# American Institute of Mathematical Sciences

October  2020, 40(10): 5831-5843. doi: 10.3934/dcds.2020248

## Existence of positive solutions of Schrödinger equations with vanishing potentials

 1 Departamento de Matemática, Universidade Federal de Juiz de Fora, Juiz de Fora, CEP 36036-330, Minas Gerais, Brazil 2 Departamento de Matematica y C. C., Universidad de Santiago de Chile, Casilla 307, Correo 2, Santiago, Chile

* Corresponding author: Pedro Ubilla

Received  October 2019 Revised  March 2020 Published  June 2020

Fund Project: The first author gratefully acknowledges financial support from Universidad de Santiago de Chile, Usach. Agradecimientos Proyecto POSTDOC_DICYT, Código 041733UL_POSTDOC, Vicerrectoría de Investigación, Desarrollo e Innovación. Partially supported by FAPEMIG CEX APQ 01745/18. The second author was supported by FONDECYT grants 1181125, 1161635 and 1171691

We prove the existence of at least one positive solution for a Schrödinger equation in
 $\mathbb{R}^N$
of type
 $- \Delta u + V(x) u = f(x, u) \ \ \text{in} \ \mathbb{R}^N$
with a vanishing potential at infinity and subcritical nonlinearity
 $f$
. Our hypotheses allow us to consider examples of nonlinearities which do not verify the Ambrosetti-Rabinowitz condition, neither monotonicity conditions for the function
 $\frac{f(x, s)}{s}$
. Our argument requires new estimates in order to prove the boundedness of a Cerami sequence.
Citation: Eduard Toon, Pedro Ubilla. Existence of positive solutions of Schrödinger equations with vanishing potentials. Discrete & Continuous Dynamical Systems, 2020, 40 (10) : 5831-5843. doi: 10.3934/dcds.2020248
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