Properties of the positive solution of a generalized Thomas-Fermi-von Weizsäcker equation
This paper is devoted to a qualitative study of the positive solutions of a generalized Thomas-Fermi-von Weizsäcker equation: \[ -\Delta u+(cu^ q-\phi)u=-\phi_ 0u \] in \({\mathbb{R}}^ 3\), where \(\phi =V(x)-\int (u^ 2(y)/| x-y|)dy\) is the total electric potential, \(V(x)=\sum^{k}_{i=1}z_ i/| x-R_ i|,\) \(q\geq 1\), \(c\geq 0\). Set \(\sum^{k}_{i=1}z_ i=Z\). For \(q>1\) and \(k=1\) it is proved that \(\int u^ 2(y)dy>Z\) a result previously known for \(q=4/3\) and arbitrary k. The paper also covers the case \(q=1\) where the validity of the above inequality is shown to depend upon c. Moreover in this case an asymptotic analysis of the solution is carried out.
scientific article; zbMATH DE number 4212561
scientific article