In this talk I consider the two parameter family of PDE´s (generalized TFWequation):\[-\Delta u + (\gamma u^{2p-2} - \phi) u=0,\]on $\mathbb{R}^3$, where\[\phi(x)= \frac{Z}{|x|}-\int_{\mathbb{R}^3} \frac{u^2(y)}{|x-y|} \, dy.\]Here, $Z>0$ is fixed and $\gamma \ge 0$, and $1<p$. The case $p=5/3$corresponds to the Thomas-Fermi-von Weizsäcker equation in the atomiccase, which was first studied by R. Benguria, H. Brezis, and E.H. Lieb in1981. The case $\gamma =0$ corresponds to the Hartree equation.Here we are interested in estimating the so called ``excess charge´´,given by $Q=\int_{\mathbb{R}^3}\|u\|_2^2 - Z$, for different values of $p$ and $\gamma$, where $u$ is thepositive solution of the generalized TFW equation. This is joint work withHeinz Siedentop (Ludwig Maximilians University).
