Abstract
This paper is concerned with the existence and multiplicity of solutions for singular Kirchhoff-type problems involving the fractional p-Laplacian operator. More precisely, we study the following nonlocal problem
where \({\cal L}_p^s\) is the generalized fractional p-Laplacian operator, N ≥ 1, s ∈ (0,1), α1, α2, β ∈ ℝ, Ω ⊂ ℝN is a bounded domain with Lipschitz boundary, and M: \(\mathbb{R}_0^ + \rightarrow \mathbb{R}_0^ + ,\,\,f:\Omega \rightarrow \mathbb{R}\) are continuous functions. Firstly, we introduce a variational framework for the above problem. Then, the existence of least energy solutions is obtained by using variational methods, provided that the nonlinear term f has (θp − 1)-sublinear growth at infinity. Moreover, the existence of infinitely many solutions is obtained by using Krasnoselskii’s genus theory. Finally, we obtain the existence and multiplicity of solutions if f has (θp − 1)-superlinear growth at infinity. The main features of our paper are that the Kirchhoff function may vanish at zero and the nonlinearity may be singular.
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The third author of this paper would like to thank Professor Giovanni Molica Bisci for helpful discussions during the preparation of manuscript.
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The first author was supported by National Natural Science Foundation of China (11601515) and Fundamental Research Funds for the Central Universities (3122017080); the second author acknowledges the support of the Slovenian Research Agency grants P1-0292, J1-8131, N1-0064, N1-0083, and N1-0114; the third author was supported by National Natural Science Foundation of China (11871199 and 12171152), Shandong Provincial Natural Science Foundation, PR China (ZR2020MA006), and Cultivation Project of Young and Innovative Talents in Universities of Shandong Province.
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Xiang, M., Rădulescu, V.D. & Zhang, B. Existence Results for Singular Fractional p-Kirchhoff Problems. Acta Math Sci 42, 1209–1224 (2022). https://doi.org/10.1007/s10473-022-0323-5
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DOI: https://doi.org/10.1007/s10473-022-0323-5