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Morse’s theory and local linking for a fractional \((p_{1}(\textrm{x}.,), p_{2}(\textrm{x}.,))\): Laplacian problems on compact manifolds

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Abstract

In this paper, we compute critical groups for establishing the existence of solutions for the following fractional \(\displaystyle (p_{1}(\textrm{x},.), p_{2}(\textrm{x},.))\)—Laplacian problem involving a singular term:

$$\begin{aligned} \begin{aligned} \left\{ \begin{array}{llll} (\mathcal {L}_{\textrm{g}} ^{s_{1}})_{p_{1}(\textrm{x},.)} \textrm{u}(\textrm{x})+ (\mathcal {L}_{\textrm{g}} ^{s_{2}})_{p_{2}(\textrm{x},.)} \textrm{u}(\textrm{x})&{}=\frac{f(\textrm{x}, \textrm{u}(\textrm{x}))}{ \textrm{u}(\textrm{x})^{\gamma (\textrm{x})}} &{} \text{ in } &{} \mathcal {U}, \\ \quad \textrm{u}&{}> 0 &{} \text{ in } &{} \mathcal {U},\\ \quad \textrm{u}&{}=0 &{} \text{ in } &{} \mathcal {M} \backslash \mathcal {U}. \end{array}\right. \end{aligned} \end{aligned}$$
(1)

More precisely, we use Morse’s relation to prove that our problem has infinitely many solutions.

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  1. These authors contributed equally to this work.

Authors and Affiliations

  1. Department of Mathematics, National School of Applied Sciences, Fez, Morocco

    Ahmed Aberqi

  2. Department of Mathematics, Faculty of Sciences Dhar El Mahraz, Atlas, Fez, 100190, Morocco

    Abdesslam Ouaziz

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  1. Ahmed Aberqi
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  2. Abdesslam Ouaziz
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Correspondence to Abdesslam Ouaziz.

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Aberqi, A., Ouaziz, A. Morse’s theory and local linking for a fractional \((p_{1}(\textrm{x}.,), p_{2}(\textrm{x}.,))\): Laplacian problems on compact manifolds. J. Pseudo-Differ. Oper. Appl. 14, 41 (2023). https://doi.org/10.1007/s11868-023-00535-5

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