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A weighted Trudinger–Moser inequalities and applications to some weighted $(N,q)-$Laplacian equation in $\mathbb {R}^N$ with new exponential growth conditions

Part of: Partial differential equations Linear function spaces and their duals Inequalities Elliptic equations and systems Representations of solutions

Published online by Cambridge University Press:  07 September 2023

Sami Aouaoui
Sami Aouaoui*
Affiliation:
University of Kairouan, High Institute of Applied Mathematics and Informatics of Kairouan, Avenue Assad Iben Fourat, 3100 Kairouan, Tunisia ([email protected])
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Abstract

In this paper, we prove some weighted sharp inequalities of Trudinger–Moser type. The weights considered here have a logarithmic growth. These inequalities are completely new and are established in some new Sobolev spaces where the norm is a mixture of the norm of the gradient in two different Lebesgue spaces. This fact allowed us to prove a very interesting result of sharpness for the case of doubly exponential growth at infinity. Some improvements of these inequalities for the weakly convergent sequences are also proved using a version of the Concentration-Compactness principle of P.L. Lions. Taking profit of these inequalities, we treat in the last part of this work some elliptic quasilinear equation involving the weighted $(N,q)-$Laplacian operator where $1 < q < N$ and a nonlinearities enjoying a new type of exponential growth condition at infinity.

Keywords

Weighted Trudinger–Moser inequalityweighted Sobolev spaceslogarithmic weightsdoubly exponential growthradial functionsconcentration-compactnessweighted (N,q)–Laplacianelliptic equation

MSC classification

Primary: 26D15: Inequalities for sums, series and integrals
Secondary: 35A15: Variational methods 35A23: Inequalities involving derivatives and differential and integral operators, inequalities for integrals 35B38: Critical points 35D30: Weak solutions 35J20: Variational methods for second-order elliptic equations 35J62: Quasilinear elliptic equations 46E35: Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems
Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , First View , pp. 1 - 52
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Copyright
Copyright © The Author(s), 2023. Published by Cambridge University Press on behalf of The Royal Society of Edinburgh

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