Construct new type solutions for the fractional Schrödinger equation

Abstract

This paper is devoted to studying the following nonlinear fractional problem: $$ \textstyle\begin{cases} (-\Delta )^{s}u+u=K( \vert x \vert )u^{p},\quad u>0, x\in {\mathbb{R}}^{N}, \\ u(x)\in H^{s}({\mathbb{R}}^{N}), \end{cases} $$ { ( − Δ ) s u + u = K ( | x | ) u p , u > 0 , x ∈ R N , u ( x ) ∈ H s ( R N ) , where $N\geq 3$ N ≥ 3 , $0< s<1$ 0 < s < 1 , $1< p<\frac{N+2s}{N-2s}$ 1 < p < N + 2 s N − 2 s , $K(|x|)$ K ( | x | ) is a positive radical function. We constructed infinitely many non-radial solutions of the new type which have a more complex concentration structure for (0.1).

Multi‐bumps solutions for the nonlinear Schrödinger equation under a slowing decaying potential

“…. Inspired by this paper, this result was applied to several equations[21][22][23].But it still remains open for the case m ≤ max{ can we find such type of solutions for (1.1) when m ≤ max {…”

“…with $$$ {V}_0,a,\kappa >0 $$$ and $$$ m>\max \left\{2,\frac{4}{p-1}\right\} $$$. Inspired by this paper, this result was applied to several equations [21–23]. But it still remains open for the case $$$ m\le \max \left\{2,\frac{4}{p-1}\right\} $$$, namely, can we find such type of solutions for () when $$$ m\le \max \left\{2,\frac{4}{p-1}\right\}?$…”