Morse index, Leray-Schauder degree and local uniqueness for multi-peak concentrating solutions of a fractional Schrödinger equation

Abstract

We study positive $k$-peak solutions of the semiclassical fractional Schrödinger equation ${\epsilon }^{2s}(-\Delta {)}^{s}u+V(x)u=u^p$ in ${\mathbb{R}}^N$, concentrating at different nondegenerate critical points ${\xi }_1^0,\dots ,{\xi }_k^0$ of $V$. For every such family satisfying the natural energy quantization, we determine the complete low spectrum of the linearized operator. The first $k$ eigenvalues remain uniformly negative, the next $kN$ eigenvalues are of order ${\epsilon }^2$ and are governed by the Hessians $D^{2}V({\xi }_j^0)$, while the remaining spectrum is uniformly separated from zero. Consequently, the Morse index equals $k$ plus the total number of negative eigenvalues of these Hessians, and every such solution is nondegenerate. Combining a unique modulation parametrization with a Leray--Schauder degree computation, we further prove that, for all sufficiently small $\epsilon$, the prescribed concentrating class contains exactly one positive solution. The result applies to the whole energy-quantized class, not only to a particular solution.

76 pages