Abstract
The aim of this paper is investigating the existence and the multiplicity of weak solutions of the quasilinear elliptic problem
where \({1 < p < + \infty, \Delta_p u = {\rm div}(|\nabla {u}|^{p-2}\nabla {u})}\), Ω is an open bounded domain of \({\mathbb{R}^N (N \geq 3)}\) with smooth boundary ∂Ω and the nonlinearity g behaves as u p−1 at infinity. The main tools of the proof are some abstract critical point theorems in Bartolo et al. (Nonlinear Anal. 7: 981–1012, 1983), but extended to Banach spaces, and two sequences of quasi–eigenvalues for the p–Laplacian operator as in Candela and Palmieri (Calc. Var. 34: 495–530, 2009), Li and Zhou (J. Lond. Math. Soc. 65: 123–138, 2002).
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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. 7(4), 539–603 (1980)
Anane, A.: Etude des valeurs propres et de la résonance pour l’opérateur p-laplacien. Thèse de Doctorat, Université Libre de Bruxelles (1987)
Anane A., Gossez J.P.: Strongly nonlinear elliptic problems near resonance: a variational approach. Comm. Partial Differ. Equ. 15, 1141–1159 (1990)
Arcoya D., Orsina L.: Landesman–Lazer conditions and quasilinear elliptic equations. Nonlinear Anal. 28, 1623–1632 (1997)
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)
Bartolo, R., Candela, A.M., Salvatore, A.: Perturbed asymptotically linear problems. Ann. Mat. Pura Appl. (to appear). doi:10.1007/s10231-012-0267-9
Benci V.: On the critical point theory for indefinite functionals in the presence of symmetries. Trans. Am. Math. Soc. 274, 533–572 (1982)
Brezis, H.: Functional Analysis, Sobolev Spaces and Partial Differential Equations. Universitext XIV, Springer, New York (2011)
Costa D.G., Magalhães C.A.: Existence results for perturbations of the p-Laplacian. Nonlinear Anal. 24, 409–418 (1995)
Candela A.M., Palmieri G.: Infinitely many solutions of some nonlinear variational equations. Calc. Var. 34, 495–530 (2009)
Dinca G., Jebelean P., Mawhin J.: Variational and topological methods for Dirichlet problems with p-Laplacian. Port. Math. 58, 339–378 (2001)
Drábek P., Robinson S.: Resonance problems for the p-Laplacian. J. Funct. Anal. 169, 189–200 (1999)
Fadell E.R., Rabinowitz P.H.: Generalized cohomological index theories for Lie group actions with an application to bifurcation questions for Hamiltonian systems. Invent. Math. 45, 139–174 (1978)
García Azorero J., Peral Alonso I.: Existence and nonuniqueness for the p-Laplacian: Nonlinear eigenvalues. Comm. Partial Differ. Equ. 12, 1389–1430 (1987)
Li G., Zhou H.S.: Asymptotically linear Dirichlet problem for the p-Laplacian. Nonlinear Anal. 43, 1043–1055 (2001)
Li G., Zhou H.S.: Multiple solutions to p-Laplacian problems with asymptotic nonlinearity as u p−1 at infinity. J. Lond. Math. Soc. 65, 123–138 (2002)
Lindqvist P.: On the equation \({ {\rm div}(|\nabla u|^{p-2}\nabla u) + \lambda|u|^{p-2}u = 0}\). Proc. Am. Math. Soc. 109, 157–164 (1990)
Lindqvist P.: On a nonlinear eigenvalue problem. Berichte Univ. Jyväskylä Math. Inst. 68, 33–54 (1995)
Liu S., Li S.: Existence of solutions for asymptotically ‘linear’ p-Laplacian equations. Bull. Lond. Math. Soc. 36, 81–87 (2004)
Perera K., Szulkin A.: p-Laplacian problems where the nonlinearity crosses an eigenvalue. Discr. Contin. Dyn. Syst. 13, 743–753 (2005)
Rabinowitz, P.H.: Minimax Methods in Critical Point Theory with Applications to Differential Equations. CBMS Regional Conference Series in Mathematics 65, American Mathematical Society, Providence (1986)
Rabinowitz, P.H.: Variational Methods for Nonlinear Eigenvalues Problems, (G. Prodi Ed.), Edizioni Cremonese, Roma, pp. 141–195 (1974)
Struwe, M.: Variational Methods. Applications to Nonlinear Partial Differential Equations and Hamiltonian Systems. 4rd Edition, Ergeb. Math. Grenzgeb. (4) 34, Springer, Berlin (2008)
Willem M.: Minimax Theorems. Birkhäuser, Boston (1996)
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Partially supported by M.I.U.R. Research Project PRIN2009 “Metodi Variazionali e Topologici nello Studio di Fenomeni Nonlineari”.
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Bartolo, R., Candela, A.M. & Salvatore, A. p-Laplacian problems with nonlinearities interacting with the spectrum. Nonlinear Differ. Equ. Appl. 20, 1701–1721 (2013). https://doi.org/10.1007/s00030-013-0226-1
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DOI: https://doi.org/10.1007/s00030-013-0226-1