Groundstate for the Schrödinger-Poisson-Slater equation involving the Coulomb-Sobolev critical exponent

Abstract In this article, we study the existence of ground state solutions for the Schrödinger-Poisson-Slater type equation with the Coulomb-Sobolev critical growth: − Δ u + 1 4 π ∣ x ∣ ∗ ∣ u ∣ 2 u = ∣ u ∣ u + μ ∣ u ∣ p − 2 u , in R 3 , -\Delta u+\left(\frac{1}{4\pi | x| }\ast | u{| }^{2}\right)u=| u| u+\mu | u{| }^{p-2}u,\hspace{1.0em}{\rm{in}}\hspace{0.33em}{{\mathbb{R}}}^{3}, where μ > 0 \mu \gt 0 and 3 < p < 6 3\lt p\lt 6 . With the help of the Nehari-Pohozaev method, we obtain a ground-state solution for the above equation by employing compactness arguments.