On critical $p$-Laplacian systems

We consider the critical $p$-Laplacian system \begin{equation}\label{92} \begin{cases}-Δ_p u-\frac{λa}{p}|u|^{a-2}u|v|^b =μ_1|u|^{p^\ast-2}u+\frac{αγ}{p^\ast}|u|^{α-2}u|v|^β, &x\inΩ,\\ -Δ_p v-\frac{λb}{p}|u|^a|v|^{b-2}v =μ_2|v|^{p^\ast-2}v+\frac{βγ}{p^\ast}|u|^α|v|^{β-2}v, &x\inΩ,\\ u,v\ \text{in } D_0^{1,p}(Ω), \end{cases} \end{equation} where $Δ_p:=\text{div}(|\nabla u|^{p-2}\nabla u)$ is the $p$-Laplacian operator defined on $D^{1,p}(\mathbb{R}^N):=\{u\in L^{p^\ast}(\mathbb{R}^N):|\nabla u|\in L^p(\mathbb{R}^N)\}$, endowed with norm $\|u\|_{D^{1,p}}:=\big(\int_{\mathbb{R}^N}|\nabla u|^p\text{d}x\big)^{\frac{1}{p}}$, $N\ge3$, $1<p<N$, $λ, μ_1, μ_2\ge 0$, $γ\neq0$, $a, b, α, β> 1$ satisfy $a + b = p, α+ β= p^\ast:=\frac{Np}{N-p}$, the critical Sobolev exponent, $Ω$ is $\mathbb{R}^N$ or a bounded domain in $\mathbb{R}^N$, $D_0^{1,p}(Ω)$ is the closure of $C_0^\infty(Ω)$ in $D^{1,p}(\mathbb{R}^N)$. Under suitable assumptions, we establish the existence and nonexistence of a positive least energy solution. We also consider the existence and multiplicity of nontrivial nonnegative solutions.

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