We consider the following nonlinear Schroodinger equations -ε^2△u ＋ u = Q（x）｜u｜^p-2u in R^N, u ∈ H^1（R^N）,where ε is a small positive parameter, N ≥ 2, 2 〈 p 〈 ∞ for N = 2 and 2 〈 p 〈2N/N-2 for N ≥ 3...We consider the following nonlinear Schroodinger equations -ε^2△u ＋ u = Q（x）｜u｜^p-2u in R^N, u ∈ H^1（R^N）,where ε is a small positive parameter, N ≥ 2, 2 〈 p 〈 ∞ for N = 2 and 2 〈 p 〈2N/N-2 for N ≥ 3. We prove that this problem has sign-changing（nodal） semi-classical bound states with clustered spikes for sufficiently small ε under some additional conditions on Q（x）.Moreover, the number of this type of solutions will go to infinity as ε→ 0^＋.展开更多
In this article, periodic Riemann boundary value problem with period 2aπ along closed smooth contours is discussed, and then singular integral equation with kernel csc t-t0/a along closed smooth contours restricted i...In this article, periodic Riemann boundary value problem with period 2aπ along closed smooth contours is discussed, and then singular integral equation with kernel csc t-t0/a along closed smooth contours restricted in the strip 0 〈 Rez 〈 aπ is discussed. Finally, the solutions with singularities of order one for the above two problems are discussed.展开更多
We study the bound states to nonlinear Schrodinger equations with electro magnetic fields ihδψ/δt=（h/i -A（x））^2ψ＋V（x）ψ-K（x）｜ψ｜^p-1ψ=0,on R＋ ×R^N. Let G（x）=[V（x）p＋1/p-1-N/2][K（x）]-2/p-1 ...We study the bound states to nonlinear Schrodinger equations with electro magnetic fields ihδψ/δt=（h/i -A（x））^2ψ＋V（x）ψ-K（x）｜ψ｜^p-1ψ=0,on R＋ ×R^N. Let G（x）=[V（x）p＋1/p-1-N/2][K（x）]-2/p-1 and suppose that G（x） has k local minimum points. For h 〉 0 small, we find multi-bump bound states ~bh （x, t） ---- e-iE~/huh （X） with Uh concentrating at the local minimum points of G（x） simultaneously as h ~ O. The potentials V（x） and K（x） are allowed to be either compactly supported or unbounded at infinity.展开更多
In the present paper a class of extended close-to-convex functions Qk,λ(α,β,ρ) defined by making use of Ruscheweyh derivatives is introduced and studied. We provide integral representations, distortion theorem, ra...In the present paper a class of extended close-to-convex functions Qk,λ(α,β,ρ) defined by making use of Ruscheweyh derivatives is introduced and studied. We provide integral representations, distortion theorem, radius of close-to-convexity and Hadamard product properties for this class.展开更多
基金the National Science Foundation of China(10461008,10661009)the Key ResearchProject of Science and Technology of MOE(207124)+1 种基金the Scientific Research Foundation of Ningxia Higher Schoolthe Natural Science Foundation of Ningxia University(LG0407,LG0409)
基金supported by NSFC(11301204)supported by program for outstanding young Technology Innovative team in universities of Hubei Province(T2014212)
文摘We consider the following nonlinear Schroodinger equations -ε^2△u ＋ u = Q（x）｜u｜^p-2u in R^N, u ∈ H^1（R^N）,where ε is a small positive parameter, N ≥ 2, 2 〈 p 〈 ∞ for N = 2 and 2 〈 p 〈2N/N-2 for N ≥ 3. We prove that this problem has sign-changing（nodal） semi-classical bound states with clustered spikes for sufficiently small ε under some additional conditions on Q（x）.Moreover, the number of this type of solutions will go to infinity as ε→ 0^＋.
基金Supported by the National Natural Science Foundation of China (19971064)
文摘In this article, periodic Riemann boundary value problem with period 2aπ along closed smooth contours is discussed, and then singular integral equation with kernel csc t-t0/a along closed smooth contours restricted in the strip 0 〈 Rez 〈 aπ is discussed. Finally, the solutions with singularities of order one for the above two problems are discussed.
基金supported by National Natural Science Foundation of China(11201132)Scientific Research Foundation for Ph.D of Hubei University of Technology(BSQD12065)the Scientific Research Project of Education Department of Hubei Province(Q20151401)
文摘We study the bound states to nonlinear Schrodinger equations with electro magnetic fields ihδψ/δt=（h/i -A（x））^2ψ＋V（x）ψ-K（x）｜ψ｜^p-1ψ=0,on R＋ ×R^N. Let G（x）=[V（x）p＋1/p-1-N/2][K（x）]-2/p-1 and suppose that G（x） has k local minimum points. For h 〉 0 small, we find multi-bump bound states ~bh （x, t） ---- e-iE~/huh （X） with Uh concentrating at the local minimum points of G（x） simultaneously as h ~ O. The potentials V（x） and K（x） are allowed to be either compactly supported or unbounded at infinity.
基金the Natural Science Foundation of Inner Mongolia (No.2009MS0113)Higher School Research Foundation of Inner Mongolia (No.NJzy08510)
文摘In the present paper a class of extended close-to-convex functions Qk,λ(α,β,ρ) defined by making use of Ruscheweyh derivatives is introduced and studied. We provide integral representations, distortion theorem, radius of close-to-convexity and Hadamard product properties for this class.
