The Choquard logarithmic equation involving fractional Laplacian operator and a nonlinearity with exponential critical growth

In the present work we investigate the existence and multiplicity of nontrivial solutions for the Choquard Logarithmic equation \((-\Delta)^{\frac{1}{2}} u + au + \lambda (\ln|\cdot|\ast |u|^{2})u = f(u) \textrm{ in } \mathbb{R}\), for \( a>0 \), \( \lambda >0 \) and a nonlinearity \(f\) with...

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Bibliographic Details
Published in:arXiv.org 2020-12
Main Authors: de Souza Böer, Eduardo, Miyagaki, Olímpio H
Format: Article
Language:English
Subjects:
Mountains
Nonlinearity
Online Access:Get full text
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Summary:In the present work we investigate the existence and multiplicity of nontrivial solutions for the Choquard Logarithmic equation \((-\Delta)^{\frac{1}{2}} u + au + \lambda (\ln|\cdot|\ast |u|^{2})u = f(u) \textrm{ in } \mathbb{R}\), for \( a>0 \), \( \lambda >0 \) and a nonlinearity \(f\) with exponential critical growth. We prove the existence of a nontrivial solution at the mountain pass level and a nontrivial ground state solution under exponential critical and subcritical growth. Morever, when \( f \) has subcritical growth we guarantee the existence of infinitely many solutions, via genus theory.
ISSN:2331-8422