Existence and decays of solutions for fractional Schrödinger equations with decaying potentials

2023/02/12 by Yinbin Deng, Shuangjie Peng, Deng, Yinbin +3
Mathematics · #Advanced Mathematical Physics Problems #Analysis of PDEs (math.AP) #FOS: Mathematics #Nonlinear Partial Differential Equations #Spectral Theory in Mathematical Physics

paper · pdf · doi:10.48550/arxiv.2302.05848

openalex publication_date 2023/02/12 · openalex created_date 2023/02/16 · openalex updated_date 2026/07/28

Abstract

We revisit the following fractional Schrödinger equation ε^2s(-Δ)^su +Vu=u^p-1, ugt;0, in \R^N, where ε>0 is a small parameter, (-Δ)^s denotes the fractional Laplacian, s∈(0,1), p∈ (2, 2_s^*), 2_s^*=\frac 2NN-2s, N>2s, V∈ C(\R^N, [0, +∞)) is a potential. Under various decay assumptions on V, we introduce a uniform penalization argument combined with a comparison principle and iteration process to detect an explicit threshold value p_*, such that the above problem admits positive concentration solutions if p∈ (p_*, 2_s^*), while it has no positive weak solutions for p∈ (2, p_*) if p_*>2, where the threshold p_*∈ [2, 2^*_s) can be characterized explicitly by p_*=\2+\frac 2sN-2s \text if lim_{|x| → ∞} (1+|x|^2s)V(x)=0,\vspace1mm 2+\frac ωN+2s-ω \text if 0 · lt;inf (1+|x|^ω)V(x)≤ sup (1+|x|^ω)V(x) · lt; ∞ \text for some ω∈ [0, 2s],\vspace1mm 2 \text if inf V(x)log(e+|x|²) · gt;0.. Moreover, corresponding to the various decay assumptions of V(x), we obtain the decay properties of the solutions at infinity.

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