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
[1] R. A. Adams, Sobolev spaces, Academic Press, New York, (1975).
[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).
[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.
Issue
Section
Artículos
-
Attribution — You must give appropriate credit, provide a link to the license, and indicate if changes were made. You may do so in any reasonable manner, but not in any way that suggests the licensor endorses you or your use.
- No additional restrictions — You may not apply legal terms or technological measures that legally restrict others from doing anything the license permits.