. H., Some critical point theorems and applications to semi-linear elliptic partial differential equations, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 5 (1978), 215–223.Google Scholar
22.Rădulescu, V., Mountain pass theorems for non-differentiable functions and applications, Proc. Japan Acad. Ser. A Math. Sci. 69 (1993), 193–198.CrossRefGoogle Scholar
23.Schechter, M., Nonlinear elliptic boundary value problems at strong resonance, Amer. J.Math. 112 (1990), 439–460.CrossRefGoogle Scholar
24.Solimini, S., On the solvability of some elliptic partial differential equations with the linear part at resonance, J. Math. Anal. Appl. 117(1986), 138–152.CrossRefGoogle Scholar
25.Thews, K., Nontrivial solutions of elliptic equations at resonance, Proc. Roy. Soc.Edinburgh Sect. A 85 (1980), 119–129.CrossRefGoogle Scholar
26.Ward, J. R. Jr, A boundary value problem with a periodic nonlinearity, Nonlinear Anal. 10 (1986), 207–213.CrossRefGoogle Scholar