Abstract
We present a new approach to solve an indefinite Schrödinger Equation autonomous at infinity, by identifying the relation between the arrangement of the spectrum of the concerned operator and the behavior of the nonlinearity at zero and at infinity. The main novelty is how to set a skillful linking structure that overcome the lack of compactness, depending on the growth of the nonlinear term and making use of information about the autonomous problem at infinity. Here no monotonicity assumption is required on the nonlinearity, which may be sign-changing as well as the potential. Furthermore, depending on the nonlinearity, the limit of the potential at infinity may be non-positive, so that zero may be an interior point in the essential spectrum of the Schrödinger operator.
Similar content being viewed by others
References
Berestycki, H., Lions, P.L.: Nonlinear scalar field equations I. Arch. Rat. Mech. Anal. 82, 313–346 (1983)
Berestycki, H., Gallouët, T., Kavian, O.: Equations de Champs scalaires euclidiens non linéaires dans le plan. CR Acad. Sci. Paris Sér. I Math. 297, 307–310 (1983)
Berezin, F.A., Shubin, M.A.: The Schrödinger Equation. Kluwer Academic Publishers, Dordrecht (1991)
Costa, D.G., Tehrani, H.: Existence and multiplicity results for a class of Schrödinger equations with indefinite nonlinearities. Adv. Differ. Equ. 8, 1319–1340 (2003)
Costa, D.G., Magalhães, C.A.: A unified approach to a class of strongly indefinite functionals. J. Differ. Equ. 125, 521–547 (1996)
Furtado, M.F., Maia, L.A., Medeiros, E.S.: Positive and nodal solutions for a nonlinear schrödinger equation with indefinite potential. Adv. Nonlinear Stud. 8, 353–373 (2008)
Li, G., Wang, C.: The existence of a nontrivial solution to a nonlinear elliptic problem of linking type without the Ambrosetti–Rabinowitz condition. Ann. Acad. Sci. Fenn. Math. 36, 461–480 (2011)
Liu, Z., Su, J., Weth, T.: Compactness results for Schrödinger equations with asymptotically linear terms. J. Differ. Equ. 231, 501–512 (2006)
Jeanjean, L., Tanaka, K.: A positive solution for an asymptotically linear elliptic problem on \({\mathbb{R}}^{N}\) autonomous at infinity. ESAIM: Cont. Opt. Calc. Var. 7, 597–614 (2002)
Jeanjean, L., Tanaka, K.: A positive solution for a nonlinear Schrödinger equation on \(\mathbb{R}^N\). Indiana Univ. Math. J. 54(2), 443–464 (2005)
de Maia, L.A., Oliveira Jr., J.C., Ruviaro, R.: A non-periodic and asymptotically linear indefinite variational problem in \(\mathbb{R}^N\). Indiana Univ. Math. J. 66(1), 31–54 (2017)
Maia, L.A., Soares, M.: A Note on Nonlinear Schrödinger Equations: Unveiling the Relation Between Spectral Gaps and the Nonlinearity. ar**v:1902.07184v1, Preprint, 2019
Pankov, A.: On decay of solutions to nonlinear Schrödinger equations. Proc. Am. Math. Soc. 136(7), 2565–2570 (2008)
Rabinowitz, P.H.: Minimax Methods in Critical Point Theory with Applications to Differential Equations. American Mathematical Society, Providence (1984)
Sato, Y., Shibata, M.: Infinitely many solutions for a nonlinear Schrödinger equation with general nonlinearity. Calc. Var. Partial Differ. Equ. 57(5), 137 (2018)
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by P. Rabinowitz.
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
Cite this article
Maia, L.A., Soares, M. An indefinite elliptic problem on \(\mathbb {R}^N\) autonomous at infinity: the crossing effect of the spectrum and the nonlinearity. Calc. Var. 59, 41 (2020). https://doi.org/10.1007/s00526-019-1683-0
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/s00526-019-1683-0