Complex Analysis and Operator Theory

, Volume 13, Issue 3, pp 801–818 | Cite as

A singular System Involving the Fractional p-Laplacian Operator via the Nehari Manifold Approach

  • Kamel SaoudiEmail author


In this work we study the fractional p-Laplacian equation with singular nonlinearity
$$\begin{aligned} \left\{ \begin{array}{ll} (-\Delta )^s_p u = \lambda a(x)|u|^{q-2}u +\frac{1-\alpha }{2-\alpha -\beta } c(x)|u|^{-\alpha }|v|^{1-\beta }, \quad \text {in }\Omega ,\\ \\ (-\Delta )^s_p v= \mu b(x)|v|^{q-2}v +\frac{1-\beta }{2-\alpha -\beta } c(x)|u|^{1-\alpha }|v|^{-\beta }, \quad \text {in }\Omega ,\\ \\ u=v = 0 ,\quad \text{ in } \,\mathbb {R}^N{\setminus }\Omega , \end{array} \right. \end{aligned}$$
where \(0<\alpha<1,\;0<\beta <1,\)\(2-\alpha -\beta<p<q<p^*_s,\)\(p^*_s=\frac{N}{N-ps}\) is the fractional Sobolev exponent, \(\lambda , \mu \) are two parameters, \(a,\, b, \,c \in C(\overline{\Omega })\) are non-negative weight functions with compact support in \(\Omega ,\) and \((-{\Delta )^{s}}_{p}\) is the fractional p-Laplace operator. We use the Nehari manifold approach and some variational techniques in order to show the existence and multiplicity of positive solutions of the above problem with respect to the parameter \(\lambda \) and \(\mu \).


Fractional p-Laplace operator Nehari manifold Singular elliptic system Multiple positive solutions 

Mathematics Subject Classification

34B15 37C25 35R20 



The author would like to thank the anonymous referees for their carefully reading this paper and their useful comments.


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© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  1. 1.College of sciences at DammamImam Abdulrahman Bin Faisal UniversityDammamKingdom of Saudi Arabia

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