POSITIVE SOLUTIONS WITH HIGH ENERGY FOR FRACTIONAL SCHR¨ODINGER EQUATIONS

In this paper, we study the Schr?dinger equations (-?)^su + V(x)u = a(x)|u|^p-2u + b(x)|u|^q-2u, x∈R^N,where 0 < s < 1, 2 < q < p < 2_s^*, 2_s^* is the fractional Sobolev critical exponent. Under suitable assumptions on V, a and b for which there may be no ground state solution, the existence of positive solutions are obtained via variational methods.     

一类奇异三阶三点边值问题的正解

In this paper,we study the existence of positive solutions for the nonlinear singular third-order three-point boundary value problemu （t） = λa（t）f（t,u（t））,u（0） = u （1） = u （η） = 0,where λ is a positive parameter and 0 ≤ η 1 2 .By using the classical Krasnosel’skii’s fixed point theorem in cone,we obtain various new results on the existence of positive solution,and the solution is strictly increasing.Finally we give an example.     

BLOWING UP AND MULTIPLICITY OF SOLUTIONS FOR A FOURTH-ORDER EQUATION WITH CRITICAL NONLINEARITY

In this paper,we consider the following nonlinear elliptic problem:△~2u=|u|~(8/(n-4))u+μ|u|~(q-1)u,in Ω,△u = u = 0 on δΩ,where Ω is a bounded and smooth domain in R~n,n ∈ {5,6,7},μ is a parameter and q ∈]4/(n- 4),(12- n)/(n- 4)[.We study the solutions which concentrate around two points of Ω.We prove that the concentration speeds are the same order and the distances of the concentration points from each other and from the boundary are bounded.For Ω =(Ω_α)α a smooth ringshaped open set,we establish the existence of positive solutions which concentrate at two points of Ω.Finally,we show that for μ 0,large enough,the problem has at least many positive solutions as the LjusternikSchnirelman category of Ω.     

The defocusing energy-supercritical Hartree equation

In this paper,we study the global well-posedness and scattering problem for the energysupercritical Hartree equation iut+Δu.(|x|.γ.|u|2)u=0 with γ4 in dimension d γ.We prove that if the solution u is apriorily bounded in the critical Sobolev space,that is,u ∈Lt∞(I;Hxsc(Rd)) with sc:= γ/2.11,then u is global and scatters.The impetus to consider this problem stems from a series of recent works for the energy-supercritical nonlinear wave equation(NLW) and nonlinear Schrdinger equation(NLS).We utilize the strategy derived from concentration compactness ideas to show that the proof of the global well-posedness and scattering is reduced to disprove the existence of three scenarios:finite time blowup;soliton-like solution and low to high frequency cascade.Making use of the No-waste Duhamel formula,we deduce that the energy of the finite time blow-up solution is zero and so get a contradiction.Finally,we adopt the double Duhamel trick,the interaction Morawetz estimate and interpolation to kill the last two scenarios.     

SIGN-CHANGING SOLUTIONS FOR SCHRDINGER EQUATIONS WITH VANISHING AND SIGN-CHANGING POTENTIALS

In this article, we study the existence of sign-changing solutions for the following Schrdinger equation-△u + λV(x)u = K(x)|u|p-2u x ∈ RN, u → 0 as |x| → +∞,where N ≥ 3, λ 0 is a parameter, 2 p 2N N-2, and the potentials V(x) and K(x) satisfy some suitable conditions. By using the method based on invariant sets of the descending flow,we obtain the existence of a positive ground state solution and a ground state sign-changing solution of the above equation for small λ, which is a complement of the results obtained by Wang and Zhou in [J. Math. Phys. 52, 113704, 2011].     

ON HERMITIAN POSITIVE DEFINITE SOLUTION OF NONLINEAR MATRIX EQUATION X＋A^＊X^-2A=Q

Based on the fixed-point theory, we study the existence and the uniqueness of the maximal Hermitian positive definite solution of the nonlinear matrix equation X＋A^＊X^-2A=Q, where Q is a square Hermitian positive definite matrix and A＊ is the conjugate transpose of the matrix A. We also demonstrate some essential properties and analyze the sensitivity of this solution. In addition, we derive computable error bounds about the approximations to the maximal Hermitian positive definite solution of the nonlinear matrix equation X＋A^＊X^-2A=Q. At last, we further generalize these results to the nonlinear matrix equation X＋A^＊X^-nA=Q, where n≥2 is a given positive integer.     

