Abstract
The aim of this paper is to investigate the existence of solutions of the semilinear elliptic problem
where Ω is an open bounded domain of \({\mathbb{R}^N}\), \({\varepsilon\in\mathbb{R}, p}\) is subcritical and asymptotically linear at infinity, and g is just a continuous function. Even when this problem has not a variational structure on \({H^1_0(\Omega)}\), suitable procedures and estimates allow us to prove that the number of distinct critical levels of the functional associated to the unperturbed problem is “stable” under small perturbations, in particular obtaining multiplicity results if p is odd, both in the non-resonant and in the resonant case.
Article PDF
Similar content being viewed by others
References
Amann H., Zehnder E.: Nontrivial solutions for a class of nonresonance problems and applications to nonlinear differential equations. Ann. Scuola Norm. Sup. Pisa 7, 539–603 (1980)
Ambrosetti A., Rabinowitz P.H.: Dual variational methods in critical point theory and applications. J. Funct. Anal. 14, 349–381 (1973)
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)
Benci V.: On the critical point theory for indefinite functionals in the presence of symmetries. Trans. Am. Math. Soc. 274, 533–572 (1982)
Benci V., Capozzi A., Fortunato D.: Periodic solutions of Hamiltonian systems with superquadratic potential. Ann. Mat. Pura Appl. CXLIII, 1–46 (1986)
Cossio J., Herrón S., Vélez C.: Existence of solutions for an asymptotically linear Dirichlet problem via Lazer–Solimini results. Nonlinear Anal. 71, 66–71 (2009)
Degiovanni M., Lancelotti S.: Perturbations of even nonsmooth functionals. Differ. Integral Equ. 8, 981–992 (1995)
Degiovanni, M., Lancelotti, S.: Perturbations of critical values in nonsmooth critical point theory. In: Sonntag, Y. (eds.) Well-posed Problems and Stability in Optimization. Serdica Math. J., vol. 22, pp. 427–450 (1996)
Degiovanni M., Rǎdulescu V.: Perturbations of nonsmooth symmetric nonlinear eigenvalue problems. C. R. Acad. Sci. Paris Sér. I 329, 281–286 (1999)
Hirano N., Zou W.: A perturbation method for multiple sign-changing solutions. Calc. Var. Partial Differ. Equ. 37, 87–98 (2010)
Krasnosel’skii M.A.: Topological methods in the theory of nonlinear integral equations. In: Armstrong, A.H., Burlak, J. (eds.) Translated from the Russian edition, Moscow, 1956, Macmillan, Pergamon, New York, NY, London (1964)
Li S., Liu Z.: Perturbations from symmetric elliptic boundary value problems. J. Differ. Equ. 185, 271–280 (2002)
Li S., Liu Z.: Multiplicity of solutions for some elliptic equations involving critical and supercritical Sobolev exponents. Topol. Methods Nonlinear Anal. 28, 235–261 (2006)
Mawhin J., Willem M.: Critical Point Theory and Hamiltonian Systems. Springer, New York, NY (1989)
Rabinowitz, P.H.: Minimax Methods in Critical Point Theory with Applications to Differential Equations, CBMS Regional Conf. Ser. in Math. vol. 65. American Mathmatical Society, Providence (1984)
Reeken M.: Stability of critical points under small perturbations. Part I: Topological theory. Manuscripta Mathematica 7, 387–411 (1972)
Struwe, M.: Variational Methods. Applications to Nonlinear Partial Differential Equations and Hamiltonian Systems, 4th edn. Ergeb. Math. Grenzgeb. (4) vol. 34. Springer, Berlin (2008)
Author information
Authors and Affiliations
Corresponding author
Additional information
Partially supported by M.I.U.R. Research project PRIN2009 “Metodi Variazionali e Topologici nello Studio di Fenomeni Nonlineari”.
Rights and permissions
About this article
Cite this article
Bartolo, R., Candela, A.M. & Salvatore, A. Perturbed asymptotically linear problems. Annali di Matematica 193, 89–101 (2014). https://doi.org/10.1007/s10231-012-0267-9
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10231-012-0267-9
Keywords
- Asymptotically linear elliptic problem
- Essential value
- Perturbed problem
- Variational methods
- Pseudo-genus
- Resonant problem