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Least Energy Sign-Changing Solution for N-Kirchhoff Problems with Logarithmic and Exponential Nonlinearities

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Abstract

In this paper, we are concerned with the existence of least energy sign-changing solutions for the following N-Laplacian Kirchhoff-type problem with logarithmic and exponential nonlinearities:

$$\begin{aligned} \left\{ \begin{array}{ll} -\left( a+b \int _{\Omega }|\nabla u|^{N} d x\right) \Delta _{N} u=|u|^{p-2} u \ln |u|^{2}+\lambda f(u), &{} \text{ in } \Omega , \\ u=0, &{} \text{ on } \partial \Omega , \end{array}\right. \end{aligned}$$

where f(t) behaves like \(\ exp\left( {\alpha |t|^{{\frac{N}{{N - 1}}}} } \right) \). Combining constrained variational method, topological degree theory and quantitative deformation lemma, we prove that the problem possesses one least energy sign-changing solution \(u_{b}\) with precisely two nodal domains. Moreover, we show that the energy of \(u_{b}\) is strictly larger than two times of the ground state energy and analyze the convergence property of \(u_{b}\) as \(b\searrow 0\).

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Funding

This study was supported in part by grants from National Natural Science Foundation of China (No. 12271443)

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  1. Yan-Ying Shang have contributed equally to this work.

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  1. School of Mathematics and Statistics, Southwest University, Chongqing, 400715, People’s Republic of China

    Ting Huang & Yan-Ying Shang

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  1. Ting Huang
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All authors contributed to the study conception and design. Material preparation, data collection and analysis were performed by [TH]and [Y-YS]. The first draft of the manuscript was written by [TH] and all authors commented on previous versions of the manuscript. All authors read and approved the final manuscript.

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Correspondence to Yan-Ying Shang.

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Huang, T., Shang, YY. Least Energy Sign-Changing Solution for N-Kirchhoff Problems with Logarithmic and Exponential Nonlinearities. Complex Anal. Oper. Theory 18, 49 (2024). https://doi.org/10.1007/s11785-024-01495-4

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