On critical $p$-Laplacian systems

Abstract

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.