Uniform estimates for elliptic equations with Carathéodory nonlinearities at the interior and on the boundary

We establish an explicit uniform a priori estimate for weak solutions to slightly subcritical elliptic problems with nonlinearities simultaneously at the interior and on the boundary. Our explicit $L^{\mathrm{\infty }}(\mathrm{\Omega })$ a priori estimates are in terms of powers of their $H^1(\mathrm{\Omega })$ norms. To prove our result, we combine a De Giorgi-Nash-Moser’s iteration scheme together with elliptic regularity and the Gagliardo-Nirenberg’s interpolation inequality.