= Solution
Let the displayed integrand be $F$. Since it has no explicit dependence on $u$ or $v$, the <Euler-Lagrange equations for two fields> are
$$
\partial_xF_{u_x}+\partial_yF_{u_y}=0,
\qquad
\partial_xF_{v_x}+\partial_yF_{v_y}=0.
$$
Its derivatives are
$$
F_{u_x}=(\lambda+2\mu)u_x+(\lambda+\mu)v_y,
\qquad F_{u_y}=\mu u_y,
$$
$$
F_{v_x}=\mu v_x,
\qquad F_{v_y}=(\lambda+2\mu)v_y+(\lambda+\mu)u_x.
$$
Thus
$$
(\lambda+2\mu)u_{xx}+\mu u_{yy}
+(\lambda+\mu)v_{xy}=0,
$$
$$
\mu v_{xx}+(\lambda+2\mu)v_{yy}
+(\lambda+\mu)u_{xy}=0.
$$
Using the <Laplacian>, <gradient>, and <divergence>, these combine into the static <Navier-Cauchy equation>
$$
\boxed{\mu\nabla^2\mathbf u
+(\lambda+\mu)\nabla(\nabla\cdot\mathbf u)=0}.
$$
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