Multiple positive solutions for a class of concave-convex Schrödinger-Poisson-Slater equations with critical exponent

In this article, we consider the multiplicity of positive solutions for a static Schrödinger-Poisson-Slater equation of the type − Δ u + u 2 ∗ 1 ∣ 4 π x ∣ u = μ f ( x ) ∣ u ∣ p − 2 u + g ( x ) ∣ u ∣ 4 u in R 3 , -\Delta u+\left({u}^{2}\ast \frac{1}{| 4\pi x| }\right)u=\mu f\left(x){| u| }^{p-2}u+g\left(x){| u| }^{4}u\hspace{1em}\hspace{0.1em}\text{in}\hspace{0.1em}\hspace{0.33em}{{\mathbb{R}}}^{3}, where μ > 0 \mu \gt 0 , 1 < p < 2 1\lt p\lt 2 , f ∈ L 6 6 − p ( R 3 ) f\in {L}^{\tfrac{6}{6-p}}\left({{\mathbb{R}}}^{3}) , and f , g ∈ C ( R 3 , R + ) f,g\in C\left({{\mathbb{R}}}^{3},{{\mathbb{R}}}^{+})