On critical $p$-Laplacian systems

Abstract

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 ${\Delta }_p:=\text{div}(|\nabla u{|}^{p-2}\nabla u)$ 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 $\Vert u{\Vert }_{D^{1,p}}:=({\int }_{{\mathbb{R}}^N}|\nabla u{|}^p\text{d}x{)}^{\frac{1}{p}}$, $N\ge 3$, $1<p<N$, $\lambda ,{\mu }_1,{\mu }_2\ge 0$, $\gamma \ne 0$, $a,b,\alpha ,\beta >1$ 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.

References in corpus (1)

• Global regularity of weak solutions to quasilinear elliptic and parabolic equations with controlled growth

Cited by in corpus (1)

• Multiple solutions for coupled gradient-type quasilinear elliptic systems with supercritical growth