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Existence and Multiplicity of Positive Solutions to a p-Kirchhoff-Type Equation

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Abstract

In this paper, a p-Kirchhoff-type elliptic equation involving a positive parameter \(\lambda \) is considered. Existence, nonexistence and multiplicity of weak solutions are obtained by the fibering maps and the mountain pass lemma. We use the nonlinear generalized Rayleigh quotient to characterize two extremal values \(\lambda _0^*\) and \(\lambda ^*\). The parameter \(\lambda _0^*\) is characterized for which the energy functional has nonnegative energy for the local minimum when \(\lambda \ge \lambda _0^*\). The parameter \(\lambda ^*\) is characterized for which the problem has no nonzero solution when \(\lambda >\lambda ^*\). Moreover, the asymptotic behavior of the solutions is also studied when \(\lambda \downarrow 0\).

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Acknowledgements

The authors wish to express their gratitude to the anonymous referee for giving a number of valuable comments and helpful suggestions, which improve the presentation of original manuscript significantly. They would like to express their sincere gratitude to Professor Wenjie Gao in Jilin University for his enthusiastic guidance and constant encouragement.

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Authors and Affiliations

  1. School of Mathematics, Jilin University, Changchun, 130012, People’s Republic of China

    Qi Li & Yuzhu Han

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  1. Qi Li
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  2. Yuzhu Han
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Correspondence to Yuzhu Han.

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Communicated by Maria Alessandra Ragusa.

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Supported by the National Key Research and Development Program of China (grant no.2020YFA0714100)

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Li, Q., Han, Y. Existence and Multiplicity of Positive Solutions to a p-Kirchhoff-Type Equation. Bull. Malays. Math. Sci. Soc. 45, 1789–1810 (2022). https://doi.org/10.1007/s40840-022-01278-0

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  • DOI: https://doi.org/10.1007/s40840-022-01278-0

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