A priori estimates of nodal solutions on the annulus for some PDE and their Morse index (Q2882905)

In this paper, the author studies radially symmetric solutions of the following semilinear elliptic Dirichlet problem: NEWLINE\[NEWLINE -\Delta u =f(u) \text{ in } \Omega, \leqno{(P)}NEWLINE\]NEWLINE where \(\Omega \subset \mathbb{R}^N\) (\(N\geq 2\)) is an annulus, \(f:\mathbb{R} \rightarrow \mathbb{R}\) is locally Lipschitz continuous with \(f(0)=0\) and superlinear in \(s\) at infinity. Under some further conditions on \(f\), the author establishes a priori estimates of radial solutions with the \(k\)-nodal of problem (P). Specially, for \(f(r,s)=K(r)g(s)\) with \(g\) odd and superlinear at infinity, the author verifies that these solutions of (P) are non-degenerate and their radial Morse index is exactly \(k\).