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Sign-Changing Solutions for Planer Kirchhoff Type Problem With Critical Exponential Growth

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Abstract

In the present paper, we study the existence of ground state sign-changing solutions for a class of Kirchhoff type problem

$$\begin{aligned} \left\{ \begin{array}{ll} -\left( a+b\int _{\Omega }|\nabla u|^{2}dx\right) \Delta u=f(u), &{} \quad x\in \Omega ; \\ u=0,&{} \quad x\in \partial \Omega , \end{array} \right. \end{aligned}$$

where \(a, b>0\), \(f\in {\mathcal {C}}({\mathbb {R}},{\mathbb {R}})\) satisfies subcritical exponential growth or critical exponential growth in a bounded domain \(\Omega \in {\mathbb {R}}^{2}\) with a smooth boundary \(\partial \Omega \). In combination with Trudinger-Moser inequality, we prove the existence of least energy sign-changing solution by variational method and obtain its concentration behaviors as \(b\searrow 0\) in both subcritical case and critical case.

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Acknowledgements

This work is supported by the National Natural Science Foundation of China (No:11971485).

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  1. School of Mathematics and Statistics, Central South University, Changsha, 410083, PR China

    Haidong Yang & Xianhua Tang

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  1. Haidong Yang
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  2. Xianhua Tang
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Yang, H., Tang, X. Sign-Changing Solutions for Planer Kirchhoff Type Problem With Critical Exponential Growth. J Geom Anal 34, 178 (2024). https://doi.org/10.1007/s12220-024-01585-x

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