Morse index of concentrated solutions for the nonlinear Schrödinger equation with a very degenerate potential (Q6568729)

The paper is concerned with solutions \(u_\varepsilon\) to the singularly perturbed nonlinear Schrödinger equation\N\[\N\begin{cases} -\varepsilon^2 \Delta u + V u = u^p \\\Nu>0 \\\Nu \in H^1(\mathbb{R}^N) \end{cases}\N\]\Nin the semiclassical regime \(\varepsilon \to 0\). Here \(N \geq 2\) and \(1 < p < 2^*-1\), where \(2^*=2N/(N-2)\) if \(N \geq 3\) and \(2^*=\infty\) if \(N=2\). The function \(V \colon \mathbb{R}^N \to \mathbb{R}\) is supposed to satisfy the following condition: there exist \(\delta>0\) and an \(M\)-dimensional compact, connected manifold \(\Gamma\) of class \(C^2\) without boundary such that \(1 \leq M \leq N-1\) and \(V(x) =1 \), \(\frac{\partial V}{\partial \nu_i} (x) = 0 \) and\N\[\N\det \left( \left( \frac{\partial^2 V}{\partial \nu_i \partial \nu_j}(x) \right)_{1\leq i,j \leq N-M} \right) \neq 0\N\]\Nfor every point \(x \in \Gamma\), where \(\nu_i\) is the \(i\)th outward normal vector of \(\Gamma\) at the point \(x \in \Gamma\), \(i=1,\dots,N-M\). Moreover \(V \in C^4\) in a tubular neighborhood \(W_\delta\) of \(\Gamma\), \(V\) is bounded in \(\mathbb{R}^N\) and \(\inf V >V_0 >0\).\N\NThe main result of the paper is a formula for computing the Morse index of a solution \(u_\varepsilon\) which concentrates at \(k\) different non-degenerate critical points \(b_1,\dots,b_k \in \Gamma\) of \(V\).