Regularity and explicit $L^{\mathrm{\infty }}$ estimates for a class of nonlinear elliptic systems

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 $L^{\mathrm{\infty }}(\mathrm{\Omega })$. Moreover, we provide an explicit $L^{\mathrm{\infty }}(\mathrm{\Omega })$- estimate of weak solutions with subcritical growth on the boundary, in terms of powers of $H^1(\mathrm{\Omega })$-norms, by combining the elliptic regularity of weak solutions with Gagliardo–Nirenberg interpolation inequality.