Existence of positive solutions to semilinear elliptic problems with nonlinear boundary condition

In this paper, a semilinear elliptic equation with a nonlinear boundary condition and a perturbation in the reaction term is studied. The existence of a positive solution and another non-zero solution to the problem is proved when \(|\lambda |\) is small enough without any specific assumptions on the perturbation term. Moreover, it is shown that the non-zero solution becomes a positive one for small \(\lambda >0\) under suitable assumptions on the perturbation term.