Parabolic approach to nonlinear elliptic eigenvalue problems

We consider the asymptotic profiles of the nonlinear parabolic flows ut=Δumut=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⊂RnΩ⊂Rn for different values of p  . In the classical case p=1p=1 we establish that log(φ)log(φ) is a strictly concave function, when 0