An interpolation approach to $L^{\mathrm{\infty }}$ a priori estimates for elliptic problems with nonlinearity on the boundary

We establish an explicit $L^{\mathrm{\infty }}(\text{Om})$ a priori estimate for weak solutions to subcritical elliptic problems with nonlinearity on the boundary, in terms of the powers of their $H^1(\text{Om})$ norms. To prove our result, we combine in a novel way Moser type estimates together with elliptic regularity and Gagliardo–Nirenberg interpolation inequality. We illustrate our result with an application to subcritical problems satisfying Ambrosetti-Rabinowitz condition.