On fundamental solutions of some singular elliptic equations of forth order (Q2713909)

Assume that \(E^+_p\) is the half-space \(x_p>0\) of the \(p\)-dimensional Euclidean space \(E_p\) of points \(x=(x',x_p)\) and \(x'=(x_1,\dots,x_{p-1})\), \(\Delta\) is the Laplace operator in \(E_p\), and \(\Delta_B\) is the operator NEWLINE\[NEWLINE \Delta_B=\Delta+{k\over x_p}{\partial\over \partial x_p}, \quad 0<k<p-2. NEWLINE\]NEWLINENEWLINENEWLINENEWLINEIn the present article, fundamental solutions are constructed in the half-space \(E^+_p\) of the operators \(\Delta\Delta_B\) and \(\Delta_B\Delta\), i.e., solutions of the equations NEWLINE\[NEWLINE \Delta\Delta_B w=\delta(x-y),\quad \Delta_B\Delta w=\delta(x-y) NEWLINE\]NEWLINE (here \(\delta\) is the Dirac function). The authors describe the character of singularities of their solutions.NEWLINENEWLINEFor the entire collection see [Zbl 0956.00039].