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

Abstract: In this paper, we deal with the following $(p,q)$-fractional problem $$ (-\Delta)^{s_{1}}_{p}u +(-\Delta)^{s_{2}}_{q}u=\lambda P(x)|u|^{k-2}u+\theta|u|^{p_{s_{1}}^{*}-2}u \, \mbox{ in }\, \Omega,\qquad u=0\, \mbox{ in }\, \mathbb{R}^{N} \setminus \Omega, $$ where $\Omega\subseteq\mathbb{R}^{N}$ is a general open set, $0<s_{2}<s_{1}\<1$, $1<q<k<p<N/s_{1}$, parameter $\lambda,\ \theta\>0$, $P$ is a nontrivial nonnegative weight, while $p_{s_{1}}^{*}=Np/(N-ps_{1})$ 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 $\lambda$, $\lambda^>0$ such that above problem admits at least $j$ distinct weak solutions with negative energy for any $\lambda\in (\lambda_,\lambda^)$. On the other hand, we show there exists $\overline{\lambda}>0$ such that for any $\lambda>\overline{\lambda}$, there exists $\theta^=\theta^(\lambda)>0$ such that the above problem admits a nonnegative weak solution with negative energy for any $\theta\in(0,\theta^*)$.