On eigenfunctions of homogeneous boundary value problems for an elliptic equation with Bessel operators (Q2713907)

Assume that \(D=\{(x,y):x^2+y^2<1\), \(x>0\), \(y>0\}\), \(\sigma=\{(x,y):x^2+y^2=1\), \(x\geq 0\), \(y\geq 0\}\), and \(I_1=\{(x,0):0<x<1\}\), and \(I_2=\{(0,y):0<y<1\}\). Considering the domain \(D\), the author addresses the degenerate equation NEWLINE\[NEWLINE \biggl({\partial^2\over \partial x^2} +{\alpha\over x}{\partial\over\partial x}+ {\partial^2\over\partial y^2}+ {\beta\over y}{\partial\over\partial y}+ \lambda^2\biggr) v(x,,y)=0\quad (\alpha,\beta-\text{const}) \tag{1} NEWLINE\]NEWLINE and poses the following problems.NEWLINENEWLINENEWLINEProblem 1. Find a solution to (1) in \(D\) satisfying the following: NEWLINE\[NEWLINE v|_{\partial D}=0. NEWLINE\]NEWLINE Problem 2. Find a solution to (1) in \(D\) satisfying the following: NEWLINE\[NEWLINE v|_\sigma=0,\quad \lim_{x\to +0}x^\alpha{\partial u\over\partial x} \bigg|_{I_2}=0,\quad \lim_{y\to +0}y^\beta{\partial v\over\partial y} \bigg|_{I_1}=0. NEWLINE\]NEWLINE Problem 3. Find a solution to (1) in \(D\) satisfying the following: NEWLINE\[NEWLINE v|_{\sigma\cup\bar I_2}=0,\quad \lim_{y\to+0}y^\beta{\partial v\over\partial y} \bigg|_{I_1}=0. NEWLINE\]NEWLINE Problem 4. Find a solution to (1) in \(D\) satisfying the following: NEWLINE\[NEWLINE v|_\sigma=0. NEWLINE\]NEWLINE The author writes out the complete systems of eigenfunctions in the class \(C(\overline D)\cap C^2(D)\) for \(\lambda>0\). For Problem~1 it is presumed that \(0<\alpha<1\), \(0<\beta<1\); for Problem~4, \(\alpha\geq 1\), \(\beta\geq 1\).NEWLINENEWLINEFor the entire collection see [Zbl 0956.00039].