Least energy sign-changing solutions for a class of fractional $(p,q)$-Laplacian problems with critical growth in ${\mathbb{R}}^N$

This paper considers the following fractional $(p,q)$-Laplacian equation:

$$(-\mathrm{\Delta }{)}_p^{s}u+(-\mathrm{\Delta }{)}_q^{s}u+V(x)\left(|u{|}^{p-2}u+|u{|}^{q-2}u\right)=\lambda f(u)+|u{|}^{q_s^{\ast }-2}u\text{in}{\mathbb{R}}^N,$$

where $s\in (0,1),\lambda >0,2<p<q<\frac{N}{s}$, $(-\mathrm{\Delta }{)}_t^s$ with $t\in \{p,q\}$ is the fractional $t$-Laplacian operator, and potential $V$ is a continuous function. Using constrained variational methods, a quantitative Deformation Lemma and Brouwer degree theory, we prove that the above problem has a least energy sign-changing solution $u_{\lambda }$ under suitable conditions on $f$, $V$ and $\lambda$. Moreover, we show that the energy of $u_{\lambda }$ is strictly larger than two times the ground state energy.