On critical $p$-Laplacian systems

TL;DR

This paper investigates the existence and nonexistence of positive least energy solutions for a critical p-Laplacian system involving nonlinear coupling terms, within both bounded and unbounded domains, under certain conditions.

Contribution

It introduces new results on the existence and multiplicity of solutions for a critical p-Laplacian system with nonlinear coupling, extending previous work to include both bounded and unbounded domains.

Findings

Established conditions for existence of positive least energy solutions.

Proved nonexistence results under certain parameter regimes.

Demonstrated multiplicity of nontrivial nonnegative solutions.

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.