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Vanational elliptic problems involving noncoercive functionals

Published online by Cambridge University Press:  14 November 2011

Miguel Ramos  and
Luis Sanchez
Miguel Ramos
Affiliation:
INIC/CMAF, Avenida Professor Gama Pinto, 2, 1699 Lisboa Codex, Portugal
Luis Sanchez
Affiliation:
INIC/CMAF, Avenida Professor Gama Pinto, 2, 1699 Lisboa Codex, Portugal
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Synopsis

We consider the nonlinear elliptic problem at resonance, Δu + λ1u + f(x, u) = h(x) in Ω, u = 0 on ∂Ω, where Ω is a bounded smooth domain in ℝN, λl is the first eigenvalue of –Δ in Ω and h(x) is orthogonal to the first eigenfunction. We give some conditions of solvability in terms of the primitive of f with respect to u.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 112 , Issue 1-2 , 1989 , pp. 177 - 185
Copyright
Copyright © Royal Society of Edinburgh 1989

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References

1.Ahmad, S.. Nonselfadjoint resonance problems with unbounded perturbations. Nonlinear Anal. 10 (1986), 147–156.CrossRefGoogle Scholar
2.Ahmad, S., Lazer, A. C. and Paul, J. L.. Elementary critical point theory and perturbations of elliptic boundary value problems at resonance. Indiana Univ. Math. J. 25 (1976), 933–944CrossRefGoogle Scholar
3.Ambrosetti, A. and Coti-Zelati, V.. Critical points with lack of compactness and singular dynamical systems. Ann. Mat. Pura Appl. (4) 149 (1987), 237–260.CrossRefGoogle Scholar
4.Berestycki, H. and de Figueiredo, D. G.. Double resonance in semilinear elliptic problems. Comm. Partial Differential Equations 6 (1981), 91–120.CrossRefGoogle Scholar
5.Brezis, H. and Nirenberg, L.. Characterization of the ranges of some nonlinear operators and application to boundary value problems. Ann. Scuola Norm. Sup. Pisa Cl. Sci (4) 5 (1978), 225–326.Google Scholar
6.Coti Zelati, V.. Periodic solutions of dynamical systems with bounded potential. J. Differential Equations 67 (1987), 400–413.CrossRefGoogle Scholar
7.de, D. G. Figueiredo and Gossez, J. P.. Nonresonance below the first eigenvalue for a semilinear elliptic problem (to appear).Google Scholar
8.de Figueiredo, D. G. and Ni, W. M.. Perturbations of second order linear elliptic problems by nonlinearities without Landesman–Lazer condition. Nonlinear Anal. 3 (1979), 629–634.CrossRefGoogle Scholar
9.Gupta, C. P.. Solvability of a boundary value problem with the nonlinearity satisfying a sign condition. J. Math. Anal. Appl. 129 (1988), 482–492.CrossRefGoogle Scholar
10.Iannacci, R. and Nkashama, M. N.. Unbounded perturbations of forced second order ordinary differential equations at resonance. J. Differential Equations 69 (1987), 289–309.CrossRefGoogle Scholar
11.Iannacci, R., Nkashama, M. N. and Ward, J. R., Jr. Nonlinear Second Order Elliptic Partial Differential Equations at Resonance (Memphis State University Report Series 87–12).Google Scholar
12.Kannan, R., Nieto, J. J. and Ray, M. B.. A class of nonlinear boundary value problems without Landesman–Lazer condition. J. Math. Anal. Appl. 105 (1985), 1–11.CrossRefGoogle Scholar
13.Mawhin, J.. Critical point theory and nonlinear differential equations. In Equadiff 6, Brno, 1985, eds Vosmansky, J. and Zlamal, M., pp. 49–58. (Berlin: Springer, 1986).Google Scholar
14.Mawhin, J., Ward, J. R. and Willem, M.. Necessary and sufficient conditions for the solvability of a nonlinear two-point boundary value problem. Proc. Amer. Math. Soc. 93 (1985), 667–674.CrossRefGoogle Scholar
15.Mawhin, J., Ward, J. R. Jr and Willem, M.. Variational methods and semilinear elliptic equations. Arch. Rational Mech. Anal. 95 (1986), 269–277.CrossRefGoogle Scholar
16.Mawhin, J. and Willem, M.. Critical points of convex perturbations of some indefinite quadratic forms and semilinear boundary value problems at resonance. Ann. Inst. H. Poincaré 3 (1986), 431–453.CrossRefGoogle Scholar
17.Mawhin, J. and Willem, M.. Critical point theory and Hamiltonian systems (Berlin: Springer, 1989).CrossRefGoogle Scholar
18.Rabinowitz, P. H.. Minimax Methods in Critical Point Theory with Applications to Differential Equations. CBMS Regional Conference Series in Mathematics 65 (Providence: American Mathematical Society, 1986).CrossRefGoogle Scholar
19.Sanchez, L.. Resonance problems with nonlinearity interfering with eigenvalues of higher order. Appl. Anal. 25 (1987), 275–286.CrossRefGoogle Scholar