量子场论中分数阶 P-Q Laplace 算子的主特征值与正解
偏微分方程分析
2020-06-08 v1
摘要
本文研究由具有不定权函数的非齐次分数阶 p&q Laplacian 算子所驱动的特征值问题正解的存在与不存在性：( − Δ p ) α u + ( − Δ q ) β u = λ [ a ( x ) ∣ u ∣ p − 2 u + b ( x ) ∣ u ∣ q − 2 u ] in \O , \left(-\Delta_p\right)^{\alpha}u + \left(-\Delta_q\right)^{\beta}u \,= \lambda\left[a(x) \left|u\right|^{p-2}u + b(x) \left|u\right|^{q-2}u \right]\quad\quad\textrm{in $\O$}, ( − Δ p ) α u + ( − Δ q ) β u = λ [ a ( x ) ∣ u ∣ p − 2 u + b ( x ) ∣ u ∣ q − 2 u ] in \O , \O \O \O R N \R^N R N \O = R N \O=\R^N \O = R N b ≡ 0 b\equiv0 b ≡ 0 1 < q < p < q β ∗ = N q N − q β 1<q<p<q^*_\beta=\frac{Nq}{N-q\beta} 1 < q < p < q β ∗ = N − q β N q 0 ≤ a ∈ L ( q β ∗ s ) ′ ( R N ) ⋂ L ∞ ( R N ) 0\leq a\in L^{\left(\frac{q_{\beta}^*}{s}\right)'}\left(\R^N\right)\bigcap L^{\infty}\left(\R^N\right) 0 ≤ a ∈ L ( s q β ∗ ) ′ ( R N ) ⋂ L ∞ ( R N ) s s s p − t p α ∗ + p ( 1 − t ) s = 1 \dfrac{p-t}{p_{\alpha}^*}+ \dfrac{p\left(1-t\right)}{s} =1 p α ∗ p − t + s p ( 1 − t ) = 1 t ∈ ( 0 , p − q p ) t\in \left(0, \sqrt{\dfrac{p-q}{p}}\right) t ∈ ( 0 , p p − q ) R N \R^N R N W α , p ( R N ) ↪ W β , q ( R N ) W^{\alpha, p}\left(\R^N\right) \hookrightarrow W^{\beta, q}\left(\R^N\right) W α , p ( R N ) ↪ W β , q ( R N )
引用
@article{arxiv.2006.03233,
title = {Principal eigenvalue and positive solutions for Fractional $P-Q$ Laplace operator in quantum field theory},
author = {Thanh-Hieu Nguyen and Hoang-Hung Vo},
journal= {arXiv preprint arXiv:2006.03233},
year = {2020}
}