AI 中文总结
本文研究分数阶与各向异性Gagliardo–Nirenberg不等式的适用条件，证明其多数尖锐条件，建立对应剖面分解，用于构造色散方程孤子解及证明极值子存在性。
AI 中文摘要
本文首先研究s,s₁,s₂∈R、1≤p₁,p₂,q≤∞、θ₁,θ₂≥0的取值范围，以确定分数阶Gagliardo–Nirenberg不等式‖D^s u‖_{L^q(R^d)}≲‖D^{s₁}u‖_{L^{p₁}(R^d)}^{θ₁}‖D^{s₂}u‖_{L^{p₂}(R^d)}^{θ₂}的有效性条件；其次考虑各向异性Gagliardo–Nirenberg不等式‖u‖_{L^q(R^d)}≲‖u‖_{L^{p₀}(R^d)}^{θ₀}∏_{j=1}^n‖D_{x_j}^{s_j}u‖_{L^{p_j}(R^d)}^{θ_j}，证明除“一种”情形外这些不等式的尖锐条件；最后，当p_j=2（0≤j≤n）时建立与上述不等式相关的剖面分解，利用这些结果建立极值子的存在性，并构造相关色散方程的孤子解。
英文摘要
In this article, we first investigate the necessary and sufficient conditions on the ranges of $s,s_1,s_2\in R$, $1\leq p_1,p_2,q\leq \infty$, $θ_1,θ_2\geq 0$ for the validity of the fractional Gagliardo--Nirenberg inequalities $$\|D^su\|_{L^q(R^d)}\lesssim \|D^{s_1}u\|_{L^{p_1}(R^d)}^{θ_1}\|D^{s_2}u\|_{L^{p_2}(R^d)}^{θ_2}.$$ Secondly, we consider the anisotropic Gagliardo--Nirenberg inequalities $$\|u\|_{L^q(R^d)}\lesssim \|u\|_{L^{p_0}(R^d)}^{θ_0} \prod_{j=1}^n\|D_{x_j}^{s_j}u\|_{L^{p_j}(R^d)}^{θ_j}.$$ We show the sharp conditions of these inequalities except ``one" case. Finally, we establish the profile decomposition associated with the above inequalities when $p_j = 2$, $0\leq j\leq n$. Using these results, we establish existence of extremizers and construct soliton solutions of relevant dispersive equations.
