Electron. J. Differential Equations, Vol. 2018 (2018), No. 164, pp. 1-21.

Quasilinear asymptotically linear Schrodinger problem in R^N without monotonicity

Olimpio Hiroshi Miyagaki, Sandra Imaculada Moreira, Ricardo Ruviaro

We establish existence and non-existence results for a quasilinear asymptotically linear Schrodinger problem. In the first result, we prove that a minimization problem constrained to the Pohozaev manifold is not achieved. In the second, the main argument consists in a splitting lemma for a functional constrained to the Pohozaev manifold. Because of the lack of the monotonicity we are not able to project to the usual Nehari manifold any longer, and this approach is crucial in order to compare the critical level to reach a contradiction. This argument was used in [21, 24, 32] for semilinear equations and in [11] for quasilinear equations.

Submitted August 27, 2017. Published September 11, 2018.
Math Subject Classifications: 35J10, 35J20, 35J60, 35Q55.
Key Words: Quasilinear Schrodinger equations; variational methods; asymptotically linear

Show me the PDF file (325 KB), TEX file for this article.

Olimpio Hiroshi Miyagaki
Universidade Federal de Juiz de Fora
Departamento de Matemática
36036-330 Juiz de Fora-MG, Brazil
email: ohmiyagaki@gmail.com
Sandra Imaculada Moreira
Universidade Estadual do Maranhão
Departamento de Matemática e Informática
65055-900 São Luís-MA, Brazil
email: ymaculada@gmail.com
Ricardo Ruviaro
Universidade de Brasília
Departamento de Matemática
70910-900 Brasília-DF, Brazil
email: ricardoruviaro@gmail.com

Return to the EJDE web page