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A theorem on the solutions of the problem (Uu0027(w)={gamma }F(U(w),w), U(w_1)=u_1, U(w_2)=u_2) is applied for finding the functional solutions of the system of partial differential equations $$begin{aligned} {nabla }cdot (a(u,w){nabla }u)= u0026 {} 0,quad u=u_1{ hbox {on}quad {{Gamma }}_1},quad u=u_2{ hbox {on}quad {{Gamma }}_2},quad frac{{partial }u}{{partial }n}=0{ hbox {on}quad {{Gamma }}_3}  {nabla }cdot (b(u,w){nabla }w)= u0026 {} 0,quad w=w_1{ hbox {on}quad {{Gamma }}_1},quad w=w_2{ hbox {on}quad {{Gamma }}_2},quad frac{{partial }w}{{partial }n}=0{ hbox {on}quad {{Gamma }}_3}. end{aligned}$$The problem of existence and uniqueness of solutions is considered.

An application of a theorem of G. Zwirner to a class of non-linear elliptic systems in divergence form