A Kirchhoff type elliptic systems with exponential growth nonlinearities

Tian, Xingliang

doi:10.12775/tmna.2021.035

Cited by 2 publications

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“…Actually new problems involved with Kirchhoff type emerged from the above researches and authors obtained the existence results of solutions for the Kirchhoff type equations involving critical exponential nonlinearities via variational methods. In relation to Kirchhoff problems, the existence and multiplicity of solutions for elliptic equations inolving critical exponential nonlinearity can be found in the literatures [20,21] and a class of elliptic equations with a small nonhomogeneous term was studied in the articles [22,23,24] in a bounded domain of R 2 . For biharmonic equations we refer to [25,26] in the whole Euclidean space R 4 .…”

Section: Introductionmentioning

confidence: 99%

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Kirchhoff Biharmonic System with Choquard Nonlinearity and Singular Weights

2024

AJOMCOR

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The aim of this paper is to find the existence of solutions for the following Kirchhoff type biharmonic system with exponential nonlinearity and singular weights \(\begin{cases}m\left(\|u\|^2+\|v\|^2\right) \Delta^2 u=\left[I_\mu * \frac{F(x, u, v)}{|x|^\alpha}\right] \frac{f_1(x, u, v)}{|x|^\alpha} & \text { in } \Omega \\ m\left(\|u\|^2+\|v\|^2\right) \Delta^2 v=\left[I_\mu * \frac{F(x, u, v)}{|x|^\alpha}\right] \frac{f_2(x, u, v)}{|x|^\alpha} & \text { in } \Omega \\ u=0, \quad v=0, \quad \nabla u=\mathbf{0}, \quad \nabla v=\mathbf{0} & \text { on } \partial \Omega\end{cases}S\) where \(\Omega\) is a bounded domain in \(\mathbb{R}^4\) containing the origin with smooth boundary, \(\mu \in(0,4), 0<\alpha<\frac{\mu}{2}\), \(I_\mu(x)=\frac{1}{|x|^4-\mu}, m\) is a Kirchhoff type function, \(\|u\|^2=\int_{\Omega}|\Delta u|^2 d x, f_i\) behaves like \(e^{\beta_{0 s^2}}\) when \(|s| \rightarrow \infty\) for some \(\beta_0>0\), and there is \(C^1\) function \(F: \mathbb{R}^2 \rightarrow \mathbb{R}\) such that \(\left(\frac{\partial F(x, u, v)}{\partial u}, \frac{\partial F(x, u, v)}{\partial v}\right)=\left(f_1(x, u, v), f_2(x, u, v)\right)\). We establish sufficient conditions for the solutions of the above system by using variational methods with Adams inequality.

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Section: Introductionmentioning

confidence: 99%

“…ε is a small positive parameter. We also refer to [21] for Kirchhoff type elliptic system involving critical exponential growth. Motivated by the above results, in this paper, we consider the existence of solutions for the Kirchhoff type biharmonic system (1.1).…”

Section: Introductionmentioning

confidence: 99%