Existence of positive solutions of Schrödinger equations with vanishing potentials

Eduard Toon; Pedro Ubilla

doi:10.3934/dcds.2020248

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2020, Volume 40, Issue 10: 5831-5843. Doi: 10.3934/dcds.2020248

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Existence of positive solutions of Schrödinger equations with vanishing potentials

Eduard Toon^1, and
Pedro Ubilla^2, ,

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
^* Corresponding author: Pedro Ubilla

Received: September 30, 2019

Revised: February 29, 2020

Published: June 2020

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

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Abstract

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.

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Mathematics Subject Classification: 35J20, 35J10, 35J91, 35J15, 35B09.

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References

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