Asymptotic Behavior of Ground States and Local Uniqueness for Fractional Schrödinger Equations with Nearly Critical Growth

We study quantitative aspects and concentration phenomena for ground states of the following nonlocal Schrödinger equation (−Δ)su+V(x)u=u2s∗−1−𝜖inℝN,\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$(-{\Delta })^{s} u+V(x)u= u^{2_{s}^{*}-1-\epsilon } \ \ \text {in}\ \ \mathbb {R}^{N},$\end{document} where 𝜖 > 0, s ∈ (0,1), 2s∗:=2NN−2s\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$2^{*}_{s}:=\frac {2N}{N-2s}$\end{document} and N > 4s, as we deal with finite energy solutions. We show that the ground state u𝜖 blows up and precisely with the following rate ∥u𝜖∥L∞(ℝN)∼𝜖−N−2s4s\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\|u_{\epsilon }\|_{L^{\infty }(\mathbb {R}^{N})}\sim \epsilon ^{-\frac {N-2s}{4s}}$\end{document}, as 𝜖→0+\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\epsilon \rightarrow 0^{+}$\end{document}. We also localize the concentration points and, in the case of radial potentials V, we prove local uniqueness of sequences of ground states which exhibit a concentrating behavior.