Positive Solutions for Boundary Value Problems in the Half-Space with Sign-Changing Nonlinearities

We study a class of semilinear elliptic equations of the form −Δv=g+μf.,v in the half-space subject to suitable boundary conditions. The source term g is assumed to be a nonnegative function in a Kato-type class, while the reaction term f may change sign. Under appropriate assumptions on f, we prove, for sufficiently small values of the parameter μ, the existence and uniqueness of a positive continuous solution that is bounded above and below by a multiple of the solution for the corresponding linear problem.