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On the solvability of semilinear differential equations at resonance

Published online by Cambridge University Press:  20 January 2009

Chung-Cheng Kuo
Chung-Cheng Kuo
Affiliation:
Department of Mathematics, Fu Jen University, Taipei, Taiwan, Republic of China
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Abstract

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In this paper we use the Leray–Schauder continuation method to study the existence of solutions for semilinear differential equations Lu + g(x, u) = h, in which the linear operator L on L2(Ω) may be non-self-adjoint, the L2(Ω)-function h belongs to N⊥(L), the nonlinear term g(x, u) ∈ O(|u|α) as |u| → ∞ for some 0 ≤ α < 1 and satisfies

for all v ∈ N(L) – {0}, where and

Keywords

Landesman-Lazer conditionLeray-Schauder continuation method
Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 43 , Issue 1 , February 2000 , pp. 103 - 112
Copyright
Copyright © Edinburgh Mathematical Society 2000

References

1.Ahmad, S., Nonselfadjoint resonance problems with unbounded perturbations, Nonlinear Analysis 10 (1986), 147–156.CrossRefGoogle Scholar
2.Berestycki, H. and De Figueiredo, D. G., Double resonance in semilinear elliptic problems, Commun. PDE 6 (1980), 91–120.CrossRefGoogle Scholar
3.Brezis, H. and Nirenberg, L., Characterizations of the ranges of some nonlinear operators and applications to boundary value problems, Ann. Scuola Norm. Sup. Pisa 5 (1978), 225–326.Google Scholar
4.Drábek, P., On the resonance problem with nonlinearity which has arbitrary linear growth, J. Math. Analysis Appl. 127 (1987), 435–442.CrossRefGoogle Scholar
5.Drábek, P., Landesman–Lazer condition for nonlinear problems with jumping nonlinearities, J. Dig. Eqns 85 (1990), 186–199.CrossRefGoogle Scholar
6.Drábek, P. and Nicolosi, F., Semilinear boundary value problems at resonance with general nonlinearities, Diff. Integ. Eqns 5 (1992), 339–355.Google Scholar
7.De Figueiredo, D. G. and Ni, W. M., Perturbations of a second order linear elliptic problem by nonlinearities without Landesman-Lazer condition, Nonlinear Analysis 3 (1979), 629–634.CrossRefGoogle Scholar
8.Gupta, C. P., Perturbations of second order linear elliptic problems by unbounded nonlinearities, Nonlinear Analysis 6 (1982), 919–933.CrossRefGoogle Scholar
9.Ha, C.-W., On the solvability of an operator equation without Landesman-Lazer condition, J. Math. Analysis Appl. 178 (1993), 547–552.CrossRefGoogle Scholar
10.Hess, P., A remark on the preceding paper of Fucik and Krbec, Math. Z. 155 (1977), 139–141.CrossRefGoogle Scholar
11.Iannacci, R. and Nkashama, M. N., Nonlinear two point boundary value problems at resonance without Landesman—Lazer condition, Proc. Am. Math. Soc. 311 (1989), 711–726.CrossRefGoogle Scholar
12.Iannacci, R. and Nkashama, M. N., Nonlinear elliptic partial differential equations at resonance: higher eigenvalues, Nonlinear Analysis 25 (1995), 455–471.CrossRefGoogle Scholar
13.Iannacci, R., Nkashama, M. N. and Ward Jr, J. R., Nonlinear second order elliptic partial differential equations at resonance, Trans. Am. Math. Soc. 311 (1989), 711–726.CrossRefGoogle Scholar
14.Kuo, C.-C., On the solvability of a nonselfadjoint resonance problems, Nonlinear Analysis 26 (1996), 887–891.CrossRefGoogle Scholar
15.Kuo, C.-C., Solvability of a nonlinear two point boundary value problem at resonance, J. Diff. Eqns 140 (1997), 1–9.CrossRefGoogle Scholar
16.Landesman, E. M. and Lazer, A. C., Nonlinear perturbations of linear elliptic boundary problems at resonance, J. Math. Mech. 19 (1970), 609–623.Google Scholar
17.Lloyd, N. G., Degree theory (Cambridge University Press, 1978).Google Scholar
18.Robinson, S. B. and Landesman, E. M., A general approach to solvability conditions for semilinear elliptic boundary value problems at resonance, Diff. Integ. Eqns 8 (1995), 1555–1569.Google Scholar