| Summary: | We are concerned with the qualitative and asymptotic analysis of solutions to the nonlocal equation
$$ (-\Delta)^su+V(|z|)u=Q(|z|)u^p\quad \text{in} \ \mathbb{R}^{N},$$
where $N\geq 3,\ 0<s<1$, and $1<p<\frac{2N}{N-2s}$. As $r\to\infty$, we assume that the potentials $V(r)$ and $Q(r)$ behave as
\begin{align*}
V(r)&=V_0+\frac{a_1}{r^\alpha}+O\left(\frac{1}{r^{\alpha+\theta_1}}\right)\\
Q(r)&=Q_0+\frac{a_2}{r^\beta}+O\left(\frac{1}{r^{\beta+\theta_2}}\right)
\end{align*}
where $a_1,\ a_2 \in{\mathbb{R}}$, $\alpha,\, \beta>\frac{N+2s}{N+2s+1}$, and $\theta_1,\ \theta_2>0,\ V_0,\ Q_0>0$. Under various hypotheses on $a_1,\, a_2,\, \alpha,\, \beta$, we establish the existence of infinitely many radial solutions. A key role in our arguments is played by the Lyapunov–Schmidt reduction method.
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