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

Abstract

In this paper, we deal with the following $(p,q)$-fractional problem $$ (-Δ)^{s_{1}}_{p}u +(-Δ)^{s_{2}}_{q}u=λP(x)|u|^{k-2}u+θ|u|^{p_{s_{1}}^{*}-2}u \, \mbox{ in }\, Ω,\qquad u=0\, \mbox{ in }\, \mathbb{R}^{N} \setminus Ω, $$ 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 $p_{s_1}^{\ast }=Np/(N-p{s}_1)$ 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 $\overline{\lambda }>0$ such that for any $\lambda >\overline{\lambda }$, there exists ${\theta }^{\ast }={\theta }^{\ast }(\lambda )>0$ such that the above problem admits a nonnegative weak solution with negative energy for any $\theta \in (0,{\theta }^{\ast })$.