## Multiplicity results for asymptotically linear elliptic problems at resonance.(English)Zbl 1290.35109

Summary: Several new multiplicity results for asymptotically linear elliptic problem at resonance are obtained via Morse theory and minimax methods. Some new observations on the critical groups of a local linking-type critical point are used to deal with the resonance case at $$0$$.

### MSC:

 35J60 Nonlinear elliptic equations 35J20 Variational methods for second-order elliptic equations 35A30 Geometric theory, characteristics, transformations in context of PDEs 47J30 Variational methods involving nonlinear operators
Full Text:

### References:

 [1] Amann, H.; Zehnder, E., Nontrivial solutions for a class of nonresonance problem and applications to nonlinear differential equations, Ann. scuola norm. sup. Pisa cl. sci., 7, 539-603, (1980) · Zbl 0452.47077 [2] Bartolo, P.; Benci, V.; Fortunato, D., Abstract critical point theorems and applications to nonlinear problems with “strong” resonance at infinity, Nonlinear anal., 7, 981-1012, (1983) · Zbl 0522.58012 [3] Bartsch, T.; Li, S.J., Critical point theory for asymptotically quadratic functionals and applications to problems with resonance, Nonlinear anal., 28, 419-441, (1997) · Zbl 0872.58018 [4] Brezis, H.; Nirenberg, L., H1 versus C1 local minimizers, C. R. acad. sci. Paris, 317, 465-475, (1993) [5] Brezis, H.; Nirenberg, L., Remarks on finding critical points, Comm. pure appl. math., 64, 939-963, (1991) · Zbl 0751.58006 [6] Chang, K.C., H1 versus C1 isolated critical points, C. R. acad. sci. Paris, 319, 441-446, (1994) [7] Chang, K.C., Infinite dimensional Morse theory and multiple solution problems, (1993), Birkhäuser Boston [8] Chang, K.C.; Li, S.J.; Liu, J.Q., Remarks on multiple solutions for asymptotically linear elliptic boundary value problems, Topol. methods nonlinear anal., 3, 179-187, (1994) · Zbl 0812.35031 [9] Costa, D.G.; Silva, E.A., On a class of resonant problems at higher eigenvalues, Differential integral equations, 8, 663-671, (1995) · Zbl 0812.35045 [10] Gromoll, D.; Meyer, W., On differential functions with isolated point, Topology, 8, 361-369, (1969) · Zbl 0212.28903 [11] Landesman, E.; Lazer, A.C., Nonlinear perturbations of linear eigenvalue problems at resonance, J. math. mech., 19, 609-623, (1970) · Zbl 0193.39203 [12] Landesman, E.; Robinson, S.; Rumbos, A., Multiple solutions of semilinear elliptic problems at resonance, Nonlinear anal., 24, 1049-1059, (1995) · Zbl 0832.35048 [13] Li, S.J.; Liu, J.Q., Nontrivial critical point for asymptotically quadratic functions, J. math. anal. appl., 165, 333-345, (1992) · Zbl 0767.35025 [14] Li, S.J.; Liu, J.Q., Computations of critical groups at degenerate critical point and application to nonlinear differential equations with resonance, Houston J. math., 25, 563-582, (1999) · Zbl 0981.58011 [15] Li, S.J.; Wang, Z.Q., Mountain pass theorem in order intervals and multiple solutions for semilinear elliptic Dirichlet problems, J. d’anal. math., 81, 373-396, (2000) · Zbl 0962.35065 [16] Li, S.J.; Willem, M., Multiple solutions for asymptotically linear boundary value problems in which the nonlinearity crosses at least one eigenvalue, Nonlinear differential equations appl., 5, 479-490, (1998) · Zbl 0933.35066 [17] Li, S.J.; Willem, M., Applications of local linking to critical point theory, J. math. anal. appl., 189, 6-32, (1995) · Zbl 0820.58012 [18] Li, S.J.; Zou, W.M., The computations of the critical groups with an application to elliptic resonant problems at higher eigenvalue, J. math. anal. appl., 235, 237-259, (1999) · Zbl 0935.35055 [19] Liu, J.Q., A Morse index for a saddle point, Systems sci. math. sci., 2, 32-39, (1989) · Zbl 0732.58011 [20] Mawhin, J.; Willem, M., Critical point theory and Hamiltonian systems, (1989), Springer-Verlag Berlin · Zbl 0676.58017 [21] Perera, K., Critical groups of critical points produced by local linking with applications, Abstr. appl. anal., 3, 437-446, (1998) · Zbl 0972.58008 [22] Rabinowitz, P., Minimax methods in critical point theory with application to differential equations, Cbms, 65, (1986), Amer. Math. Soc. Providence [23] Schechter, M., Elliptic resonance problems with unequal limits at infinity, Math. ann., 300, 629-642, (1994) · Zbl 0819.35059 [24] Schechter, M., Bounded resonance problem for semilinear elliptic equations, Nonlinear anal., 24, 1471-1482, (1995) · Zbl 0829.35040 [25] Su, J.B., Semilinear elliptic resonant problems at higher eigenvalue with unbounded nonlinear terms, Acta math. sinica (N.S.), 14, 411-419, (1998) · Zbl 0909.35050 [26] Su, J.B., Semilinear elliptic boundary value problems with double resonance between two consecutive eigenvalues, Nonlinear anal., 48, 6, 881-895, (2002) · Zbl 1018.35037 [27] Su, J.B.; Tang, C.L., Multiplicity results for semilinear elliptic equations with resonance at higher eigenvalues, Nonlinear anal., 44, 3, 311-321, (2001) · Zbl 1153.35336
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.