A fractional $$p(x,\cdot )$$ p ( x, · ) -Laplacian problem involving a singular term

A. Mokhtari (), K. Saoudi () and N. T. Chung ()
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A. Mokhtari: University of M’sila
K. Saoudi: University of Imam Abdulrahman Bin Faisal
N. T. Chung: Quang Binh University

Indian Journal of Pure and Applied Mathematics, 2022, vol. 53, issue 1, 100-111

Abstract: Abstract This paper deals with a class of singular problems involving the fractional $$p(x,\cdot )$$ p ( x , · ) -Laplace operator of the form $$\begin{aligned} \left\{ \begin{array}{ll} (-\Delta )^{s}_{p(x,\cdot )}u(x)= \frac{\lambda }{u^{\gamma (x)}}+u^{q(x)-1} &{} \hbox {in }\Omega , \\ u>0, \;\;\text {in}\;\; \Omega &{} \hbox {} \\ u=0 \;\;\text {on}\;\;{\mathbb {R}}^N\setminus \Omega , &{} \hbox {} \end{array} \right. \end{aligned}$$ ( - Δ ) p ( x , · ) s u ( x ) = λ u γ ( x ) + u q ( x ) - 1 in Ω , u > 0 , in Ω u = 0 on R N \ Ω , where $$\Omega $$ Ω is a smooth bounded domain in $${\mathbb {R}}^N$$ R N ( $$N\ge 3$$ N ≥ 3 ), $$0 0$$ λ > 0 small enough. To our best knowledge, this paper is one of the first attempts in the study of singular problems involving fractional $$p(x,\cdot )$$ p ( x , · ) -Laplace operators.

Keywords: Fractional $$p(x; \cdot )$$ p ( x; · ) -Laplace operators; Singular equations; Minimization methods; Fractional Sobolev spaces; 35J60; 35J91, 35S30, 46E35, 58E30 (search for similar items in EconPapers)
Date: 2022
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DOI: 10.1007/s13226-021-00037-4

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