Uniqueness of positive radially symmetric solutions for a Schrödinger–Poisson–Slater equation

The paper deals with radially symmetric solutions of the following nonlinear Schrödinger–Poisson–Slater equation <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" display="block"> <m:mo>−</m:mo> <m:mi mathvariant="normal">Δ</m:mi> <m:mi>u</m:mi> <m:mo>+</m:mo> <m:mfenced open="(" close=")"> <m:mrow> <m:mo stretchy="false">|</m:mo> <m:mi>x</m:mi> <m:msup> <m:mrow> <m:mo stretchy="false">|</m:mo> </m:mrow> <m:mrow> <m:mo>−</m:mo> <m:mn>1</m:mn> </m:mrow> </m:msup> <m:mo>∗</m:mo> <m:msup> <m:mrow> <m:mi>u</m:mi> </m:mrow> <m:mrow> <m:mn>2</m:mn> </m:mrow> </m:msup> </m:mrow> </m:mfenced> <m:mi>u</m:mi> <m:mo>=</m:mo> <m:mi>μ</m:mi> <m:mo stretchy="false">|</m:mo> <m:mi>u</m:mi> <m:msup> <m:mrow> <m:mo stretchy="false">|</m:mo> </m:mrow> <m:mrow> <m:mi>p</m:mi> <m:mo>−</m:mo> <m:mn>2</m:mn> </m:mrow> </m:msup> <m:mi>u</m:mi> <m:mo>,</m:mo> <m:mspace width="0.3333em" class="nbsp"/> <m:mi mathvariant="normal">i</m:mi> <m:mi mathvariant="normal">n</m:mi> <m:mspace width="0.3333em" class="nbsp"/> <m:msup> <m:mrow> <m:mi mathvariant="double-struck">R</m:mi> </m:mrow> <m:mrow> <m:mn>3</m:mn> </m:mrow> </m:msup> <m:mo>,</m:mo> </m:math> -Δ u+( x ⁻¹ u²)u=μ u ᵖ⁻²u, in R³, where (| x | −1 ∗ u 2 ) is the repulsive Coulomb potential and μ > 0 is the Slater constant. When p = 18/5, we prove that the radially symmetric ground state solution of above equation is unbroken. Then, we obtain the uniqueness and smoothness of this positive radially symmetric solution and determine the expression for the best constant of the Coulomb–Sobolev inequality.