id	author	title	date	pages	extension	mime	words	sentence	flesch	summary	cache	txt
ejde-1361	Zhang, Lei; Liu, Lintao; Chen, Haibo	Minimizers for fractional Schrodinger equations with inhomogeneous perturbation	2025	20	.pdf	application/pdf	9508	503	84	M − C5A p+1 M α̃2s M α̃ (t+1)[N−(N+2s)(p+1)] M + 4s N(p− 1) Ap+1 M α̃2s M a∗ (M a∗ ) M a∗ ∫ RN |ξ|s|Q̌(ξ)|2dξ ≤ C 1 + C̃α̃ −(t+1)(N+4s) M a∗ α̃ s(1−t) M ∫ RN (1 + |ξ|2s)|Q̌(ξ)|2dξ ≤ C(1 + C̃α̃ −(t+1)(N+4s) M )α̃ s(1−t) M (3.24) EJDE-2025/59 MINIMIZERS FOR FRACTIONAL SCHRÖDINGER EQUATIONS 11 as M → ∞, where Q̌ denote the Fourier transform of Q. By the Hölder inequality, (1.7), (3.18) and (3.20), we obtain that |T4| ≤ C A2 M α̃ 2s M a∗ (∫ RN Q2(x)|(−∆)s/2φ(α̃−t−1 M x)|2dx )1/2 × (∫ RN φ2(α̃−t−1 M x)|(−∆)s/2Q|2dx )1/2 ≤ C A2 M α̃ 2s M a∗ ( C3α̃ −2s(t+1) M ∫ RN Q2(x)dx )1/2(∫ RN |(−∆)s/2Q|2dx )1/2 ≤ C(1 + C̃α̃ −(t+1)(N+4s) M )α̃ s(1−t) M as M → ∞. (3.25) From the Hölder inequality, (1.7), (3.18), (3.20) and (3.21), it follows that |T5| ≤ C A2 M α̃ 2s M a∗ (∫ RN Q2(x)|(−∆)s/2φ(α̃−t−1 M x)|2dx )1/2 × (∫ RN B2(φ(α̃−t−1 M x), Q(x))dx )1/2 ≤ C(1 + C̃α̃ −(t+1)(N+4s) M )α̃2s M α̃ −s(t+1) M α̃ − s(t+1) 2 M ≤ C(1 + C̃α̃ −(t+1)(N+4s) M )α̃ s( 1 2− 3 2 t) M asM → ∞. (3.26) Similarly, |T6| ≤ C A2 M α̃ 2s M a∗ (∫ RN φ2(α̃−t−1 M x)|(−∆)s/2Q|2dx )1/2(∫ RN B2(φ(α̃−t−1 M x), Q(x))dx )1/2 ≤ C(1 + C̃α̃ −(t+1)(N+4s) M )α̃2s M α̃ − s(t+1) 2 M ≤ C(1 + C̃α̃ −(t+1)(N+4s) M )α̃ s( 3 2− 1 2 t) M asM → ∞. (3.27)	cache/ejde-1361.pdf	txt/ejde-1361.txt