Critical $(p,q)$-fractional problems involving a sandwich type nonlinearity

In this paper, we deal with the following $(p,q)$-fractional problem $$(-\mathrm{\Delta }{)}_p^{s_1}u+(-\mathrm{\Delta }{)}_q^{s_2}u=\lambda P(x)\mathrm{\mid }u{\mathrm{\mid }}^{k-2}u+\theta \mathrm{\mid }u{\mathrm{\mid }}^{p_{s_1}^{\ast }-2}uin\mathrm{\Omega },u=0in{\mathbb{R}}^N\setminus \mathrm{\Omega },$$ where $\mathrm{\Omega }\subseteq {\mathbb{R}}^N$ is a general open set, $0<s_2<s_1<1$, $1<q<k<p<N\mathrm{/}{s}_1$, parameter $\lambda ,\theta >0$, $P$ is a nontrivial nonnegative weight, while $p_{s_1}^{\ast }=Np\mathrm{/}(N-p{s}_1)$ is the critical exponent. We prove that there exists a decreasing sequence $\{{\theta }_j{\}}_j$ such that for any $j\in \mathbb{N}$ and with $\theta \in (0,{\theta }_j)$, there exist ${\lambda }_{\ast }$, ${\lambda }^{\ast }>0$ such that above problem admits at least $j$ distinct weak solutions with negative energy for any $\lambda \in ({\lambda }_{\ast },{\lambda }^{\ast })$. On the other hand, we show there exists $\bar{\lambda }>0$ such that for any $\lambda >\bar{\lambda }$, there exists ${\theta }^{\ast }={\theta }^{\ast }(\lambda )>0$ such that the above problem admits a nonnegative weak solution with negative energy for any $\theta \in (0,{\theta }^{\ast })$.