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

Abstract

We prove that, if E is the Engel group and u is a stable solution of ∆Eu = f(u), then ˆ {∇Eu 6=0}  |∇Eu|2 {( p + 〈 (Hu) ν, v 〉 |∇Eu| )2 + h } − J   η ≤ ˆ E |∇Eη||∇Eu| 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 Poincare inequality, extending the work of [21, 22, 13] to stratified groups of step 3. As an application, we provide a non-existence result.