Hamiltonian elliptic system involving nonlinearities with supercritical exponential growth

Yony Raúl Santaria Leuyacc; Yony Raúl Santaria Leuyacc

doi:10.3934/math.2023976

AIMS Mathematics

2023, Volume 8, Issue 8: 19121-19141. doi: 10.3934/math.2023976

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Hamiltonian elliptic system involving nonlinearities with supercritical exponential growth

Yony Raúl Santaria Leuyacc ^,

Universidad Nacional Mayor de San Marcos, Lima, Perú

Received: 06 February 2023 Revised: 29 April 2023 Accepted: 07 May 2023 Published: 07 June 2023
MSC : 35J20, 35J47, 35J50, 26D10

In this paper, we deal with the existence of nontrivial solutions to the following class of strongly coupled Hamiltonian systems:

$ \begin{equation*} \quad \left\{ \begin{array}{rclll} -{\rm div} \big(w(x)\nabla u\big) \ = \ g(x,v),&\ & x \in B_1(0), \\[5pt] - {\rm div}\big(w(x) \nabla v\big)\ = \ f(x,u),&\ & x \in B_1(0), \\[5pt] u = v = 0&\ & x \in \partial B_1(0), \end{array} \right. \end{equation*} $

where $ w(x) = \big(\log 1/|x|\big)^{\gamma} $, $ 0\leq\gamma < 1 $, and the nonlinearities $ f $ and $ g $ possess exponential growth ranges above the exponential critical hyperbola. Our approach is based on Trudinger-Moser type inequalities for weighted Sobolev spaces and variational methods.
- Hamiltonian system,
- Trudinger-Moser inequality,
- supercritical exponential growth,
- variational methods,
- linking theorem
Citation: Yony Raúl Santaria Leuyacc. Hamiltonian elliptic system involving nonlinearities with supercritical exponential growth[J]. AIMS Mathematics, 2023, 8(8): 19121-19141. doi: 10.3934/math.2023976

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Abstract

In this paper, we deal with the existence of nontrivial solutions to the following class of strongly coupled Hamiltonian systems:

$ \begin{equation*} \quad \left\{ \begin{array}{rclll} -{\rm div} \big(w(x)\nabla u\big) \ = \ g(x,v),&\ & x \in B_1(0), \\[5pt] - {\rm div}\big(w(x) \nabla v\big)\ = \ f(x,u),&\ & x \in B_1(0), \\[5pt] u = v = 0&\ & x \in \partial B_1(0), \end{array} \right. \end{equation*} $

where $ w(x) = \big(\log 1/|x|\big)^{\gamma} $, $ 0\leq\gamma < 1 $, and the nonlinearities $ f $ and $ g $ possess exponential growth ranges above the exponential critical hyperbola. Our approach is based on Trudinger-Moser type inequalities for weighted Sobolev spaces and variational methods.

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