H\"older continuity of bounded, weak solutions of a variational system in the critical case

Let $\mathrm{\Omega }\subset {\mathbb{R}}^2$ be a bounded, Lipschitz domain. We consider bounded, weak solutions ($u\in W^{1,2}\cap L^{\mathrm{\infty }}(\mathrm{\Omega };{\mathbb{R}}^N)$) of the vector-valued, Euler-Lagrange system: $\text{div }$ \big( A(x, u)Du\big)=g(x, u, Du)\quad$\text{in }$\Omega. Under natural growth conditions on the principal part and the inhomogeneity, but without any further restriction on the growth of the inhomogeneity (for example, via a smallness condition), we use a blow-up argument to prove that every bounded, weak solution of the system is H"older continuous. Since the dimension of $\mathrm{\Omega }$ is $2$ and $u\in W^{1,2}(\mathrm{\Omega };{\mathbb{R}}^N)$, we are in the critical setting, and hence, cannot use the Sobolev embedding theorem to deduce H"older continuity. Our results are connected to a particular case of the open problem of whether all solutions (and not just extremals) of variational systems are H"older continuous in the critical setting.