Multi-peak positive solutions of a nonlinear Schrödinger-Newton type system (Q2305516)

This paper deals with the nonlinear Schrödinger-Newton type system \[ \begin{cases} -\varepsilon^2\Delta u + u -\Phi(x)u = Q(x)|u|u, &x \in\mathbb{R}^3,\\ -\varepsilon\Delta\Phi =u^2, &x \in\mathbb{R}^3, \end{cases}\tag{1} \] where \(\varepsilon > 0\) and \(Q\) is a bounded positive continuous function such that \begin{itemize} \item \(Q\) has a strict local minimum at some point \(x_0 \in\mathbb{R}^3\); \item there exist two constants \(C,\,\theta>0\) such that \(|Q(x)-Q(z)|\le C|x-z|^\theta\) for all \(x, \, z\) sufficiently close to \(x_0\). \end{itemize} System (1) describes the interaction of a particle with its own gravitational field, \(\Phi\) is the time independent potential of the particle. The authors prove that in correspondence of each strict local minimum point \(x_0\) of \(Q\) and each positive integer \(k\), (1) has a positive solution with \(k\) peaks concentrating near \(x_0\), provided that \(\varepsilon > 0\) is sufficiently small, namely a solution with \(k\) maximum points converging to \(x_0\) as \(\varepsilon \to 0\).