We use De Giorgi-Nash-Moser iteration scheme to establish that weak solutions to a coupled system of elliptic equations with critical growth on the boundary are in L ∞ (Ω). Moreover, we provide an explicit L ∞ (Ω)- estimate of weak solutions with subcritical growth on the boundary, in terms of powers of H 1 (Ω)-norms, by combining the elliptic regularity of weak solutions with Gagliardo–Nirenberg interpolation inequality.
Chhetri et al. (Mon,) studied this question.