We review the concept of congruence of null geodesics, the Raychaudhuri equation for the expansion, its harmonic oscillator version and associated “quantum” propagator, the role of the equation in the derivation of ...We review the concept of congruence of null geodesics, the Raychaudhuri equation for the expansion, its harmonic oscillator version and associated “quantum” propagator, the role of the equation in the derivation of the Penrose singularity theorem, the definition of trapped surfaces, and the derivation of the theorem itself.展开更多
We establish the existence of positive periodic solutions of the second-order singular coupled systems{x′′＋ p_1（t）x′＋ q_1（t）x = f_1（t, y（t）） ＋ c_1（t）,y′′＋ p_2（t）y′＋ q_2（t）y = f_2（t, x（t）） ...We establish the existence of positive periodic solutions of the second-order singular coupled systems{x′′＋ p_1（t）x′＋ q_1（t）x = f_1（t, y（t）） ＋ c_1（t）,y′′＋ p_2（t）y′＋ q_2（t）y = f_2（t, x（t）） ＋ c_2（t）,where pi, qi, ci ∈ C（R/T Z; R）, i = 1, 2; f_1, f_2 ∈ C（R/T Z ×（0, ∞）, R） and may be singular near the zero. The proof relies on Schauder＇s fixed point theorem and anti-maximum principle.Our main results generalize and improve those available in the literature.展开更多
In this paper N-dimensional singular, p-Laplace equations of the following form △pu：=N↑∑↑i=1Di（｜Du｜^p-2Diu）=f（｜x｜,u,｜Du｜u^-β,x∈R^N（N≥3） are considered, where p≥N,β〉0,and f：[0,∞）×[0,∞）&#...In this paper N-dimensional singular, p-Laplace equations of the following form △pu：=N↑∑↑i=1Di（｜Du｜^p-2Diu）=f（｜x｜,u,｜Du｜u^-β,x∈R^N（N≥3） are considered, where p≥N,β〉0,and f：[0,∞）×[0,∞）×[0,∞）is a continuous tunctlon. Some sufficient conditions are obtained for the existence of infinitely many radially positive entire solutions of the equation which are asymptotic to positive constant multiples of ｜x｜^（p-N）/（p-1） for p〉N or log｜x｜ for N-p as ｜x｜→∞.展开更多
文摘We review the concept of congruence of null geodesics, the Raychaudhuri equation for the expansion, its harmonic oscillator version and associated “quantum” propagator, the role of the equation in the derivation of the Penrose singularity theorem, the definition of trapped surfaces, and the derivation of the theorem itself.
基金Supported by the Scientific Research Funds for the Ningxia Universities(Grant No.NGY2015141)
文摘We establish the existence of positive periodic solutions of the second-order singular coupled systems{x′′＋ p_1（t）x′＋ q_1（t）x = f_1（t, y（t）） ＋ c_1（t）,y′′＋ p_2（t）y′＋ q_2（t）y = f_2（t, x（t）） ＋ c_2（t）,where pi, qi, ci ∈ C（R/T Z; R）, i = 1, 2; f_1, f_2 ∈ C（R/T Z ×（0, ∞）, R） and may be singular near the zero. The proof relies on Schauder＇s fixed point theorem and anti-maximum principle.Our main results generalize and improve those available in the literature.
基金The work is supported by the National Natural Science Foundation of China (10271056)the Natural Science Foundation of Fujian Province (F00018).
文摘In this paper N-dimensional singular, p-Laplace equations of the following form △pu：=N↑∑↑i=1Di（｜Du｜^p-2Diu）=f（｜x｜,u,｜Du｜u^-β,x∈R^N（N≥3） are considered, where p≥N,β〉0,and f：[0,∞）×[0,∞）×[0,∞）is a continuous tunctlon. Some sufficient conditions are obtained for the existence of infinitely many radially positive entire solutions of the equation which are asymptotic to positive constant multiples of ｜x｜^（p-N）/（p-1） for p〉N or log｜x｜ for N-p as ｜x｜→∞.
