A geometric inequality for stable solutions of semilinear elliptic problems in the Engel group

We prove that, if E is the Engel group and u is a stable solution of ΔEu = f(u), then for any test function η ∈ C∞0 (E). Here above, h is the horizontal mean curvature, p is the imaginary curvature and J:= 2(X3X2uX1u - X3X1uX2u) + (X4u)(X1u - X2u) This can be interpreted as a geometric Poincaré inequality, extending the work of [21,22, 13] to stratified groups of step 3. As an application, we provide a non-existence result.