×

zbMATH — the first resource for mathematics

Multiplicity results for some nonlinear elliptic problems with asymptotically \(p\)-linear terms. (English) Zbl 1380.35083
The paper under review deals with the study of a class of nonlinear PDEs with gradient term and Dirichlet boundary condition. The authors are interested in the existence of multiple nontrivial bounded solutions in the non-resonant case. The proofs combine refined topological and variational techniques, such as the cohomological index theory, pseudo-index theory, or elliptic estimates.

MSC:
35J60 Nonlinear elliptic equations
35J35 Variational methods for higher-order elliptic equations
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Amann, H; Zehnder, E, Nontrivial solutions for a class of nonresonance problems and applications to nonlinear differential equations, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4), 7, 539-603, (1980) · Zbl 0452.47077
[2] Anane, A; Gossez, JP, Strongly nonlinear elliptic problems near resonance: a variational approach, Commun. Partial Differ. Equ., 15, 1141-1159, (1990) · Zbl 0715.35029
[3] Arcoya, D; Boccardo, L, Critical points for multiple integrals of the calculus of variations, Arch. Ration. Mech. Anal., 134, 249-274, (1996) · Zbl 0884.58023
[4] Arcoya, D; Orsina, L, Landesman-lazer conditions and quasilinear elliptic equations, Nonlinear Anal., 28, 1623-1632, (1997) · Zbl 0871.35037
[5] Bartolo, P; Benci, V; Fortunato, D, Abstract critical point theorems and applications to some nonlinear problems with “strong” resonance at infinity, Nonlinear Anal., 7, 981-1012, (1983) · Zbl 0522.58012
[6] Bartolo, R; Candela, AM; Salvatore, A, \(p\)-Laplacian problems with nonlinearities interacting with the spectrum, Nonlinear Differ. Equ. Appl., 20, 1701-1721, (2013) · Zbl 1280.35059
[7] Benci, V, On the critical point theory for indefinite functionals in the presence of symmetries, Trans. Am. Math. Soc., 274, 533-572, (1982) · Zbl 0504.58014
[8] Boccardo, L; Murat, F; Puel, JP, Existence of bounded solutions for nonlinear elliptic unilateral problems, Ann. Mat. Pura Appl. IV Ser., 152, 183-196, (1988) · Zbl 0687.35042
[9] Brézis, H; Coron, JM; Nirenberg, L, Free vibrations for a nonlinear wave equation and a theorem of P. Rabinowitz, Commun. Pure Appl. Math., 33, 667-689, (1980) · Zbl 0484.35057
[10] Browder, F.E.: Existence theorems for nonlinear partial differential equations. In: Chern, S.S., Smale, S. (eds.) Proceedings of the Symposia in Pure Mathematics, vol. XVI. AMS, Providence, pp. 1-60 (1970) · Zbl 0211.17204
[11] Candela, AM; Palmieri, G, Infinitely many solutions of some nonlinear variational equations, Calc. Var. Partial Differ. Equ., 34, 495-530, (2009) · Zbl 1160.49007
[12] Candela, A.M., Palmieri, G.: Some abstract critical point theorems and applications. In: Hou, X., Lu, X., Miranville, A., Su, J., Zhu, J. (eds.) Dynamical Systems, Differential Equations and Applications, Discrete Contin. Dynam. Syst. Suppl. 2009, pp. 133-142 (2009) · Zbl 0818.35029
[13] Candela, A.M., Palmieri, G.: Multiple solutions for \(p\)-Laplacian type problems with an asymptotically \(p\)-linear term. In: de Figueiredo, D.G., do Ó, J.M., Tomei, C. (eds.) Analysis and Topology in Nonlinear Differential Equations, Progr. Nonlinear Differential Equations Appl., vol. 85, pp. 175-186 (2014) · Zbl 1319.35030
[14] Candela, AM; Palmieri, G; Perera, K, Multiple solutions for \(p\)-Laplacian type problems with asymptotically \(p\)-linear terms via a cohomological index theory, J. Differ. Equ., 259, 235-263, (2015) · Zbl 1317.35056
[15] Cerami, G, Un criterio di esistenza per i punti critici su varietà illimitate, Istit. Lombardo Accad. Sci. Lett. Rend. A, 112, 332-336, (1978) · Zbl 0436.58006
[16] Costa, DG; Magalhães, CA, Existence results for perturbations of the \(p\)-Laplacian, Nonlinear Anal., 24, 409-418, (1995) · Zbl 0818.35029
[17] Drábek, P; Robinson, S, Resonance problems for the \(p\)-Laplacian, J. Funct. Anal., 169, 189-200, (1999) · Zbl 0940.35087
[18] Fadell, ER; Rabinowitz, PH, Generalized cohomological index theories for Lie group actions with an application to bifurcation questions for Hamiltonian systems, Invent. Math., 45, 139-174, (1978) · Zbl 0403.57001
[19] Ladyzhenskaya, O.A., Ural’tseva, N.N.: Linear and Quasilinear Elliptic Equations. Academic Press, New York (1968) · Zbl 0164.13002
[20] Liu, S; Li, S, Existence of solutions for asymptotically ‘linear’ \(p\)-Laplacian equations, Bull. Lond. Math. Soc., 36, 81-87, (2004) · Zbl 1088.35025
[21] Li, G; Zhou, HS, Asymptotically linear Dirichlet problem for the \(p\)-Laplacian, Nonlinear Anal., 43, 1043-1055, (2001) · Zbl 0983.35046
[22] Li, G; Zhou, HS, Multiple solutions to \(p\)-Laplacian problems with asymptotic nonlinearity as \(u^{p-1}\) at infinity, J. Lond. Math. Soc., 65, 123-138, (2002) · Zbl 1171.35384
[23] Palais, RS, Critical point theory and the minimax principle, Proc. Symp. Pure Math., 15, 185-212, (1970) · Zbl 0212.28902
[24] Perera, K., Agarwal, R.P., O’Regan, D.: Morse Theoretic Aspects of \(p\)-Laplacian Type Operators, Math. Surveys Monogr., vol. 161, Am. Math. Soc., Providence RI (2010) · Zbl 1192.58007
[25] Perera, K; Szulkin, A, \(p\)-Laplacian problems where the nonlinearity crosses an eigenvalue, Discrete Contin. Dyn. Syst., 13, 743-753, (2005) · Zbl 1094.35052
[26] Rabinowitz, P.H.: Minimax Methods in Critical Point Theory with Applications to Differential Equations, CBMS Reg. Conf. Ser. Math., vol. 65, Providence (1986)
[27] Struwe, M.: Variational Methods. Applications to Nonlinear Partial Differential Equations and Hamiltonian Systems, 4rd edn, Ergeb. Math. Grenzgeb. (4), vol. 34. Springer, Berlin (2008) · Zbl 1284.49004
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.