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Concentration phenomena for a mixed local/nonlocal Schrödinger equation with Dirichlet datum

Serena Dipierro, Xifeng Su, Enrico Valdinoci, Jiwen Zhang

Abstract

We consider the mixed local/nonlocal semilinear equation \begin{equation*} -ε^{2}Δu +ε^{2s}(-Δ)^s u +u=u^p\qquad \text{in } Ω \end{equation*} with zero Dirichlet datum, where $ε>0$ is a small parameter, $s\in(0,1)$, $p\in(1,\frac{n+2}{n-2})$ and $Ω$ is a smooth, bounded domain. We construct a family of solutions that concentrate, as $ε\rightarrow 0$, at an interior point of $Ω$ having uniform distance to $\partialΩ$ (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 $ε$, due to the difficulty caused by the different scales coming from the mixed operator.

Concentration phenomena for a mixed local/nonlocal Schrödinger equation with Dirichlet datum

Abstract

We consider the mixed local/nonlocal semilinear equation \begin{equation*} -ε^{2}Δu +ε^{2s}(-Δ)^s u +u=u^p\qquad \text{in } Ω \end{equation*} with zero Dirichlet datum, where is a small parameter, , and is a smooth, bounded domain. We construct a family of solutions that concentrate, as , at an interior point of having uniform distance to (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 , due to the difficulty caused by the different scales coming from the mixed operator.

Paper Structure

This paper contains 12 sections, 28 theorems, 357 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Energy estimates and functional expansion at $\bar{u}_\xi$
  3. Estimates on $\mathcal{H}_\varepsilon(\xi)$
  4. Functional expansion
  5. Decay estimates for the ground state $w$
  6. Some regularity estimates
  7. The Lyapunov-Schmidt reduction
  8. Linear theory
  9. The nonlinear project problem
  10. Reduction to finite dimensions
  11. Proof of Theorem \ref{['th main theorem']}
  12. Uniform $L^\infty$ estimates

Key Result

Theorem 1.1

If $\varepsilon>0$ is sufficiently small, there exist a point ${\xi}_\varepsilon\in\Omega$ with dist$({\xi}_\varepsilon,\partial\Omega)\geqslant c$, and a solution ${u}_\varepsilon$ of problem vfveffd such that for a suitable constant $\gamma_1>0$ depending only on $n$, $s$ and $p$. Here, $c$ and $C$ are positive constants depending only on $n$, $s$, $p$ and $\Omega$.

Figures (1)

  • Figure 1: The set $\mho_p$ in the proof of Lemma \ref{['lemma regularity']}

Theorems & Definitions (55)

  • Theorem 1.1
  • Remark 1.2
  • Remark 1.3
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • proof
  • Lemma 2.4
  • ...and 45 more