A system of equations involving the fractional p-Laplacian and doubly critical nonlinearities

Abstract

This article deals with existence of solutions to the following fractional pp-Laplacian system of equations: (−Δp)su=∣u∣ps*−2u+γαps*∣u∣α−2u∣v∣βinΩ,(−Δp)sv=∣v∣ps*−2v+γβps*∣v∣β−2v∣u∣αinΩ,\left\{\begin{array}{l}{\left(-{\Delta }_{p})}^{s}u={| u| }^{{p}_{s}^{* }-2}u+\frac{\gamma \alpha }{{p}_{s}^{* }}{| u| }^{\alpha -2}u{| v| }^{\beta }\hspace{0.33em}\hspace{0.33em}\hspace{0.1em}\text{in}\hspace{0.1em}\hspace{0.33em}\Omega ,\hspace{1.0em}\\ {\left(-{\Delta }_{p})}^{s}v={| v| }^{{p}_{s}^{* }-2}v+\frac{\gamma \beta }{{p}_{s}^{* }}{| v| }^{\beta -2}v{| u| }^{\alpha }\hspace{0.33em}\hspace{0.33em}\hspace{0.1em}\text{in}\hspace{0.1em}\hspace{0.33em}\Omega ,\hspace{1.0em}\end{array}\right. where s∈(0,1)s\in \left(0,1), p∈(1,∞)p\in \left(1,\infty ) with N>spN\gt sp, α,β>1\alpha ,\beta \gt 1 such that α+β=ps*≔NpN−sp\alpha +\beta ={p}_{s}^{* }:= \frac{Np}{N-sp} and Ω=RN\Omega ={{\mathbb{R}}}^{N} or smooth bounded domains in RN{{\mathbb{R}}}^{N}. When Ω=RN\Omega ={{\mathbb{R}}}^{N} and γ=1\gamma =1, we show that any ground state solution of the aforementioned system has the form (λU,τλV)\left(\lambda U,\tau \lambda V) for certain τ>0\tau \gt 0 and UU and VV are two positive ground state solutions of (−Δp)su=∣u∣ps*−2u{\left(-{\Delta }_{p})}^{s}u={| u| }^{{p}_{s}^{* }-2}u in RN{{\mathbb{R}}}^{N}. For all γ>0\gamma \gt 0, we establish existence of a positive radial solution to the aforementioned system in balls. When Ω=RN\Omega ={{\mathbb{R}}}^{N}, we also establish existence of positive radial solutions to the aforementioned system in various ranges of γ\gamma .

Keywords

• fractional p-laplacian
• doubly critical
• ground state
• existence
• system
• least energy solution
• nehari manifold
• 35b09
• 35b33
• 35e20
• 35d30
• 35j50
• 45k05