A Morse lemma for degenerate critical points of solutions of nonlinear equations in \(\mathbb{R}^2\) (Q2305512)

The author considers the equation \(-\Delta u=f(u)\) within the unit ball in two dimensions assuming that \(f\) is a smooth nonlinearity. He proves a Morse Lemma for the degenerate critical points for a \(C^2\) function \(u\) defined on the unit ball satisfying the above equation. Further, he shows results on the non-degeneracy of such critical points. Additionally, the shapes of the level sets are given.