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

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 $\mu$.