Multi-bump bound states of Schrödinger equations with shrinking self-focusing core

We consider the equation −Δu+u=Qn(x)up−2u,x∈RN, where Qn are concrete bounded functions whose self-focusing core Qn > 0 shrinks to the set {xi|i = 1, …, k}, k ≥ 3, as n → ∞. It is known that the nonlinear Schrödinger equation possesses a rich physical background. The potential Qn(x) is utilized in nonlinear optics and quantum electrodynamics. It is worthwhile to explore the concentration of solutions, which represents the intensity of waves in physics. We investigate the existence of the solution which concentrates simultaneously on {xi|i = 1, …, k}. Furthermore, the energy of each concentration point {xi|i = 1, …, k} is strictly positive, meaning that the solution may concentrate at each point of the limiting set of the self-focusing core. In our paper, the extension of region and the shrinkage of self-focusing core are simultaneous. For this scenario, we employ a penalty function approach to solve the problem, which is commonly used in concentration problems. However, most properties of the limit equation remain unknown. To address this, we aim to provide some of these properties in our paper.