Uniform estimates for elliptic equations with Carath\'eodory nonlinearities at the interior and on the boundary

Abstract

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 $L^{\infty}(\Omega )$ a priori estimates are in terms of powers of their $H^{1}(\Omega )$ 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.