Complex Analysis and Operator Theory

, Volume 13, Issue 3, pp 801–818

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

• Kamel Saoudi
Article

## Abstract

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 $$\mu$$.

## Keywords

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

## Mathematics Subject Classification

34B15 37C25 35R20

## Notes

### Acknowledgements

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

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