An interpolation approach to 𝐿^{∞} a priori estimates for elliptic problems with nonlinearity on the boundary
Maya Chhetri, Nsoki Mavinga, Rosa Pardo San Gil · Proceedings of the American Mathematical Society · 2024
We establish an explicit L ∞ ( Ω ) L^\infty (\Omega ) a priori estimate for weak solutions to subcritical elliptic problems with nonlinearity on the boundary, in terms of the powers of their H 1 ( Ω ) H^1(\Omega ) 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.