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Nontrivial solutions of a semilinear elliptic problem via variational methods

Published online by Cambridge University Press:  17 April 2009

Zhi-Qing Han
Zhi-Qing Han
Affiliation:
Department of Applied Mathematics, Dalian University of Technology, Dalian 116023, Liaoning, Peoples Republic of China, e-mail: [email protected]
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Using variational methods, we investigate the existence of nontrivial solutions of a nonlinear elliptic boundary value problem at resonance under generalised Ahmad-Lazer-Paul conditions. Some new results are obtained and some results in the literature are improved.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 69 , Issue 2 , April 2004 , pp. 267 - 275
Copyright
Copyright © Australian Mathematical Society 2004

References

[1]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), 933–944.CrossRefGoogle Scholar
[2]Amann, H. and Zehnder, E., ‘Nontrivial solutions for a class of non-resonance problems and applications to nonlinear differential equations’, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 7 (1980), 539–603.Google Scholar
[3]Bartolo, P., Benci, V. and Fortunato, D., ‘Abstract critical point theorems and applications to nonlinear problems with “strong” resonance at infinity’, Nonlinear Anal. 7 (1983), 981–1012.CrossRefGoogle Scholar
[4]Capozzi, A., Lupo, D. and Solimini, S., ‘On the existence of a nontrivial solution to nonlinear problems at resonance’, Nonlinear Anal. 13 (1989), 151–163.CrossRefGoogle Scholar
[5]Chang, K.C. and Liu, J.Q., ‘A strong resonance problem’, Chinese Ann. Math. Ser. B 11 (1990), 191–210.Google Scholar
[6]Dancer, E.N., ‘On the Dirichlet problem for weakly non-linear elliptic partial differential equations’, Proc. Roy. Soc. Edinburgh Sect. A 76 (1977), 283–300.CrossRefGoogle Scholar
[7]De Figueiredo, D.G. and Gossez, J.-P, ‘Nonresonance below the first eigenvalue for a semilinear elliptic problem’, Math. Ann. 281 (1988), 589–610.CrossRefGoogle Scholar
[8]Gilbarg, D. and Trudinger, N.S., Elliptic partial differential equations of second order (Springer-Verlag, Berlin, Heidelberg, New York, Tokyo, 1983).Google Scholar
[9]Han, Z.Q., ‘An improvement on the Ahmad-Lazer-Paul's theorem’, J. Math. (Wuhan) 16 (1996), 512–518.Google Scholar
[10]Han, Z.Q., ‘Solvability of elliptic boundary value problems without standard Landes-man-Lazer conditions’, Acta Math. Sin. Engl. Ser. 16 (2000), 349–360.CrossRefGoogle Scholar
[11]Hirano, N., ‘Existence of nontrivial solutions of semilinear elliptic equations’, Nonlinear Anal. 13 (1989), 695–705.CrossRefGoogle Scholar
[12]Landesman, E.M. and Lazer, A., ‘Nonlinear perturbations of a linear elliptic boundary value problems at resonance’, J. Math. Mech. 19 (1970), 609–623.Google Scholar
[13]Li, S.J. and Zou, W.M., ‘The computations of the critical groups with an application to elliptic resonant problems at a higher eigenvalue’, J. Math. Anal. Appl. 235 (1999), 237–259.CrossRefGoogle Scholar
[14]Mizoguchi, N., ‘Asymptotically linear elliptic equations without nonresonance conditions’, J. Differential Equations 113 (1994), 150–165.CrossRefGoogle Scholar
[15]Rabinowitz, P., Minimax methods in critical point theory with applications to differential equations,CBMS Regional Conference Series in Mathematics 65 (American Mathematical Society,Providence, RI,1986)).CrossRefGoogle Scholar
[16]Tang, C.L. and Gao, Q.J., ‘Elliptic resonant problems at higher eigenvalues with an unbounded nonlinear term’, J. Differential Equations 146 (1998), 56–66.CrossRefGoogle Scholar