Abstract
We consider the following fractional Schrödinger equation:
where s ∈ (0, 1), \(1 < p < {{N + 2s} \over {N - 2s}}\), 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 (y1, y2)-plane with k and the radius large enough. Then we show that uk is non-degenerate in our special symmetric workspace, and glue it with an n-spike solution, whose centers lie in another circle in the (y3, y4)-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.
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Acknowledgements
Qing Guo was supported by National Natural Science Foundation of China (Grant No. 11771469). Yuxia Guo was supported by National Natural Science Foundation of China (Grant No. 11771235). Shuangjie Peng was supported by National Natural Science Foundation of China (Grant No. 11831009).
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Guo, Q., Guo, Y. & Peng, S. New existence of multi-spike solutions for the fractional Schrödinger equations. Sci. China Math. 66, 977–1002 (2023). https://doi.org/10.1007/s11425-021-1991-3
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DOI: https://doi.org/10.1007/s11425-021-1991-3