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Applicable Analysis
An International Journal
Volume 102, 2023 - Issue 15
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Articles

On concentration of solutions for quasilinear Schrödinger equations with critical growth in the plane

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Pages 4058-4091 | Received 23 Feb 2022, Accepted 17 May 2022, Published online: 22 Jul 2022
 

Abstract

In this work, we study the existence of positive ground states and concentration phenomena for the following class of quasilinear Schrödinger equations: ε2div(g2(u)u)+ε2g(u)g(u)|u|2+V(x)u=Q(x)f(u)inR2, where ε>0 is a parameter, g:RR+ is a continuously differentiable function, V(x) and Q(x) are positive and bounded continuous potentials and the nonlinearity f(u) can exhibit critical exponential growth. In order to prove our concentration result, we exploit variational arguments that take into account an interaction between the potentials V and Q, in combination with the Trudinger–Moser inequality and Moser iteration method.

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Acknowledgements

We would like to thank the referees for careful reading of the paper with many useful comments and important suggestions, which substantially helped improving the quality of the paper.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

Research partially supported by CNPq grant 310747/2019-8 and Grant 2019/0014 Paraíba State Research Foundation (Fapesq), Brazil.

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