We consider the critical $p$-Laplacian system \begin{ equation}\label{ 92} \begin{ cases}-\Delta_p u-\frac{ \lambda a}{ p}|u|^{ a-2}u|v|^b =\mu_1|u|^{ p^\ast-2}u+\frac{ \alpha\gamma}{ p^\ast}|u|^{ \alpha-2}u|v|^{ \beta}, &x\in\Omega,\\ -\Delta_p v-\frac{ \lambda b}{ p}|u|^a|v|^{ b-2}v =\mu_2|v|^{ p^\ast-2}v+\frac{ \beta\gamma}{ p^\ast}|u|^{ \alpha}|v|^{ \beta-2}v, &x\in\Omega,\\ u,v\ \text{ in } D_0^{ 1,p}(\Omega), \end{ cases} \end{ equation} where $\Delta_p:=\text{ div}(|\nabla u|^{ p-2}\nabla u)$ 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 $\|u\|_{ D^{ 1,p}}:=\big(\int_{ \mathbb{ R}^N}|\nabla u|^p\text{ d}x\big)^{ \frac{ 1}{ p}}$, $N\ge3$, $1<p<N$, $\lambda, \mu_1, \mu_2\ge 0$, $\gamma\neq0$, $a, b, \alpha, \beta > 1$ satisfy $a + b = p, \alpha + \beta = p^\ast:=\frac{ Np}{ N-p}$, the critical Sobolev exponent, $\Omega$ is $\mathbb{ R}^N$ or a bounded domain in $\mathbb{ R}^N$, $D_0^{ 1,p}(\Omega)$ is the closure of $C_0^\infty(\Omega)$ 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.
Categories: math.AP