Concentration on points for a nonlinear Schrödinger problem with Dirichlet boundary condition
Liqin Zhang · Discrete and Continuous Dynamical Systems · 2025
We considered the problem$ \begin{align*} \begin{cases} \left(\dfrac{\varepsilon}{i} abla-\boldsymbol{A}(x)\right)^2v(x)+v(x) = f\big(v(x)\big),&\text{in}\ \Omega, \\ v(x) = 0,&\text{on}\ \partial\Omega, \end{cases} \end{align*} \quad\quad\quad (\ast) $where $ \Omega $ was a bounded smooth domain in $ {\mathbb R}^3 $, $ \varepsilon>0 $ was a small parameter, and $ f(v) = |v|^2v $. In this paper, we constructed some solutions of problem (*) and proved that if we change the magnetic potential $ \boldsymbol{A} $ to $ \varepsilon \boldsymbol{A} $, the solutions will concentrate at some points. By careful estimates of the energy of an approximate solution and solving a nonlinear projected problem, we used the Liapunov-Schmidt reduction method to reduce the problem to a finite dimensional minimization problem. Then, we used the min-maxing procedure to locate multiple interior spikes.