Elliptic equations in $ \mathbb{R}^2 $ involving supercritical exponential growth

Yony Raúl Santaria Leuyacc; Yony Raúl Santaria Leuyacc

doi:10.3934/era.2024247

Electronic Research Archive

2024, Volume 32, Issue 9: 5341-5356. doi: 10.3934/era.2024247

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Research article

Elliptic equations in $ \mathbb{R}^2 $ involving supercritical exponential growth

Yony Raúl Santaria Leuyacc ^,

Universidad Nacional Mayor de San Marcos, Lima, Perú

Received: 16 July 2024 Revised: 09 September 2024 Accepted: 10 September 2024 Published: 18 September 2024

In this work, we investigated the existence of nontrivial weak solutions for the equation
$ -{\rm div}(w(x)\nabla u) \ = \ f(x,u),\qquad x \in \mathbb{R}^2, $
where $ w(x) $ is a positive radial weight, the nonlinearity $ f(x, s) $ possesses growth at infinity of the type $ {\rm \exp}\big((\alpha_0+h(|x|)\big)|s|^{2/(1-\beta)}) $, with $ \alpha_0 > 0 $, $ 0 < \beta < 1 $ and $ h $ is a continuous radial function that may be unbounded at infinity. To show the existence of weak solutions, we used variational methods and a new type of the Trudinger-Moser inequality defined on the whole two-dimensional space.
- Trudinger-Moser inequality,
- supercritical exponential growth,
- mountain pass theorem,
- elliptic equation,
- variational method
Citation: Yony Raúl Santaria Leuyacc. Elliptic equations in $ \mathbb{R}^2 $ involving supercritical exponential growth[J]. Electronic Research Archive, 2024, 32(9): 5341-5356. doi: 10.3934/era.2024247

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Abstract

In this work, we investigated the existence of nontrivial weak solutions for the equation

$ -{\rm div}(w(x)\nabla u) \ = \ f(x,u),\qquad x \in \mathbb{R}^2, $

where $ w(x) $ is a positive radial weight, the nonlinearity $ f(x, s) $ possesses growth at infinity of the type $ {\rm \exp}\big((\alpha_0+h(|x|)\big)|s|^{2/(1-\beta)}) $, with $ \alpha_0 > 0 $, $ 0 < \beta < 1 $ and $ h $ is a continuous radial function that may be unbounded at infinity. To show the existence of weak solutions, we used variational methods and a new type of the Trudinger-Moser inequality defined on the whole two-dimensional space.

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