We study linear and quasi-linear elliptic equations containing the Bessel operator with respect to a selected variable (so-called special variable). The well-posedness of the nonclassical Dirichlet problem (with the additional condition of evenness with respect to the special variable) in the half-space is proved, an integral representation of the solution is constructed, and a necessary and sufficient condition of stabilization is established. The stabilization is understood as follows: the solution has a finite limit as the independent variable tends to infinity along the direction orthogonal to the boundary hyperplane.
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