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

TL;DR

This paper studies solutions to a mixed local/nonlocal Schrödinger equation that concentrate at interior points as a small parameter tends to zero, revealing unique polynomial behavior influenced by nonlocal effects.

Contribution

It constructs solutions concentrating at interior points for a mixed Schrödinger equation, highlighting the polynomial nature of the reduced energy functional due to nonlocal effects.

Findings

Solutions concentrate at interior points with uniform distance from boundary.

The reduced energy functional is polynomial, not exponential, due to nonlocal effects.

Uniform estimates are achieved despite multiple scales from the mixed operator.

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.