A nonresonance between non-consecutive eigenvalues of semilinear elliptic equations : Variational methods
DOI:
https://doi.org/10.4067/S0716-09172001000100004Keywords:
Eigenvalue, Resonance, Nonresonance, Variational method.Abstract
We study the solvability of the problem
??u = f(x, u) + h in ? ; u = 0 on ??
when the nonlinearity f is assumed to lie asymptotically between two non- consecutive eigenvalues of ??. We show that this problem is nonresonant.
References
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[10] J. Mawhin, J. R. Ward, Nonresonance and existence for nonlinear elliptic boundary value problems, Nonlinear Analysis TMA, pp. 677-694, (1981).
[11] J. Mawhin, J. R. Ward, M. Willem, Variational methods of semilinear elliptic equations. Arch. Rat. Mech. An 95, pp. 269-277, (1986).
[12] P. H. Rabinowitz, Some minimax theorems and applications to nonlinear partial diffirential equations, Nonlinear Analysis, Cesari, Kannan and Weinberger. Eds, 161-177, Academic Press, Orlando, Fl. (1978).
[13] P. H. Rabinowitz, Some minimax methods in critical point theory with applications to differential equations, CBMS, Regional conf. Ser. Math., vol 65 AMS, Providence Ri. (1986).
[14] M. Schechter, Nonlinear elliptic boundary value problems at strong resonance, Amer. J. Math., 112, pp. 439-460, (1990).
[15] E. A. B. Silva, Linking theorems and applications to semilinear elliptic problems at resonance, Nonlinear Analysis TMA, 16, pp. 455-477, (1991).
[2] L. Boccardo, P. Drabek , D. Giachetti, M. Kucera, Generalisation of Fredholm alternative for nonlinear differential operator, Nonli. An. Th. Math. Appl. 10, pp. 1083-1103, (1986).
[3] H. Berestycki, D. G. de Figueiredo, Double resonance in semilinear elliptic problems. Comm. Partial Differential Equations 6, pp. 91 ? 120, (1981).
[4] D. G. Costa, A. S. Oliveira, Existence of solutions for a class of semilinear elliptic problems at double resonance. Bol. Soc. BRAS. Mat., vol 19, pp. 21-37, (1988).
[5] D. G. Costa, E. A. Deb and E. Silva, Existence of solutions for a class of resonant elliptic problems. J. Math. Anal. Appl. 175, pp. 411-424, (1993).
[6] D. G. Costa, C. A. Magalhces, Variational elliptic problems which are nonquadratic at infinity. Nonlinear Analysis, vol 23. No 11, pp. 1401-1412, (1994).
[7] D. G. de Figueiredo, J. P. Gossez, Conditions de non- résonance pour certains probèmes elliptiques semi-linéaire, C. R. Acad. Sci. Paris 302, pp. 543-545, (1986).
[8] D. G. de Figueiredo, J. P. Gossez, Nonresonance below the first eigenvalue for a semilinear elliptic problems, Math. An. 281, pp. 589-610, (1988).
[9] D. G. de Figueiredo, J. P. Gossez, Strict monotonicity of eigenvalues and unique contination, Comm. Part. Diff. Eq. , 17 , pp. 339-346, (1992)
[10] J. Mawhin, J. R. Ward, Nonresonance and existence for nonlinear elliptic boundary value problems, Nonlinear Analysis TMA, pp. 677-694, (1981).
[11] J. Mawhin, J. R. Ward, M. Willem, Variational methods of semilinear elliptic equations. Arch. Rat. Mech. An 95, pp. 269-277, (1986).
[12] P. H. Rabinowitz, Some minimax theorems and applications to nonlinear partial diffirential equations, Nonlinear Analysis, Cesari, Kannan and Weinberger. Eds, 161-177, Academic Press, Orlando, Fl. (1978).
[13] P. H. Rabinowitz, Some minimax methods in critical point theory with applications to differential equations, CBMS, Regional conf. Ser. Math., vol 65 AMS, Providence Ri. (1986).
[14] M. Schechter, Nonlinear elliptic boundary value problems at strong resonance, Amer. J. Math., 112, pp. 439-460, (1990).
[15] E. A. B. Silva, Linking theorems and applications to semilinear elliptic problems at resonance, Nonlinear Analysis TMA, 16, pp. 455-477, (1991).
Published
2017-04-24
How to Cite
[1]
M. Moussaoui, “A nonresonance between non-consecutive eigenvalues of semilinear elliptic equations : Variational methods”, Proyecciones (Antofagasta, On line), vol. 20, no. 1, pp. 53-63, Apr. 2017.
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