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

Abstract

We study the nonlocal scalar field equation with a vanishing parameter \[ \left\{\begin{array}{lll} (-Δ)^s u+εu &=|u|^{p-2}u -|u|^{q-2}u \quad\text{in}\quad\mathbb{R}^N \\ u >0, & u \in H^s(\mathbb{R}^N), \end{array} \right. \] 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 $2^{\ast }=\frac{2N}{N-2s}$. For $p<2^{\ast }$, the solution asymptotically coincides with unique positive ground state solution of $(-\Delta {)}^{s}u+u=u^p$. On the other hand, for $p=2^{\ast }$ the asymptotic behaviour of the solutions is given by the unique positive solution of the nonlocal critical Emden-Fowler type equation. For $p>2^{\ast }$, 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 $p\le 2^{\ast }$.

49 pages

References in corpus (2)

• Existence and symmetry results for a Schrödinger type problem involving the fractional Laplacian
• Profile of solutions for nonlocal equations with critical and supercritical nonlinearities