New existence of multi-spike solutions for the fractional Schrödinger equations

• Published in: Science China. Mathematics 2023-05, Vol.66 (5), p.977-1002
• Main Authors: Guo, Qing, Guo, Yuxia, Peng, Shuangjie
• Format: Article
• Language: English
• Subjects: Apexes; Applications of Mathematics; Estimates; Identities; Infinity; Mathematical analysis; Mathematics; Mathematics and Statistics; Operators (mathematics); Schrodinger equation
• ISSN: 1674-7283; 1869-1862
• DOI: 10.1007/s11425-021-1991-3

We consider the following fractional Schrödinger equation: (0.1) ( − Δ ) s u + V ( y ) u = u p , u > 0 in ℝ N , where s ∈ (0, 1), 1 < p < N + 2 s N − 2 s , and V ( y ) is a positive potential function and satisfies some expansion condition at infinity. Under the Lyapunov-Schmidt reduction framework, we construct two kinds of multi-spike solutions for (0.1). The first k-spike solution u k is concentrated at the vertices of the regular k -polygon in the ( y 1 , y 2 )-plane with k and the radius large enough. Then we show that u k is non-degenerate in our special symmetric workspace, and glue it with an n -spike solution, whose centers lie in another circle in the ( y 3 , y 4 )-plane, to construct infinitely many multi-spike solutions of new type. The nonlocal property of (−Δ) s is in sharp contrast to the classical Schrödinger equations. A striking difference is that although the nonlinear exponent in (0.1) is Sobolev-subcritical, the algebraic (not exponential) decay at infinity of the ground states makes the estimates more subtle and difficult to control. Moreover, due to the non-locality of the fractional operator, we cannot establish the local Pohozaev identities for the solution u directly, but we address its corresponding harmonic extension at the same time. Finally, to construct new solutions we need pointwise estimates of new approximation solutions. To this end, we introduce a special weighted norm, and give the proof in quite a different way.