Ground State Solutions of Nehari-Pohozaev Type for Schrödinger-Poisson Equation with Zero-Mass and Weighted Hardy Sobolev Subcritical Exponent

This paper is concerned with the following Schrödinger-Poisson equation with zero-mass and weighted Hardy Sobolev subcritical exponent \(\begin{aligned} -\Delta u+\left( \frac{1}{4\pi }|x|^{-1}*u^2\right) u=|x|^bf(u), \ \ \ \ x\in {\mathbb {R}}^{3}, \end{aligned}\) $$\begin{array}{c}\hfill -\mathrm{\Delta }u+\left(\frac{1}{4\pi },{|x|}^{-1},,\ast ,u^2\right)u={|x|}^{b}f(u),x\in {\mathbb{R}}^3,\end{array}$$ where \(-2<b<\infty \) $-2<b<\infty$ . Based on the known embedding results in Ruiz (Arch Ration Mech Anal 198:349–368, 2010) and Su et al. (Commun Contemp Math 9:571–583, 2007), we establish a new version of embedding theorem within the working space associated with the energy functional relevant to the aforementioned problem, which is different with Wang and Su (Appl Math Lett 107:106484, 2020). By combining the variational methods and some new analytical techniques, we prove the above problem admits a ground state solution of Nehari-Pohozaev type and a least energy solution under mild assumptions on f.