## On interacting bumps of semi-classical states of nonlinear Schrödinger equations.(English)Zbl 1217.35065

Summary: We study concentrated positive bound states of the following nonlinear Schrödinger equation $h^2 \Delta u \Gamma V (x)u + u^p = 0;~ u> 0;~ x\in\mathbb{R}^N$ where $$p$$ is subcritical. We prove that, at a local maximum point $$x_0$$ of the potential function $$V (x)$$ and for arbitrary positive integer $$K$$ ($$K>1$$), there always exist solutions with $$K$$ interacting bumps concentrating near $$x_0$$ . We also prove that at a nondegenerate local minimum point of $$V (x)$$ such solutions do not exist.

### MSC:

 35J60 Nonlinear elliptic equations 35B50 Maximum principles in context of PDEs 35Q55 NLS equations (nonlinear Schrödinger equations)