Asymptotic behavior of ground state solutions to nonlinear elliptic problems with the fractional Laplacian

TL;DR

This paper studies how ground state solutions to a nonlinear fractional Laplacian equation behave as the fractional parameter approaches critical values, revealing convergence, blow-up, and concentration phenomena.

Contribution

It characterizes the asymptotic behavior of solutions as the fractional order varies, including existence, convergence, blow-up, and concentration at potential minima.

Findings

Solutions exist for s in (s_0, 1)

As s approaches 1, solutions converge to a limit function

As s approaches s_0, solutions blow up and concentrate at minima of V

Abstract

In this paper, we consider the asymptotic behavior of the ground state solution $u_s$ of the nonlinear fractional Laplacian equation \begin{equation}\label{eq:0.1a} (-\Delta)^su+Vu=|u|^{p-2}u\quad x\in \mathbb{R}^n \end{equation} by taking $s$ as a parameter, where $n\geq 4$, $2<p<\frac{2n}{n-2}$, $V$ is a potential function. We show that for a fixed $p$, there exists $s_0\in(0,1)$ such that equation \eqref{eq:0.1a} admits a ground state solution $u_s$ if and only if $s_0<s<1$. Our main results give a description of the asymptotic behavior of $u_s$ as $s\uparrow1$ and $s\downarrow s_0$: $u_s$ converges to a function as $s\uparrow1$, and it blows up as $s\downarrow s_0$. Particularly, we prove that $u_s$ concentrates at a minimum point of the function $V$ as $s\downarrow s_0$. The local uniqueness of $u_s$ is also given.