Singular weighted sharp Trudinger-Moser inequalities defined on $ \mathbb{R}^N $ and applications to elliptic nonlinear equations

Sami Aouaoui; Rahma Jlel

doi:10.3934/dcds.2021137

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2022, Volume 42, Issue 2: 781-813. Doi: 10.3934/dcds.2021137

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Singular weighted sharp Trudinger-Moser inequalities defined on $ \mathbb{R}^N $ and applications to elliptic nonlinear equations

Sami Aouaoui^1, , and
Rahma Jlel^2,

1.
University of Kairouan, High Institute of Applied Mathematics, and Informatics of Kairouan, , Avenue Assad Iben Fourat, Kairouan, 3100, Tunisia
2.
University of Monastir, Faculty of Sciences of Monastir, Avenue de l'environnement 5019 Monastir, Tunisia

^*Corresponding author: Sami Aouaoui
^*Corresponding author: Sami Aouaoui

Received: March 31, 2021

Early access: September 2021

Published: February 2022

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Abstract

This work comes to complete some previous ones of ours. Actually, in this paper, we establish some singular weighted inequalities of Trudinger-Moser type for radial functions defined on the whole euclidean space $ \mathbb{R}^N,\ N \geq 2. $ The weights considered are of logarithmic type. The singularity plays a capital role to prove the sharpness of the inequalities. These inequalities are later improved using some concentration-compactness arguments. The last part in this work is devoted to the application of the inequalities established to some singular elliptic nonlinear equations involving a new growth conditions at infinity of exponential type.

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Mathematics Subject Classification: Primary: 26D15, 35A21, 35A23, 35B33, 35D30; Secondary: 35J20, 35J62, 35J75.

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