Concentration phenomena for a mixed local/nonlocal Schr\"{o}dinger equation with Dirichlet datum

Abstract

We consider the mixed local/nonlocal semilinear equation \begin{equation*} -\epsilon^{2}\Delta u +\epsilon^{2s}(-\Delta)^s u +u=u^p\qquad \text{in } \Omega \end{equation*} with zero Dirichlet datum, where $\epsilon>0$ is a small parameter, $s\in(0,1)$, $p\in(1,\frac{n+2}{n-2})$ and $\Omega$ is a smooth, bounded domain. We construct a family of solutions that concentrate, as $\epsilon\rightarrow 0$, at an interior point of $\Omega$ having uniform distance to $\partial\Omega$ (this point can also be characterized as a local minimum of a nonlocal functional). In spite of the presence of the Laplace operator, the leading order of the relevant reduced energy functional in the Lyapunov-Schmidt procedure is polynomial rather than exponential in the distance to the boundary, in light of the nonlocal effect at infinity. A delicate analysis is required to establish some uniform estimates with respect to $\epsilon$, due to the difficulty caused by the different scales coming from the mixed operator.