Abstract:
This paper deals with the following singular problem:
\begin{align*}
\begin{cases}
(-\Delta)^s_p u+ \mu(-\Delta)^s_q u =\frac{a(x)}{ u^\gamma} +\lambda f(x,u)
&\text{ in }\,\Omega,\\
u = 0,&\text{ in }\,\mathbb{R}^N\setminus\Omega,
\end{cases}
\end{align*}
where $\Omega\subset\mathbb{R}^N$
($N\geq 3$) are a bounded smooth domain, $f\in
C(\Omega\times
\mathbb{R}, \mathbb{R})$
is positively homogeneous
of degree $r-1$,
$a\in L^\infty(\Omega)$,
$a(x)>0$
for almost every $x\in \Omega$,
$\lambda$,
$\mu >0$,
$s\in(0,1)$,
$N> ps$,
and $0<\gamma<1<q<p<r<p^*_s$.
Under
appropriate conditions on the function $f$,
we establish the existence of multiple
solutions by using the Nehari manifold method.
