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: −Δu+14π∣x∣∗∣u∣2u=∣u∣u+μ∣u∣p−2u,inR3,-\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<63\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.

Keywords

• schrödinger-poisson-slater equation
• coulomb-sobolev critical exponent
• ground state
• 35j20
• 35j61
• 35b33