Geometric properties of the Gelfand problem through a parabolic approach

We consider the asymptotic profiles of the nonlinear parabolic flows ( e u ) t = Δ u + λ e u to show the geometric properties of minimal solutions of the following elliptic nonlinear eigenvalue problems known as the Gelfand problem: Δ φ + λ e φ = 0 , φ > 0 in Ω φ = 0 on Ω posed in a strictly convex domain Ω ⊂ R n . In this work, we show that there is a strictly increasing function f ( s ) such that f − 1 ( φ ( x ) ) is convex for 0 < λ ⩽ λ ⁎ , i.e., we prove that level set of φ is convex. Moreover, we also present the boundary condition of φ which guarantees the f-convexity of solution φ.