2022Topological Methods in Nonlinear AnalysisOpen access

The Choquard logarithmic equation involving a nonlinearity with exponential growth

Eduardo de Souza Böer, Olı́mpio H. Miyagaki

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Abstract

In the present work, we are concerned with the Choquard Logarithmic equation $-\Delta u + au + \lambda (\ln|\cdot|\ast |u|^{2})u = f(u)$ in $ \mathbb{R}^2$, 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. Also, we provide these results under a symmetric setting, taking into account subgroups of $ O(2) $.

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In the present work, we are concerned with the Choquard Logarithmic equation $-\Delta u + au + \lambda (\ln|\cdot|\ast |u|^{2})u = f(u)$ in $ \mathbb{R}^2$, 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. Also, we provide these results under a symmetric setting, taking into account subgroups of $ O(2) $.

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Available abstract

In the present work, we are concerned with the Choquard Logarithmic equation $-\Delta u + au + \lambda (\ln|\cdot|\ast |u|^{2})u = f(u)$ in $ \mathbb{R}^2$, 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. Also, we provide these results under a symmetric setting, taking into account subgroups of $ O(2) $.

Key concepts: Mathematics, Logarithm, Lambda, Exponential function, Nonlinear system, Ground state, Mathematical analysis, Exponential growth

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