A log–exp elliptic equation in the plane

Giovany Figueiredo; Marcelo Montenegro; Matheus F. Stapenhorst

doi:10.3934/dcds.2021125

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2022, Volume 42, Issue 1: 481-504. Doi: 10.3934/dcds.2021125

This issue Previous Article Exact null-controllability of interconnected abstract evolution equations with unbounded input operators Next Article

A log–exp elliptic equation in the plane

1.
Universidade de Brasília, Departamento de Matemática, Campus Darcy Ribeiro, 01, CEP 70910-900, Brasília, DF, Brasil
2.
Universidade Estadual de Campinas, IMECC, Departamento de Matemática, Rua Sérgio Buarque de Holanda, 651, CEP 13083-859, Campinas, SP, Brasil

^* Corresponding author
^* Corresponding author

Received: November 30, 2020

Revised: May 31, 2021

Early access: September 2021

Published: January 2022

The author G.F. is supported by CNPq and FAP-DF, M.M. is supported by CNPq and FAPESP and M.F.S. is supported by CAPES

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Abstract

In this paper we show the existence of a nonnegative solution for a singular problem with logarithmic and exponential nonlinearity, namely $ -\Delta u = \log(u)\chi_{\{u>0\}} + \lambda f(u) $ in $ \Omega $ with $ u = 0 $ on $ \partial\Omega $, where $ \Omega $ is a smooth bounded domain in $ \mathbb{R}^{2} $. We replace the singular function $ \log(u) $ by a function $ g_\epsilon(u) $ which pointwisely converges to -$ \log(u) $ as $ \epsilon \rightarrow 0 $. When the parameter $ \lambda>0 $ is small enough, the corresponding energy functional to the perturbed equation $ -\Delta u + g_\epsilon(u) = \lambda f(u) $ has a critical point $ u_\epsilon $ in $ H_0^1(\Omega) $, which converges to a nontrivial nonnegative solution of the original problem as $ \epsilon \rightarrow 0 $.

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Mathematics Subject Classification: Primary: 35A15; Secondary: 35B20, 35D30, 35J20, 35J60, 35J75.

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