Analogy to the Gellerstedt problem for an equation of mixed type with two inner lines of degeneracy in an unlimited domain (Q2742981)

The following mixed type equation with two inner lines of degeneracy NEWLINE\[NEWLINE\operatorname{sgn } y y^mu_{xx}+ x^nu_{yy}=0,\quad m,n=\operatorname{const}>0\tag{*} NEWLINE\]NEWLINE in the unbounded domain \(\Omega=\{x\geq 0,y\geq 0\} \cup\{\frac 1q x^q>\frac 1p(-y)^p\), \(y\leq 0\}\) is considered. An analogy of the Gellerstedt problem for the equation (*), named as problem \(\Gamma^\infty\), is formulated. The existence and uniqueness theorem is stated. The considered problem is reduced to the system of two Fredholm integral equations having a weak singularity.