Let $\Omega \subseteq \mathbb{R}^2$ be a bounded convex domain with piecewise smooth boundary. Let $\phi :\partial \Omega \to \mathbb R$ be a continuous function. Let $L$ be a quasilinear elliptic operator in divergence form (in particular I am interested in the case of the capillary equation, i.e. of prescribed mean curvature) without any zero order terms.

Is there any general existence result for classical solutions of the following Dirichlet problem? $$ \begin{cases} L u = 0 \qquad \text{in } \Omega;\\ u|_{\partial \Omega} = \phi. \end{cases} $$

I know that the answer is positive in the case of the minimal surface equation, but I was wonder if the result is still true even in the presence of a (nonlinear) first order term. Any help would be very appreciated!