We consider the asymptotic profiles of the nonlinear parabolic flows ut=Δum to show the geometric properties of the following elliptic nonlinear eigenvalue problems:Δφ+λφp=0,φ>0inΩ,φ=0onΩposed in a strictly convex domain Ω⊂Rn for different values of p. In the classical case p=1 we establish that log(φ) is a strictly concave function, when 0<p<1 the function φ1−p2 is strictly concave, while for 1<p<n+2n−2 there is at least a solution such that φ1−p2 is strictly convex. Moreover, eventual convexity properties for the parabolic flows are proved.