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

Abstract

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

21 pages, 3 figures