In this article, the following concave and convex nonlinearities elliptic equations involving critical growth is considered,{-△u=g（x）｜u｜2＊-2u＋λf（x）｜u｜q-2u,x∈Ω u=0,x∈δΩ where Ω RN（N ≥ 3） is an op...In this article, the following concave and convex nonlinearities elliptic equations involving critical growth is considered,{-△u=g（x）｜u｜2＊-2u＋λf（x）｜u｜q-2u,x∈Ω u=0,x∈δΩ where Ω RN（N ≥ 3） is an open bounded domain with smooth boundary, 1 〈 q 〈 2, λ 〉 0. 2＊= 2N/N-2 is the critical Sobolev exponent, f ∈L2＊/2N/N-2 is nonzero and nonnegative, and g E （Ω） is a positive function with k local maximum points. By the Nehari method and variational method, k ＋ 1 positive solutions are obtained. Our results complement and optimize the previous work by Lin [MR2870946, Nonlinear Anal. 75（2012） 2660-26711.展开更多
Ordered zinc oxide （ZnO） rod arrays with very high orientation were fabricated on Si substrates by using a solution method. The substrate surfaces were functionalized by Self-Assembly Monolayers （SAMs）. In the ver...Ordered zinc oxide （ZnO） rod arrays with very high orientation were fabricated on Si substrates by using a solution method. The substrate surfaces were functionalized by Self-Assembly Monolayers （SAMs）. In the very early growth stage, the oriented ZnO crystals had already grown, which appeared to be the main reason why ZnO nanorods showed very high orientation. The un-dense and un-uniform SAMs provided a surface that was heterogeneous to ZnO nucleation. Consequently, highly oriented ZnO rods were selectively grown on the ＂coin-like＂ SAM-uncovered regions. The route developed here can provide some helpful information to control the nucleation and orientation of ZnO in aqueous solution. Also, the site-selective growth mechanisms can indicate a clue to grow patterned highly oriented ZnO nanorod arrays by the organic template.展开更多
A new model of dendritic growth and solute distribution of Fe-0.04%C binary alloys was developed, which is based on the sharp interface model of dendritic growth. This innovative model improved the cellular automaton ...A new model of dendritic growth and solute distribution of Fe-0.04%C binary alloys was developed, which is based on the sharp interface model of dendritic growth. This innovative model improved the cellular automaton method, combined with the finite difference method, considered concentration field, temperature field and the shape of molten pool. This model simulated the growth morphologies of single equiaxial crystal, the relationship between tip growth velocity and time, multi-equiaxed crystals’ growth morphologies and solute distribution, the growth of columnar crystals, columnar-to-equiaxed transition after coupling temperature field, and compared with experimental results. The results indicate that crystallographic orientation has certain influence on dendritic morphologies, that the tip growth velocity tends to be stable with the extension of time in the end, that the shape of molten pool influences the growth morphologies of columnar crystals and that the solute mainly concentrates in dendritic roots and among grain boundaries. The simulated results are in accord with experimental results.展开更多
The steady-state dendritic growth from the undercooled binary alloy melt with the far field flow is considered. By neglecting the interface energy, interface kinetics and buoyancy effects in the system, we obtaine the...The steady-state dendritic growth from the undercooled binary alloy melt with the far field flow is considered. By neglecting the interface energy, interface kinetics and buoyancy effects in the system, we obtaine the steady-state solution for the case of the large Schmidt number, in terms of the multiple variable expansion method. The changes of the temperature and concentration fields, the morphology of the interface, the normalization parameter and the Peclet number of the system induced by uniform external flow are derived. The results show that, compared with the system of dendritic growth from undercooled pure melt, the convective flow in the system of growth from undercooled binary alloy has stronger effects on the morphology of the interface. Nevertheless, the shape of the interface still remains nearly a paraboloid.展开更多
基金Supported by National Natural Science Foundation of China(11471267)the Doctoral Scientific Research Funds of China West Normal University(15D006 and 16E014)+1 种基金Meritocracy Research Funds of China West Normal University(17YC383)Natural Science Foundation of Education of Guizhou Province(KY[2016]046)
文摘In this article, the following concave and convex nonlinearities elliptic equations involving critical growth is considered,{-△u=g（x）｜u｜2＊-2u＋λf（x）｜u｜q-2u,x∈Ω u=0,x∈δΩ where Ω RN（N ≥ 3） is an open bounded domain with smooth boundary, 1 〈 q 〈 2, λ 〉 0. 2＊= 2N/N-2 is the critical Sobolev exponent, f ∈L2＊/2N/N-2 is nonzero and nonnegative, and g E （Ω） is a positive function with k local maximum points. By the Nehari method and variational method, k ＋ 1 positive solutions are obtained. Our results complement and optimize the previous work by Lin [MR2870946, Nonlinear Anal. 75（2012） 2660-26711.
基金the National Natural Science Foundation of China(No.50702029)Shandong Provincial Education Department(No.J05D08)Natural Science Foundation of Qingdao(No.05-1-JC-89)
文摘Ordered zinc oxide （ZnO） rod arrays with very high orientation were fabricated on Si substrates by using a solution method. The substrate surfaces were functionalized by Self-Assembly Monolayers （SAMs）. In the very early growth stage, the oriented ZnO crystals had already grown, which appeared to be the main reason why ZnO nanorods showed very high orientation. The un-dense and un-uniform SAMs provided a surface that was heterogeneous to ZnO nucleation. Consequently, highly oriented ZnO rods were selectively grown on the ＂coin-like＂ SAM-uncovered regions. The route developed here can provide some helpful information to control the nucleation and orientation of ZnO in aqueous solution. Also, the site-selective growth mechanisms can indicate a clue to grow patterned highly oriented ZnO nanorod arrays by the organic template.
文摘A new model of dendritic growth and solute distribution of Fe-0.04%C binary alloys was developed, which is based on the sharp interface model of dendritic growth. This innovative model improved the cellular automaton method, combined with the finite difference method, considered concentration field, temperature field and the shape of molten pool. This model simulated the growth morphologies of single equiaxial crystal, the relationship between tip growth velocity and time, multi-equiaxed crystals’ growth morphologies and solute distribution, the growth of columnar crystals, columnar-to-equiaxed transition after coupling temperature field, and compared with experimental results. The results indicate that crystallographic orientation has certain influence on dendritic morphologies, that the tip growth velocity tends to be stable with the extension of time in the end, that the shape of molten pool influences the growth morphologies of columnar crystals and that the solute mainly concentrates in dendritic roots and among grain boundaries. The simulated results are in accord with experimental results.
基金Supported by the National Basic Research Program of China (Grant No. 2006CB605205)the National Natural Science Foundation of China (Grant No. 10672019)
文摘The steady-state dendritic growth from the undercooled binary alloy melt with the far field flow is considered. By neglecting the interface energy, interface kinetics and buoyancy effects in the system, we obtaine the steady-state solution for the case of the large Schmidt number, in terms of the multiple variable expansion method. The changes of the temperature and concentration fields, the morphology of the interface, the normalization parameter and the Peclet number of the system induced by uniform external flow are derived. The results show that, compared with the system of dendritic growth from undercooled pure melt, the convective flow in the system of growth from undercooled binary alloy has stronger effects on the morphology of the interface. Nevertheless, the shape of the interface still remains nearly a paraboloid.
