Standing waves for quasilinear Schrödinger equations involving double exponential growth

Yony Raúl Santaria Leuyacc; Yony Raúl Santaria Leuyacc

doi:10.3934/math.2023086

AIMS Mathematics

2023, Volume 8, Issue 1: 1682-1695. doi: 10.3934/math.2023086

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Research article

Standing waves for quasilinear Schrödinger equations involving double exponential growth

Yony Raúl Santaria Leuyacc ^,

Faculty of Mathematical Sciences, National University of San Marcos, Lima, Peru

Received: 05 September 2022 Revised: 30 September 2022 Accepted: 04 October 2022 Published: 24 October 2022
MSC : 35J62, 35A15, 35J20

We will focus on the existence of nontrivial, nonnegative solutions to the following quasilinear Schrödinger equation

$ \begin{equation*} \left\lbrace\begin{array}{rcll} -{\rm div} \Big(\log \dfrac{e}{|x|}\nabla u\Big) -{\rm div} \Big(\log \dfrac{e}{|x|}\nabla (u^2)\Big) u \ & = &\ g(x, u), &\ x \in B_1, \\ u \ & = &\ 0, &\ x \in \partial B_1, \end{array}\right. \end{equation*} $

where $ B_1 $ denotes the unit ball centered at the origin in $ \mathbb{R}^2 $ and $ g $ behaves like $ {\rm exp}(e^{s^4}) $ as $ s $ tends to infinity, the growth of the nonlinearity is motivated by a Trudinder-Moser inequality version, which admits double exponential growth. The proof involves a change of variable (a dual approach) combined with the mountain pass theorem.
- quasilinear Schrödinger equation,
- double exponential growth,
- mountain pass theorem,
- Trudinger-Moser inequality,
- dual approach
Citation: Yony Raúl Santaria Leuyacc. Standing waves for quasilinear Schrödinger equations involving double exponential growth[J]. AIMS Mathematics, 2023, 8(1): 1682-1695. doi: 10.3934/math.2023086

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Abstract

We will focus on the existence of nontrivial, nonnegative solutions to the following quasilinear Schrödinger equation

$ \begin{equation*} \left\lbrace\begin{array}{rcll} -{\rm div} \Big(\log \dfrac{e}{|x|}\nabla u\Big) -{\rm div} \Big(\log \dfrac{e}{|x|}\nabla (u^2)\Big) u \ & = &\ g(x, u), &\ x \in B_1, \\ u \ & = &\ 0, &\ x \in \partial B_1, \end{array}\right. \end{equation*} $

where $ B_1 $ denotes the unit ball centered at the origin in $ \mathbb{R}^2 $ and $ g $ behaves like $ {\rm exp}(e^{s^4}) $ as $ s $ tends to infinity, the growth of the nonlinearity is motivated by a Trudinder-Moser inequality version, which admits double exponential growth. The proof involves a change of variable (a dual approach) combined with the mountain pass theorem.

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