An interpolation approach to L^ a priori estimates for elliptic problems with nonlinearity on the boundary

We establish an explicit L^ () a priori estimate for weak solutions to subcritical elliptic problems with nonlinearity on the boundary, in terms of the powers of their H¹ () 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.