Supercritical Trudinger-Moser inequalities with logarithmic weights in dimension two

Yony Raúl Santaria Leuyacc; Yony Raúl Santaria Leuyacc

doi:10.3934/math.2023933

AIMS Mathematics

2023, Volume 8, Issue 8: 18354-18372. doi: 10.3934/math.2023933

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Research article

Supercritical Trudinger-Moser inequalities with logarithmic weights in dimension two

Yony Raúl Santaria Leuyacc ^,

Universidad Nacional Mayor de San Marcos, Lima, Perú

Received: 08 March 2023 Revised: 01 May 2023 Accepted: 09 May 2023 Published: 29 May 2023
MSC : 35J15, 35J20, 26D10

In this work, we are interested in studying the existence of nontrivial weak solutions to the following class of Schrödinger equations

$ \left\lbrace\begin{array}{rcll} -{\rm div}(w(x)\nabla u) \ & = &\ f(x, u), &\ x \in B_1(0), \\ u \ & = &\ 0, &\ x \in \partial B_1(0), \end{array}\right. $

where $ w(x) = \big(\ln (1/|x|)\big)^{\beta} $ for some $ \beta \in [0, 1) $, the nonlinearity $ f(x, s) $ behaves like $ {\rm \exp}((1+h(|x|))|s|^{2/(1-\beta)}) $ and $ h $ is a continuous radial function such that $ h(r) $ tends to infinity as $ r $ tends to $ 1 $. The proof involves variational methods and a new version of Trudinger-Moser inequality.
- Trudinger-Moser inequality,
- supercritical exponential growth,
- logarithmic weight variational methods,
- Schrödinger equation
Citation: Yony Raúl Santaria Leuyacc. Supercritical Trudinger-Moser inequalities with logarithmic weights in dimension two[J]. AIMS Mathematics, 2023, 8(8): 18354-18372. doi: 10.3934/math.2023933

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Abstract

In this work, we are interested in studying the existence of nontrivial weak solutions to the following class of Schrödinger equations

$ \left\lbrace\begin{array}{rcll} -{\rm div}(w(x)\nabla u) \ & = &\ f(x, u), &\ x \in B_1(0), \\ u \ & = &\ 0, &\ x \in \partial B_1(0), \end{array}\right. $

where $ w(x) = \big(\ln (1/|x|)\big)^{\beta} $ for some $ \beta \in [0, 1) $, the nonlinearity $ f(x, s) $ behaves like $ {\rm \exp}((1+h(|x|))|s|^{2/(1-\beta)}) $ and $ h $ is a continuous radial function such that $ h(r) $ tends to infinity as $ r $ tends to $ 1 $. The proof involves variational methods and a new version of Trudinger-Moser inequality.

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