Existence and multiplicity of solutions to perturbed \(\Delta ^{\alpha ,\beta }_{\alpha _1,\beta _1}\)-Laplacian system in \(\mathbb R^N\) involving critical nonlinearity

In this article, we consider a \(\Delta ^{\alpha ,\beta }_{\alpha _1,\beta _1}\) ${\mathrm{\Delta }}_{{\alpha }_1,{\beta }_1}^{\alpha ,\beta }$ -Laplacian system with critical nonlinearity in \(\mathbb R^N\) ${\mathbb{R}}^N$ \(\begin{aligned} -\varepsilon ^2\Delta ^{\alpha ,\beta }_{\alpha _1,\beta _1}u+b(X)u= & {} g(X)|u|^{\widetilde{2}^*-2}u+ F_u(X,u,v),\quad X\in \mathbb {R}^N,\\ -\varepsilon ^2\Delta ^{\alpha ,\beta }_{\alpha _1,\beta _1}v+b(X)v= & {} g(X)|v|^{\widetilde{2}^*-2}v+F_v(X,u,v),\quad X\in \mathbb {R}^N,\\{} & {} \quad u(X),\quad v(X)\rightarrow 0 \quad \text {as } |X|\rightarrow \infty , \end{aligned}\) $$\begin{array}{cc}\hfill -{\epsilon }^2{\mathrm{\Delta }}_{{\alpha }_1,{\beta }_1}^{\alpha ,\beta }u+b(X)u=& {g(X)|u|}^{{\tilde{2}}^{\ast }-2}u+F_u(X,u,v),X\in {\mathbb{R}}^N,\hfill \\ \hfill -{\epsilon }^2{\mathrm{\Delta }}_{{\alpha }_1,{\beta }_1}^{\alpha ,\beta }v+b(X)v=& {g(X)|v|}^{{\tilde{2}}^{\ast }-2}v+F_v(X,u,v),X\in {\mathbb{R}}^N,\hfill \\ \hfill & u(X),v(X)\to 0\text{as}|X|\to \infty ,\hfill \end{array}$$ where \(\Delta ^{\alpha ,\beta }_{\alpha _1,\beta _1}\) ${\mathrm{\Delta }}_{{\alpha }_1,{\beta }_1}^{\alpha ,\beta }$ is the subelliptic operator of the type \(\begin{aligned}&\Delta ^{\alpha ,\beta }_{\alpha _1,\beta _1}: =\Delta _x +\Delta _y +\left| x\right| ^{2\alpha }\left| y\right| ^{2\beta }\left( \left| x\right| ^{\alpha _1}+\left| y\right| ^{\beta _1}\right) ^2\Delta _z; x\in \mathbb R^{N_1}; y\in \mathbb R^{N_2}, z\in \mathbb R^{N_3};\\&N=N_1+N_2+N_3, \alpha , \beta , \alpha _1, \beta _1\ge 0, \widetilde{N}:=N_1+N_2+N_3(1+\alpha +\alpha _1+\beta +\beta _2),\\&\widetilde{2}^*=2\widetilde{N}/(\widetilde{N}-2), X=(x,y,z), (\widetilde{N}> 2). \end{aligned}\) $$\begin{array}{cc}& {\mathrm{\Delta }}_{{\alpha }_1,{\beta }_1}^{\alpha ,\beta }:={\mathrm{\Delta }}_x+{\mathrm{\Delta }}_y+{\left|x\right|}^{2\alpha }{\left|y\right|}^{2\beta }{\left({\left|x\right|}^{{\alpha }_1},+,{\left|y\right|}^{{\beta }_1}\right)}^2{\mathrm{\Delta }}_z;x\in {\mathbb{R}}^{N_1};y\in {\mathbb{R}}^{N_2},z\in {\mathbb{R}}^{N_3};\hfill \\ \hfill & N=N_1+N_2+N_3,\alpha ,\beta ,{\alpha }_1,{\beta }_1\ge 0,\tilde{N}:=N_1+N_2+N_3(1+\alpha +{\alpha }_1+\beta +{\beta }_2),\hfill \\ \hfill & {\tilde{2}}^{\ast }=2\tilde{N}/(\tilde{N}-2),X=(x,y,z),(\tilde{N}>2).\hfill \end{array}$$ Under some proper conditions, we obtain the existence of standing wave solutions \((u_\varepsilon , v_\varepsilon )\) $(u_{\epsilon },v_{\epsilon })$ which tend to the trivial solutions as \(\varepsilon \rightarrow 0\) $\epsilon \to 0$ . Moreover, we get m pairs of solutions for the above system under some extra assumptions. Our results improve and supplement some existing relevant results.