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On a nonlinear elliptic boundary-value problem

Published online by Cambridge University Press:  17 April 2009

E.N. Dancer
Affiliation:
Department of Mathematics, University of New England, Armidale, New South Wales.
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Abstract

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We announce a number of results on the existence of solutions of nonlinear elliptic boundary value problems in the case where the dominating linear part is not invertible. Our theorems improve recent results of Landesman and Lazer, and Williams.

Type
Research Article
Copyright
Copyright © Australian Mathematical Society 1975

References

[1]Ambrosetti, Antonio, Prodi, Giovanni, “On the inversion of some differentiable mappings with singularities between Banach spaces”, Theory of nonlinear operators, 928 (Proc. Summer School, September 1971, Babylon, Czechoslovakia. Academia Publishing House of the Czechoslovak Academy of Sciences, Prague; Academic Press, New York and London; 1973).Google Scholar
[2]Fučík, Svatopluk, “Further remark on a theorem of E.M. Landesman and A.C. Lazer”, Comment. Math. Univ. Carolinae 15 (1974), 259271.Google Scholar
[3]Fučík, Svatopluk, Lovicar, Vladimír, “Boundary value and periodic problem for the equation x″(t) + g(x(t)) = p(t)”, Comment. Math. Univ. Carolinae 15 (1974), 351355.Google Scholar
[4]Hess, Peter, “On sami-coercive nonlinear problems”, J. Math. Mech. 23 (1974), 645654.Google Scholar
[5]Landesman, E.M. & Lazer, A.C., “Nonlinear perturbations of linear elliptic boundary value problems at resonance”, J. Math. Mech. 19 (1970), 609623.Google Scholar
[6]Landesman, E.M. and Lazer, A.C., “Linear eigenvalues and a nonlinear boundary value problem”, Pacific J. Math. 33 (1970), 311328.CrossRefGoogle Scholar
[7]Lazer, A.C. and Leach, D.E., “Bounded perturbations of forced harmonic oscillators at resonance”, Ann. Mat. Pura Appl. 82 (1969), 4968.CrossRefGoogle Scholar
[8]Nirenberg, L., Topics in nonlinear functional analysis (Courant Institute of Mathematical Sciences, New York University, New York, 1973–1974).Google Scholar
[9]Schatzman, Mlle Michelle, “Problèmes aux limites non linéaires semi-coercifs”, C.R. Acad. Sci. Paris 275 (1972), 13051308.Google Scholar
[10]Trudinger, N.S., “Linear elliptic operators with measurable coefficients”, Ann. Scuola. Norm. Sup. Pisa 27 (1973), 265308.Google Scholar
[11]Williams, S.A., “A sharp sufficient condition for solution of a nonlinear elliptic boundary value problem”, J. Differential Equations 8 (1970), 580586.Google Scholar