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

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} (-Δ)^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 $s\in (0,1)$, $N>2s$, $1<p<2_s^{\ast }-1:=\frac{N+2s}{N-2s}$, $0<a\in L^{\infty }({\mathbb{R}}^N)$ and $f\in H^{-s}({\mathbb{R}}^N)$ is a nonnegative functional i.e., $⟨f,u⟩\ge 0$ 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 $a(x)\ge 1$, $a(x)\to 1$ as $|x|\to \infty$ and $\Vert f{\Vert }_{H^{-s}({\mathbb{R}}^N)}$ 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 $a(x)\le 1$ with $a(x)\to 1$ as $|x|\to \infty$ and $\Vert f{\Vert }_{H^{-s}({\mathbb{R}}^N)}$ is small enough (but $f\not\equiv 0$).

35 pages