Uniform estimates for elliptic equations with Carathéodory 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 $L^{\infty }(\hat{I}\text{c◯})$ a priori estimates are in terms of powers of their $H^1(\hat{I}\text{c◯})$ 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.

key words: A priori estimates, slightly subcritical non-linearities, $L^{\infty }$ a priori bounds, nonlinear boundary conditions