Construct new type solutions for the fractional Schrödinger equation

Abstract

Abstract This paper is devoted to studying the following nonlinear fractional problem: 0.1 { ( − Δ ) s u + u = K ( | x | ) u p , u > 0 , x ∈ R N , u ( x ) ∈ H s ( R N ) , $$ \textstyle\begin{cases} (-\Delta )^{s}u+u=K( \vert x \vert )u^{p},\quad u>0, x\in {\mathbb{R}}^{N}, \\ u(x)\in H^{s}({\mathbb{R}}^{N}), \end{cases} $$ where N ≥ 3 $N\geq 3$ , 0 < s < 1 $0< s<1$ , 1 < p < N + 2 s N − 2 s $1< p<\frac{N+2s}{N-2s}$ , 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).

Keywords

• Fractional Schrödinger equation
• Lyapunov–Schmidt reduction method
• New type solutions