Some inequalities associated with semilinear elliptic equations with variable coefficients and applications

In this paper starting from some inequalities associated with stable solutions of semilinear equations with variable coefficients, we introduce new inequalities that we use revisiting some wellknown results concerning a De Giorgi's conjecture. Indeed, instead of studying the curvatures of the level sets, we consider some invariants naturally associated with the Hessian matrix of the solutions. Further applications of such weighted inequalities are briefly discussed in a couple of degenerate elliptic cases and, as far as some Poincaré type inequalities concerns, we deduce a few inequalities in the framework of the hyperbolic plane.