This paper studies the existence of positive solutions of the Dirichlet problem for the nonlinear equation involving p-Laplacian operator: -△pu = λf(u) on a bounded smooth domain Ω in Rn. The authors extend part of...This paper studies the existence of positive solutions of the Dirichlet problem for the nonlinear equation involving p-Laplacian operator: -△pu = λf(u) on a bounded smooth domain Ω in Rn. The authors extend part of the Crandall-Rabinowitz bifurcation theory to this problem. Typical examples are checked in detail and multiplicity of the solutions are illustrated. Then the stability for the associated parabolic equation is considered and a Fujita-type result is presented.展开更多
In this paper,a super version of the Hopf quiver theory is developed.The notion of Hopf superquivers is introduced.It is shown that only the path supercoalgebras of Hopf superquivers admit graded Hopf superalgebra str...In this paper,a super version of the Hopf quiver theory is developed.The notion of Hopf superquivers is introduced.It is shown that only the path supercoalgebras of Hopf superquivers admit graded Hopf superalgebra structures.A complete classification of such graded Hopf superalgebras is given.A superquiver setting for general pointed Hopf superalgebras is also built up.In particular,a super version of the Gabriel type theorem and the Cartier-Gabriel decomposition theorem is given.展开更多
The famous three-body problem can be traced back to Isaac Newton in the 1680 s. In the 300 years since this "three-body problem"was first recognized, only three families of periodic solutions had been found,...The famous three-body problem can be traced back to Isaac Newton in the 1680 s. In the 300 years since this "three-body problem"was first recognized, only three families of periodic solutions had been found, until 2013 when ˇSuvakov and Dmitraˇsinovi′c [Phys.Rev. Lett. 110, 114301(2013)] made a breakthrough to numerically find 13 new distinct periodic orbits, which belong to 11 new families of Newtonian planar three-body problem with equal mass and zero angular momentum. In this paper, we numerically obtain 695 families of Newtonian periodic planar collisionless orbits of three-body system with equal mass and zero angular momentum in case of initial conditions with isosceles collinear configuration, including the well-known figure-eight family found by Moore in 1993, the 11 families found by ˇSuvakov and Dmitraˇsinovi′c in 2013, and more than 600 new families that have never been reported, to the best of our knowledge. With the definition of the average period T = T=Lf, where Lf is the length of the so-called "free group element", these 695 families suggest that there should exist the quasi Kepler's third law T* ≈ 2:433 ± 0:075 for the considered case, where T*= T|E|^(3/2) is the scale-invariant average period and E is its total kinetic and potential energy,respectively. The movies of these 695 periodic orbits in the real space and the corresponding close curves on the "shape sphere"can be found via the website: http://numericaltank.sjtu.edu.cn/three-body/three-body.htm.展开更多
We are concerned with the existence of least energy solutions of nonlinear Schrodinger equations involving the fractional Laplacian（-△）%s u（x）＋λV（x）u（x）=u（x）^（p-1）,u（x）〉=0,x∈R^N,for sufficiently lar...We are concerned with the existence of least energy solutions of nonlinear Schrodinger equations involving the fractional Laplacian（-△）%s u（x）＋λV（x）u（x）=u（x）^（p-1）,u（x）〉=0,x∈R^N,for sufficiently large λ,2〈p〈N-2s^-2N for N≥2. V（x） is a real continuous function on RN. Using variational methods we prove the existence of least energy solution uλ（x） which localizes near the potential well int V-1 （0） for A large. Moreover, if the zero sets int V-1 （0） of V（x） include more than one isolated component, then ux（x） will be trapped around all the isolated components. However, in Laplacian case s = 1, when the parameter A is large, the corresponding least energy solution will be trapped around only one isolated component and become arbitrarily small in other components of int V^-1（0）. This is the essential difference with the Laplacian problems since the operator （-△）s is nonlocal.展开更多
基金Project supported by the 973 Project of the Ministry of Science and Technology of China (No.G1999075107) a Scientific Grant of Tsinghua University.
文摘This paper studies the existence of positive solutions of the Dirichlet problem for the nonlinear equation involving p-Laplacian operator: -△pu = λf(u) on a bounded smooth domain Ω in Rn. The authors extend part of the Crandall-Rabinowitz bifurcation theory to this problem. Typical examples are checked in detail and multiplicity of the solutions are illustrated. Then the stability for the associated parabolic equation is considered and a Fujita-type result is presented.
基金Project supported by the National Natural Science Foundation of China (No. 10601052)the Shandong Provincial Natural Science Foundation of China (Nos. YZ2008A05, 2009ZRA01128)
文摘In this paper,a super version of the Hopf quiver theory is developed.The notion of Hopf superquivers is introduced.It is shown that only the path supercoalgebras of Hopf superquivers admit graded Hopf superalgebra structures.A complete classification of such graded Hopf superalgebras is given.A superquiver setting for general pointed Hopf superalgebras is also built up.In particular,a super version of the Gabriel type theorem and the Cartier-Gabriel decomposition theorem is given.
基金supported by the National Natural Science Foundation of China(Grant No.11432009)
文摘The famous three-body problem can be traced back to Isaac Newton in the 1680 s. In the 300 years since this "three-body problem"was first recognized, only three families of periodic solutions had been found, until 2013 when ˇSuvakov and Dmitraˇsinovi′c [Phys.Rev. Lett. 110, 114301(2013)] made a breakthrough to numerically find 13 new distinct periodic orbits, which belong to 11 new families of Newtonian planar three-body problem with equal mass and zero angular momentum. In this paper, we numerically obtain 695 families of Newtonian periodic planar collisionless orbits of three-body system with equal mass and zero angular momentum in case of initial conditions with isosceles collinear configuration, including the well-known figure-eight family found by Moore in 1993, the 11 families found by ˇSuvakov and Dmitraˇsinovi′c in 2013, and more than 600 new families that have never been reported, to the best of our knowledge. With the definition of the average period T = T=Lf, where Lf is the length of the so-called "free group element", these 695 families suggest that there should exist the quasi Kepler's third law T* ≈ 2:433 ± 0:075 for the considered case, where T*= T|E|^(3/2) is the scale-invariant average period and E is its total kinetic and potential energy,respectively. The movies of these 695 periodic orbits in the real space and the corresponding close curves on the "shape sphere"can be found via the website: http://numericaltank.sjtu.edu.cn/three-body/three-body.htm.
基金supported by National Natural Science Foundation of China (Grant No. 11171028)
文摘We are concerned with the existence of least energy solutions of nonlinear Schrodinger equations involving the fractional Laplacian（-△）%s u（x）＋λV（x）u（x）=u（x）^（p-1）,u（x）〉=0,x∈R^N,for sufficiently large λ,2〈p〈N-2s^-2N for N≥2. V（x） is a real continuous function on RN. Using variational methods we prove the existence of least energy solution uλ（x） which localizes near the potential well int V-1 （0） for A large. Moreover, if the zero sets int V-1 （0） of V（x） include more than one isolated component, then ux（x） will be trapped around all the isolated components. However, in Laplacian case s = 1, when the parameter A is large, the corresponding least energy solution will be trapped around only one isolated component and become arbitrarily small in other components of int V^-1（0）. This is the essential difference with the Laplacian problems since the operator （-△）s is nonlocal.
