Semilinear elliptic problems involving exponential critical growth in the half-space

Diego D. Felix; Marcelo F. Furtado; Everaldo S. Medeiros

doi:10.3934/cpaa.2020219

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2020, Volume 19, Issue 10: 4937-4953. Doi: 10.3934/cpaa.2020219

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Semilinear elliptic problems involving exponential critical growth in the half-space

1.
Universidade Federal da Paraíba, Departamento de Matemática, 58051-900, João Pessoa-PB, Brazil
2.
Universidade de Brasília, Departamento de Matemática, 70910-900, Brasília-DF, Brazil

^* Corresponding author
^* Corresponding author

Received: December 31, 2019

Revised: April 30, 2020

Published: July 2020

All the authors were supported by CNPq/Brazil. The second author was also supported by FAP-DF/Brazil. The third author was also supported by Grant 2019/2014 Paraíba State Research Foundation (FAPESQ)

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Abstract

We perform an weighted Sobolev space approach to prove a Trudinger-Moser type inequality in the upper half-space. As applications, we derive some existence and multiplicity results for the problem

$ \begin{cases} -\Delta u+h(x)|u|^{q-2}u = a(x) f(u), &\mbox{in } \mathbb{R}^2_+,\\ \dfrac{\partial u}{\partial \nu}+u = 0, &\mbox{on } \partial\mathbb{R}^2_+, \end{cases} $

under some technical condition on $ a $, $ b $ and the the exponential nonlinearity $ f $. The ideas can also be used to deal with Neumann boundary conditions.

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Mathematics Subject Classification: Primary:35J20;Secondary:35J25.

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