Infinitely many new solutions for singularly perturbed Schr\"{o}dinger equations

Abstract

This paper deals with the existence of solutions for the following perturbed Schr\"{o}dinger equation \begin{equation*} -\varepsilon^{2} \Delta u + V(x)u= |u|^{p-2}u, \, \, \text{ in } \, \, \r^{N}, \end{equation*} where $\varepsilon$ is a parameter, $N \geq 3$, $p \in (2, \frac{2N}{N-2})$, and $V(x)$ is a potential function in $\r^{N}$. We demonstrate an interesting ``dichotomy'' phenomenon for concentrating solutions of the above Schr\"{o}dinger 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.