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

作者：Guo, Qing^*; Guo, Yuxia; Peng, Shuangjie

来源：Science China Mathematics, 2023, 66(5): 977-1002.

DOI：10.1007/s11425-021-1991-3

• non-degeneracy
• fractional Schrodinger equations
• Pohozaev identity
• Lyapunov-Schmidt reduction

We consider the following fractional Schrodinger equation: @@@ (-Delta)(s)u + V(y)u = u(p), u > 0 in R-N, (0, 1) @@@ where s is an element of (0, 1), 1 < p < N+2s/N-2s, 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 uk 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 (-Delta)(8) is in sharp contrast to the classical Schrodinger 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.

• 单位
  清华大学