Existence and Multiplicity of positive solutions of certain nonlocal scalar field equations

TL;DR

This paper investigates the existence and multiplicity of positive solutions for a class of nonlocal scalar field equations involving fractional Laplacians, demonstrating conditions under which one, two, or three positive solutions exist.

Contribution

It provides new results on positive solutions for nonlocal scalar field equations with variable coefficients and external forces, including multiplicity results under specific asymptotic conditions.

Findings

Existence of a positive solution when the forcing term is zero.

At least two positive solutions when the coefficient approaches 1 and the forcing is small but nonzero.

Existence of three positive solutions under certain coefficient bounds and small forcing.

Abstract

We study existence and multiplicity of positive solutions of the following class of nonlocal scalar field equations: \begin{equation} \tag{$\mathcal{P}$} \left\{\begin{aligned} (-\Delta)^s u + u &= a(x) |u|^{p-1}u+f(x)\;\;\text{in}\;\mathbb{R}^{N}, u &\in H^{s}{(\mathbb{R}^{N})} \end{aligned} \right. \end{equation} where $s \in (0,1)$, $N>2s$, $1<p<2_s^*-1:=\frac{N+2s}{N-2s}$, $0< a\in L^\infty(\mathbb{R}^N)$ and $f\in H^{-s}(\mathbb{R}^N)$ is a nonnegative functional i.e., ${\langle}f,u{\rangle} \geq 0$ whenever $u$ is a nonnegative function in $H^s(\mathbb{R}^N)$. We prove existence of a positive solution when $f\equiv 0$ under certain asymptotic behavior on the function $a.$ Moreover, when $a(x)\geq 1$, $a(x)\to 1$ as $|x|\to\infty$ and $\|f\|_{H^{-s}(\mathbb{R}^N)}$ is small enough (but $f\not\equiv 0$), then we show that the above equation admits at least two positive solutions. Finally, we establish existence of three positive solutions to the above equation, under the condition that $a(x)\leq 1$ with $a(x)\to 1$ as $|x|\to\infty$ and $\|f\|_{H^{-s}(\mathbb{R}^N)}$ is small enough (but $f\not\equiv 0$).