Nonlinear elliptic problems with resonance at the two first eigenvalue : A variational approach
DOI:
https://doi.org/10.4067/S0716-09172001000100003Keywords:
p-laplacian, Eigenvalue, Resonance, Variational method.Abstract
We study the nonlinear elliptic problems with Dirichlet boundary condition
{
-Dpu
=¦(x, u) in W
u
=0 on ¶W
Resonance conditions at the first or at the second eigenvalue will be considered.
References
[1] R.A. Adams, Sobolev spaces. Academic Press, New York, (1975).
[2] A. Anane Simplicité et isolation de la première valeur propre du p-laplacien avec poids. C. R. Ac. Sc. Paris, 305, pp. 725 ? 728, (1987).
[3] A. Anane, J. P. Gossez Strongly nonlinear elliptic problems near resonance: A variational approach. Comm. Part. Diff. Eq. 15(8), pp. 1141-1159, (1990).
[4] A. Anane, N. Tsouli On the second eigenvalue of the p-laplacian, Nonlinear Partial Differential Equations, Pitman Research Notes 343, pp. 1-9, (1996).
[5] Bartolo P., Benci and Fortunato D. Abstrat critical point theorems and applications to some nonlinear problems with strong resonance at infinity, Nonlinear Analysis 7, pp. 981-1012, (1983).
[6] D. G. Costa, A. S. Oliveira Existence of solution for a class of semilinear elliptic problems at double resonance. Bol. Soc. BRAS. Mat., vol 19, pp. 21-37, (1988).
[7] D.G. COSTA, C.A. Magalh¸ces Variational elliptic problems which are nonquadratic at infinity. Nonlinear Analysis, vol 23. No 11, pp. 1401-1412, (1994).
[8] M. Cuesta, D. de Figueiredo, J. P. Gossez The beginning of the Fucik spectrum for the p-laplacian. J. Diff. Equat., 159 , pp. 212- 238, (1999).
[9] D. G. de Figueiredo, J. P. Gossez, Conditions de non résonance pour certains problèmes elliptiques semi-linéaire, C. R. Acad. Sci. Paris 302, pp. 543-545, (1986).
[10] D. G. de Figueireido, J. P. Gossez, Nonresonance below the first eigenvalue for a semilinear elliptic problems. Math. An. 281, pp. 589-610 (1988).
[11] D. G. de Figueiredo, J.P. Gossez, Strict monotonicity of eigenvalues and unique contination. Comm. Part. Diff. Eq., 17, pp. 339-346, (1992).
[12] A. R. EL Amrouss, M. Moussaoui Non resonance entre les deux premières valeurs propres d’un problème quasi-lineaire. Bul. Bel. Math . Soc. 4, (1997).
[13] J. Mawhin, J.R. Ward, M. Willem, Variational methods of semilinear elliptic equations. Arch. Rat. Mech. An 95, pp. 269-277, (1986).
[14] Joao. Marcos. B. do O. Solution to perturbed eigenvalue problems of the p-laplacian in N? . J. D. E. N 11, pp. 1-15 (1997).
[15] M. Moussaoui, J.P. Gossez, A note on noresonance between consecutive eigenvalues for a semilinear elliptic problem Pitman Res. Notes in Math., 343, pp. 155 ? 166, (196).
[16] 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).
[17] P.H. Rabinowitz, Some minimax methods in critical point theory with application to differential equations, CBMS, Regional conf. Ser. Math., vol 65 AMS, Providence Ri. (1986).
[18] M Schechter, Nonlinear elliptic boundary value problems at strong resonance, Amer. J. Math., 112, pp. 439-460, (1990).
[19] E. A. B. Silva, Linking theorems and applications to semilinear elliptic problems at resonance, Nonlinear Analysis TMA, 16, pp. 455-477, (1991).
[2] A. Anane Simplicité et isolation de la première valeur propre du p-laplacien avec poids. C. R. Ac. Sc. Paris, 305, pp. 725 ? 728, (1987).
[3] A. Anane, J. P. Gossez Strongly nonlinear elliptic problems near resonance: A variational approach. Comm. Part. Diff. Eq. 15(8), pp. 1141-1159, (1990).
[4] A. Anane, N. Tsouli On the second eigenvalue of the p-laplacian, Nonlinear Partial Differential Equations, Pitman Research Notes 343, pp. 1-9, (1996).
[5] Bartolo P., Benci and Fortunato D. Abstrat critical point theorems and applications to some nonlinear problems with strong resonance at infinity, Nonlinear Analysis 7, pp. 981-1012, (1983).
[6] D. G. Costa, A. S. Oliveira Existence of solution for a class of semilinear elliptic problems at double resonance. Bol. Soc. BRAS. Mat., vol 19, pp. 21-37, (1988).
[7] D.G. COSTA, C.A. Magalh¸ces Variational elliptic problems which are nonquadratic at infinity. Nonlinear Analysis, vol 23. No 11, pp. 1401-1412, (1994).
[8] M. Cuesta, D. de Figueiredo, J. P. Gossez The beginning of the Fucik spectrum for the p-laplacian. J. Diff. Equat., 159 , pp. 212- 238, (1999).
[9] D. G. de Figueiredo, J. P. Gossez, Conditions de non résonance pour certains problèmes elliptiques semi-linéaire, C. R. Acad. Sci. Paris 302, pp. 543-545, (1986).
[10] D. G. de Figueireido, J. P. Gossez, Nonresonance below the first eigenvalue for a semilinear elliptic problems. Math. An. 281, pp. 589-610 (1988).
[11] D. G. de Figueiredo, J.P. Gossez, Strict monotonicity of eigenvalues and unique contination. Comm. Part. Diff. Eq., 17, pp. 339-346, (1992).
[12] A. R. EL Amrouss, M. Moussaoui Non resonance entre les deux premières valeurs propres d’un problème quasi-lineaire. Bul. Bel. Math . Soc. 4, (1997).
[13] J. Mawhin, J.R. Ward, M. Willem, Variational methods of semilinear elliptic equations. Arch. Rat. Mech. An 95, pp. 269-277, (1986).
[14] Joao. Marcos. B. do O. Solution to perturbed eigenvalue problems of the p-laplacian in N? . J. D. E. N 11, pp. 1-15 (1997).
[15] M. Moussaoui, J.P. Gossez, A note on noresonance between consecutive eigenvalues for a semilinear elliptic problem Pitman Res. Notes in Math., 343, pp. 155 ? 166, (196).
[16] 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).
[17] P.H. Rabinowitz, Some minimax methods in critical point theory with application to differential equations, CBMS, Regional conf. Ser. Math., vol 65 AMS, Providence Ri. (1986).
[18] M Schechter, Nonlinear elliptic boundary value problems at strong resonance, Amer. J. Math., 112, pp. 439-460, (1990).
[19] E. A. B. Silva, Linking theorems and applications to semilinear elliptic problems at resonance, Nonlinear Analysis TMA, 16, pp. 455-477, (1991).
Published
2001-05-01
How to Cite
[1]
M. Moussaoui, “Nonlinear elliptic problems with resonance at the two first eigenvalue : A variational approach”, Proyecciones (Antofagasta, On line), vol. 20, no. 1, pp. 33-51, May 2001.
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