Infinitely Many Solutions for the Fractional Nonlinear Schrödinger Equations of a New Type (Journal of Partial Differential Equations, 2022)
This paper, we study the multiplicity of solutions for the fractional Schrödinger equation\begin{equation*}(-Δ)^su+V(x)u=u^p,\ \ u>0,\ \ x\in\mathbb{R}^N,\ \ u\in H^s(\mathbb{R}^N),\end{equation*}with $s\in(0,1),\ N≥3,\ p\in(1,\frac{2N}{N-2s}-1)$ and $\lim_{|y|→+∞}V(y)>0$. By assuming suitable decay property of the radial potential $V(y)=V(|y|)$, we construct another type of solutions concentrating at infinite vertices of two similar equilateral polygonal with infinitely large length of sides. Hence, besides the length of each polygonal, we must consider one more parameter, that is the height of the podetium, simultaneously. Another difficulty lies in the non-local property of the operator $(-Δ)^s$ and the algebraic decay involving the approximation solutions make the estimates become more subtle.
Keywords: Fractional Schrödinger equations; infinitely many solutions; reduction method.