Acta mathematica scientia,Series A ›› 2022, Vol. 42 ›› Issue (3): 1209-1224.doi: 10.1007/s10473-022-0323-5

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Mingqi XIANG1, Vicenţiu D. RǍDULESCU2,3,4, Binlin ZHANG5   

  1. 1. College of Science, Civil Aviation University of China, Tianjin, 300300, China;
    2. Faculty of Applied Mathematics, AGH University of Science and Technology, al. Mickiewicza 30, 30-059, Kraków, Poland;
    3. Department of Mathematics, University of Craiova, Street A. I. Cuza No. 13, 200585, Craiova, Romania;
    4. Institute of Mathematics, Physics and Mechanics, Jadranska 19, 1000, Ljubljana, Slovenia;
    5. College of Mathematics and Systems Science, Shandong University of Science and Technology, Qingdao, 266590, China
  • Received:2021-04-07 Online:2022-06-26 Published:2022-06-24
  • Contact: Binlin ZHANG,
  • Supported by:
    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.

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: \begin{align*} \begin{cases} M\left(\displaystyle\iint_{\mathbb{R}^{2N}}\frac{|x|^{\alpha_1p}|y|^{\alpha_2p}|u(x)-u(y)|^{p}}{|x-y|^{N+ps}}{\rm d}x{\rm d}y\right) \mathcal{L}^{s}_pu= |x|^{\beta} f(u)\,\, \ &{\rm in}\ \Omega,\\ u=0\ \ \ \ &{\rm in}\ \mathbb{R}^N\setminus \Omega, \end{cases} \end{align*} where $\mathcal{L}^{s}_p$ is the generalized fractional $p$-Laplacian operator, $N\geq1$, $s\in(0,1)$, $\alpha_1,\alpha_2,\beta\in\mathbb{R}$, $\Omega\subset \mathbb{R}^N$ 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 $(\theta p-1)$-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 $(\theta p-1)$-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.

Key words: Fractional Kirchhoff equation, singular problems, variational and topological methods

CLC Number: 

  • 35R11