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} (-\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 $f\not\equiv 0$).