Infinitely many new solutions for singularly perturbed Schrödinger equations

This paper deals with the existence of solutions for the following perturbed Schrödinger equation \begin{equation*} -\varepsilon^{2} Δu + V(x)u= |u|^{p-2}u, , , $\text{ in }$ , , \r^{N}, \end{equation*} where $\epsilon$ is a parameter, $N\ge 3$, $p\in (2,\frac{2N}{N-2})$, and $V(x)$ is a potential function in \r^{N}. We demonstrate an interesting ``dichotomy’’ phenomenon for concentrating solutions of the above Schrödinger equation. More specifically, we construct infinitely many new solutions with peaks locating both in the bounded domain and near infinity, which fulfills the profile of the concentration compactness. Moreover, this approach can be extended to solve other related problems.