Semiclassical standing waves with clustering peaks, for nonlinear Schrödinger equations (Q2925674)

The paper aims to produce a rigorous proof of the existence of ground-state (lowest-energy) solutions of a generalized equation of the Gross-Pitaevskii equation in the multidimensional space, NEWLINE\[NEWLINE -\epsilon^2\Delta v +V(x)v = f(v), NEWLINE\]NEWLINE where \(\Delta\) is the Laplacian, \(V(x)\) is a trapping potential with a set of local-minimum points, \(f(v)\) is a nonlinear term, and the ground-state solution sought for must be positive and localized, i.e., vanishing at \(|x|\to\infty\). In previous works on this topic, it was rigorously proved that, in the case of a single local minimum of \(V(x)\), the ground state solution reduces, in the limit of \(\epsilon\to 0\) (in the case of the self-repulsive nonlinearity, it corresponds to the Thomas-Fermi approximation), to a single peak, i.e., a singular solution concentrated at the minimum point. The previous proofs were based on the Lyapunov-Schmidt reduction and mountain-pass theorem (the steepest-descent method), and required information about properties of solutions for small perturbations linearized around the ground state, which may be difficult to establish. The present work presents a proof for the potential with multiple local minima, which give rise to a cluster of multiple peaks, and the proof does not requires the knowledge of the spectral properties of the linearized equation. The proof is based on a variational method.