Solution blow-up for a new stationary Sobolev-type equation

A new nonlinear stationary Sobolev-type equation with a parameter $\eta\in\mathbb{R}^1$ is derived. For $\eta>0$, global solvability in the weak generalized sense is proved in the entire waveguide $\mathbb{S}\otimes\mathbb{R}_+^1$. For $\eta<0$, the strong generalized solution is shown to blow up in a certain waveguide cross section $z=R_0>0$. An upper bound for $R_0$ in terms of the original parameters of the problem is obtained.