Nonlocal scalar field equations: qualitative properties, asymptotic profiles and local uniqueness of solutions

TL;DR

This paper investigates nonlocal scalar field equations with a small parameter, establishing existence, symmetry, decay, asymptotic behavior, and local uniqueness of solutions across different Sobolev regimes.

Contribution

It provides new results on existence, symmetry, decay, asymptotics, and local uniqueness of solutions for nonlocal scalar field equations with a vanishing parameter.

Findings

Existence of ground state solutions for small epsilon.

Solutions are radially symmetric and decreasing.

Asymptotic profiles depend on the criticality of p.

Abstract

We study the nonlocal scalar field equation with a vanishing parameter \[ \left\{\begin{array}{lll} (-\Delta)^s u+\epsilon u &=|u|^{p-2}u -|u|^{q-2}u \quad\text{in}\quad\mathbb{R}^N \\ u >0, & u \in H^s(\mathbb{R}^N), \end{array} \right. \] where $s\in(0,1)$, $N>2s$, $q>p>2$ are fixed parameters and $\epsilon>0$ is a vanishing parameter. For $\epsilon>0$ small, we prove the existence of a ground state solution and show that any positive solution of above problem is a classical solution and radially symmetric and symmetric decreasing. We also obtain the decay rate of solution at infinity. Next, we study the asymptotic behavior of ground state solutions when $p$ is subcritical, supercritical or critical Sobolev exponent $2^*=\frac{2N}{N-2s}$. For $p<2^*$, the solution asymptotically coincides with unique positive ground state solution of $(-\Delta)^s u+u=u^p$. On the other hand, for $p=2^*$ the asymptotic behaviour of the solutions is given by the unique positive solution of the nonlocal critical Emden-Fowler type equation. For $p>2^*$, the solution asymptotically coincides with a ground-state solution of $(-\Delta)^s u=u^p-u^q$. Furthermore, using these asymptotic profile of solutions, we prove the \textit{local uniqueness} of solution in the case $p\leq 2^*$.