A planar Schrodinger-Newton system with Trudinger-Moser critical growth

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Date
2023-03-20
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Mark
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Springer Nature
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Abstract
In this paper, we focus on the existence of positive solutions to the following planar Schrodinger-Newton system with general critical exponential growth $-\Delta u + u + \phi u = f (u) in R^2, \Delta \phi = u^2 in R^2 $, where $f$ is an element of $ C^1( R, R)$. We apply a variational approach developed in [36] to study the above problem in the Sobolev space $H^1(R^2)$. The analysis developed in this paper also allows to investigate the relation between a Riesz-type of Schrodinger-Newton systems and a logarithmic-type of Schrodinger-Poisson systems. Furthermore, this approach can overcome some difficulties resulting from either the nonlocal term with sign-changing and unbounded logarithmic integral kernel, or the critical nonlinearity, or the lack of monotonicity of $ f(t)/t(3)$. We emphasize that it seems much difficult to use the variational framework developed in the existed literature to study the above problem.
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CALCULUS OF VARIATIONS AND PARTIAL DIFFERENTIAL EQUATIONS. 2023, vol. 62, issue 4, p. 1-31.
https://link.springer.com/article/10.1007/s00526-023-02463-0
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Peer-reviewed
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en
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Creative Commons Attribution 4.0 International
http://creativecommons.org/licenses/by/4.0/
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