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Multiplicity results for semilinear elliptic boundary value problems in Besov and Triebel-Lizorkin spaces

Published online by Cambridge University Press:  20 January 2009

L. Päivärinta  and
T. Runst
L. Päivärinta
Affiliation:
Department of MathematicsUniversity of HelsinkiHallituskatu 15SF-00100 Helsinki 10Finland
T. Runst
Affiliation:
Section MathematikUniversität JenaUniversitätschochhausDDR-6900 JenaGermany
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Abstract

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The paper deals with superlinear elliptic boundary value problems depending on a parameter. Given appropriate hypotheses concerning the asymptotic behaviour of the nonlinearity, we prove lower bounds on the number of solutions. The results generalize a theorem due to Lazer and McKenna within the framework of quasi-Banach spaces of Besov and Triebel-Lizorkin spaces.

Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 34 , Issue 3 , October 1991 , pp. 393 - 410
Copyright
Copyright © Edinburgh Mathematical Society 1991

References

REFERENCES

1.Amman, H. and Hess, P., A multiplicity result for a class of elliptic boundary value problems, Proc. Roy. Soc. Edinburgh Sect. A (1979), 145–151.CrossRefGoogle Scholar
2.Ambrosetti, A. and Prodi, G., On the inversion of some differentiable mappings with singularities between Banach spaces, Ann. Mat. Pura Appl. 93 (1972), 231–246.CrossRefGoogle Scholar
3.Berger, M. S. and Podolak, E., On the solutions of a nonlinear Dirichlet problem, Indiana Univ. Math. J. 24 (1975), 837–849.CrossRefGoogle Scholar
4.Dancer, E. N., On the ranges of certain weakly nonlinear elliptic partial differential equations, J. Math. Pure Appl. 57 (1978), 351–366.Google Scholar
5.Deimling, K., Nonlinear Functional Analysis (Springer-Verlag, Berlin/Heidelberg/New York/ Tokyo, 1985).CrossRefGoogle Scholar
6.Dolph, C. L., Nonlinear integral equations of the Hammerstein type, Trans. Amer. Math. Soc. 60 (1949), 289–307.CrossRefGoogle Scholar
7.Drabek, P. and Runst, T., On the existence of solutions of a semilinear elliptic boundary value problem with superlinear nonlinearities, Z. Anal. Anwendungen, to appear.Google Scholar
8.Franke, J., Regular elliptic boundary value problems in Besov and Triebel-Lizorkin spaces. The case 0<p≦∞, 0<q≦∞, preprint.Google Scholar
9.Franke, J. and Runst, T., On the admissibility of function spaces of type Bp, qs and Fp, qs, and boundary value problems for non-linear partial differential equations, Anal. Math. 13 (1987), 3–27.CrossRefGoogle Scholar
10.Franke, J. and Runst, T., Non-linear perturbations of linear non-invertible boundary value problems in function spaces of type Bp, qs, and Fp, qs, Czechoslovak Math. J. 38 (113) (1988), 623–641.CrossRefGoogle Scholar
11.Fucik, S., Solvability of Nonlinear Equations and Boundary Value Problems (Soc. Czechoslovak Math. Phys., Prague, 1980).Google Scholar
12.Geisler, M. and Runst, T., On a superlinear Ambrosetti-Prodi problem in Besov and Triebel-Lizorkin spaces, to appear.Google Scholar
13.Hess, P. and Ruf, B., On a superlinear elliptic boundary value problem, Math. Z. 164 (1978), 9–14.CrossRefGoogle Scholar
14.Klee, V., Leray-Schauder theory without local convexity, Math. Ann. 141 (1960), 281–285.CrossRefGoogle Scholar
15.Köthe, G., Topologische lineare Räume, I (Springer-Verlag, Berlin/Heidelberg/New York, 1960).CrossRefGoogle Scholar
16.Kazdan, J. L. and Warner, F. W., Remarks on some quasi-linear elliptic equations, Comm. Pure Appl. Math. 28 (1975), 567–597.CrossRefGoogle Scholar
17.Lazer, A. C. and McKenna, P. J., On the number of solutions of a nonlinear Dirichlet problem, J. Math. Anal. Appl. 84 (1981), 282–294.CrossRefGoogle Scholar
18.Lazer, A. C. and McKenna, P. J., Multiplicity of solutions of nonlinear boundary value problems with nonlinearities crossing several higher eigenvalues, J. Reine Angew. Math. 368 (1986), 184–200.Google Scholar
19.Manes, A. and Micheletti, A. M., Un' estensione della teoria variazionale classica degli autovalori per operatori ellittici del secondo ordine, Boll. Un. Mat. Ital. 7 (1973), 285–301.Google Scholar
20.Riedrich, T., Vorlesungen über nichtlineare Operatorengleichungen (Teubner-Texte Math. Teubner, Leipzig, 1976).Google Scholar
21.Ruf, B., On nonlinear elliptic boundary value problems with jumping nonlinearities, Ann. Math. Pura Appl. 128 (1980), 131–151.Google Scholar
22.Runst, T., Mapping properties of non-linear operators in spaces of Triebel-Lizorkin and Besov type, Analysis Math. 12 (1986), 313–346.CrossRefGoogle Scholar
23.Solimini, S., Some remarks on the number of solutions of some nonlinear elliptic problems, Ann. Inst. H. Poincaré 2 (1985), 143–156.CrossRefGoogle Scholar
24.Triebel, H., Theory of Function Spaces (Geest and Porting, Leipzig, 1983; Birkhäuser, Basel, 1983).CrossRefGoogle Scholar
25.Triebel, H., Mapping properties of non-linear operators generated by Φ(u) = |u|p and by holomorphic Φ(u) in function spaces of Besov-Hardy-Sobolev type. Boundary value problems for elliptic differential equations of type Δu = f(x) + Φ(u), Math. Nachr. 117 (1984), 193–213.CrossRefGoogle Scholar
26.Williamson, J. H., Compact linear operators in linear topological spaces. J. London Math. Soc. 29 (1954), 149–156.CrossRefGoogle Scholar
27.Zeidler, E., Vorlesungen über nichtlinear Funktionalanalysis I-Fixpunktsätze (Teubner-Texte Math., Teubner, Leipzig, 1976).Google Scholar