Concentrating solutions for double critical fractional Schrödinger-Poisson system with p-Laplacian in ℝ3 - Liang Shuaishuai; Song Yueqiang; Shi Shaoyun (Advances in Nonlinear Analysis)
In this article, we consider the following double critical fractional Schrödinger-Poisson system involving p-Laplacian in R3{{\mathbb{R}}}^{3} of the form: εsp(−Δ)psu+V(x)∣u∣p−2u−ϕ∣u∣ps♯−2u=∣u∣ps*−2u+f(u)inR3,εsp(−Δ)sϕ=∣u∣ps♯inR3,\left\{\begin{array}{l}{\varepsilon }^{sp}{\left(-\Delta )}_{p}^{s}u+{\mathcal{V}}\left(x){| u| }^{p-2}u-\phi {| u| }^{{p}_{s}^{\sharp }-2}u={| u| }^{{p}_{s}^{* }-2}u+f\left(u)\hspace{1em}\hspace{0.1em}\text{in}\hspace{0.1em}\hspace{0.33em}{{\mathbb{R}}}^{3},\\ {\varepsilon }^{sp}{\left(-\Delta )}^{s}\phi ={| u| }^{{p}_{s}^{\sharp }}\hspace{1em}\hspace{0.1em}\text{in}\hspace{0.1em}\hspace{0.33em}{{\mathbb{R}}}^{3},\end{array}\right. where ε>0\varepsilon \gt 0 is a positive parameter, s∈(34,1)s\in \left(\frac{3}{4},1), (−Δ)ps{\left(-\Delta )}_{p}^{s} is the fractional p-Laplacian operator, p∈(32,3)p\in \left(\frac{3}{2},3), ps*=3p3−sp{p}_{s}^{* }=\frac{3p}{3-sp} is the Sobolev critical exponent, ps♯=p2(3+2s)(3−sp){p}_{s}^{\sharp }=\frac{p}{2}\frac{\left(3+2s)}{\left(3-sp)} is the upper exponent in the sense of the Hardy-Littlewood-Sobolev inequality, V(x):R3→R{\mathcal{V}}\left(x):{{\mathbb{R}}}^{3}\to {\mathbb{R}} symbolizes a continuous potential function with a local minimum, and the continuous function ff possesses subcritical growth. With the help of well-known penalization methods and Ljusternik-Schnirelmann category theory, we use the topological arguments to attain the multiplicity and concentration of the positive solutions for the above system.