In this paper, we investigate HUA’s Theorem for short intervals under GRH. Let E k(x)=#{{n≤x;2|n,k is odd, n≠p 1+p k 2}∪{n≤x;2|n,2|k,(p-1)|k, n1(modp),n≠p 1+p k 2}}. Assume GRH. For any k≥2, any A】0 ...In this paper, we investigate HUA’s Theorem for short intervals under GRH. Let E k(x)=#{{n≤x;2|n,k is odd, n≠p 1+p k 2}∪{n≤x;2|n,2|k,(p-1)|k, n1(modp),n≠p 1+p k 2}}. Assume GRH. For any k≥2, any A】0 and any 0【ε【14,E k(x+H)-E k(x)≤H(log x) -Aholds for x 12-14k+ε≤H≤x, here the implies constant depends at most on A and ε.展开更多
In this paper,a system of elliptic equations is investigated,which involves Hardy potential and multiple critical Sobolev exponents.By a global compactness argument of variational method and a fine analysis on the Pal...In this paper,a system of elliptic equations is investigated,which involves Hardy potential and multiple critical Sobolev exponents.By a global compactness argument of variational method and a fine analysis on the Palais-Smale sequences created from related approximation problems,the existence of infinitely many solutions to the system is established.展开更多
A monotonicity formula for stable solutions to a class of weighted semilinear elliptic equations with ＂negative exponent＂ is established. It is well known that such a monotonieity formula plays an essential role in ...A monotonicity formula for stable solutions to a class of weighted semilinear elliptic equations with ＂negative exponent＂ is established. It is well known that such a monotonieity formula plays an essential role in the study of finite Morse index solutions of equations with ＂positive exponent＂. Unlike the positive exponent case, we will see that both the monotonicity formula and the sub-harmonicity play crucial roles in classifying positive finite Morse index solutions to the equations with negative exponent and obtaining sharp results for their asymptotic behaviors.展开更多
In this paper,we mainly discuss a priori bounds of the following degenerate elliptic equation,a^ ij（x）δij u＋b^ i（x）δiu＋f（x,u）=0,in Ω belong to belong toR^n,（＊）where a^ijδiφδjφ=0 on δΩ,andφis the d...In this paper,we mainly discuss a priori bounds of the following degenerate elliptic equation,a^ ij（x）δij u＋b^ i（x）δiu＋f（x,u）=0,in Ω belong to belong toR^n,（＊）where a^ijδiφδjφ=0 on δΩ,andφis the defining function of δΩ.Imposing suitable conditions on the coefficients and f（x,u）,one can get the L^∞-estimates of（＊）via blow up method.展开更多
In this paper,a system of elliptic equations is investigated,which involves multiple critical Sobolev exponents and symmetric multi-polar potentials.By variational methods and analytic techniques,the relevant best con...In this paper,a system of elliptic equations is investigated,which involves multiple critical Sobolev exponents and symmetric multi-polar potentials.By variational methods and analytic techniques,the relevant best constants are studied and the existence of(Zk×SO(N.2))2-invariant solutions to the system is established.展开更多
文摘In this paper, we investigate HUA’s Theorem for short intervals under GRH. Let E k(x)=#{{n≤x;2|n,k is odd, n≠p 1+p k 2}∪{n≤x;2|n,2|k,(p-1)|k, n1(modp),n≠p 1+p k 2}}. Assume GRH. For any k≥2, any A】0 and any 0【ε【14,E k(x+H)-E k(x)≤H(log x) -Aholds for x 12-14k+ε≤H≤x, here the implies constant depends at most on A and ε.
基金supported by National Natural Science Foundation of China(Grant Nos.10771219 and 11071092)the PhD Specialized Grant of the Ministry of Education of China(Grant No.20110144110001)
文摘In this paper,a system of elliptic equations is investigated,which involves Hardy potential and multiple critical Sobolev exponents.By a global compactness argument of variational method and a fine analysis on the Palais-Smale sequences created from related approximation problems,the existence of infinitely many solutions to the system is established.
基金supported by National Natural Science Foundation of China(Grant Nos.11171092 and 11271133)Innovation Scientists and Technicians Troop Construction Projects of Henan Province(Grant No.114200510011)
文摘A monotonicity formula for stable solutions to a class of weighted semilinear elliptic equations with ＂negative exponent＂ is established. It is well known that such a monotonieity formula plays an essential role in the study of finite Morse index solutions of equations with ＂positive exponent＂. Unlike the positive exponent case, we will see that both the monotonicity formula and the sub-harmonicity play crucial roles in classifying positive finite Morse index solutions to the equations with negative exponent and obtaining sharp results for their asymptotic behaviors.
基金supported by National Natural Science Foundation of China(Grant No.11131005)
文摘In this paper,we mainly discuss a priori bounds of the following degenerate elliptic equation,a^ ij（x）δij u＋b^ i（x）δiu＋f（x,u）=0,in Ω belong to belong toR^n,（＊）where a^ijδiφδjφ=0 on δΩ,andφis the defining function of δΩ.Imposing suitable conditions on the coefficients and f（x,u）,one can get the L^∞-estimates of（＊）via blow up method.
基金supported by the Science Foundation of State Ethnic Affairs Commission of the People’s Republic of China(Grant No.12ZNZ004)
文摘In this paper,a system of elliptic equations is investigated,which involves multiple critical Sobolev exponents and symmetric multi-polar potentials.By variational methods and analytic techniques,the relevant best constants are studied and the existence of(Zk×SO(N.2))2-invariant solutions to the system is established.