POSITIVE PERIODIC SOLUTION TO A CLASS OF NONLINEAR PERIODIC DIFFERENTIAL EQUATION WITH IMPULSES AND DELAY

In this paper, by using a fixed point theorem of Krasnoselskii, we study the positive periodic solution for a class of nonlinear periodic differential equation with impulses and delay. Firstly, definition of periodic solution and some lemmas are stated. Then some results of the existence of positive periodic solution about the equation are obtained.     

Unconditional Uniqueness of Solution for H~(s_c) Critical 4th Order NLS in High Dimensions

In this paper, we study the unconditional uniqueness of solution for the Cauchy problem of H~(s_c)(0 ≤ s_c 2) critical nonlinear fourth-order Schrdinger equations i?_t u + Δ~2 u-∈u = λ|u|~αu. By employing paraproduct estimates and Strichartz estimates, we prove that unconditional uniqueness of solution holds in C_t(I;H~(s_c)(R~d)) for d ≥ 11 and min{1-,8/(d-4)} ≥α (-(d-4)+((d-4)~2+64)~(1/2))/4.     

Local Hamilton type Gradient Estimates and Harnack Inequalities for Nonlinear Parabolic Equations on Riemannian Manifolds

In this paper,we prove a local Hamilton type gradient estimate for positive solution of the nonlinear parabolic equation u_t(x,t)=Δu(x,t)+au(x,t) ln u(x,t)+bu^α(x,t),on M×(-∞,∞) with α∈R,where a and b are constants.As application,the Harnack inequalities are derived.     

L~p-decay rates to nonlinear diffusion waves for p-system with nonlinear damping

In this paper, we study the Lp (2≤p≤ ∞) convergence rates of the solutions to the Cauchy problem of the so-called p-system with nonlinear damping. Precisely, we show that the corresponding Cauchy problem admits a unique global solution (v (x,t), u(x, t)) and such a solution tends time-asymptotically to the corresponding nonlinear diffusion wave ((v|-)(x,t),(u|-)(x,t)) governed by the classical Darcy's law provided that the corresponding prescribed initial error function lies in and is sufficiently small. Furthermore, the Lp (2≤p≤ ∞) convergence rates of the solutions are also obtained.     

A SOBOLEV-HARDY INEQUALITY WITH APPLICATION TO A NONLINEAR ELLIPTIC EQUATION

In this paper, when μ< 1/4, and 2 0 and q=2(3-σ),the method is coming from the idea of Pohozaev.     

Global Existence and Blow-up of Solutions for Parabolic Systems Involving Cross-Diffusions and Nonlinear Boundary Conditions

Abstract In this paper, we investigate the positive solutions of strongly coupled nonlinear parabolic systems with nonlinear boundary conditions： {ut-a（u, v）△u=g（u, v）, vt-b（u, v）△v=h（u, v）, δu/δη=d（u, v）, δu/δη=f（u, v）.Under appropriate hypotheses on the functions a, b, g, h, d and f, we obtain that the solutions may exist globally or blow up in finite time by utilizing upper and lower solution techniques.     

Long time dynamics of the 3D radial NLS with the combined terms

In this paper,we study the scattering and blow-up dichotomy result of the radial solution to nonlinear Schrodinger equation(NLS) with the combined terms iu_t+△u=-|u|~4u+|4|~(p-1)u,1+4/3p5 in energy space H~1(R~3).The threshold energy is the energy of the ground state W of the focusing,energy critical NLS,which means that the subcritical perturbation does not affect the determination of threshold,but affects the scattering and blow-up dichotomy result with subcritical threshold energy.This extends algebraic perturbation in a previous work of Miao,Xu and Zhao[Comm.Math.Phys.,318,767-808(2013)]to all mass supercritical,energy subcritical perturbation.     

非线性分数微分方程边值问题正解的存在性(英文)

In this paper,we study a Dirichlet-type boundary value problem（BVP） of nonlinear fractional differential equation with an order α∈(3,4],where the fractional derivative D^α_o⁺is the standard Riemann-Liouville fractional derivative.By constructing the Green function and investigating its properties,we obtain some criteria for the existence of one positive solution and two positive solutions for the above BVP.The Krasnosel’skii fixedpoint theorem in cones is used here.We also give an example to illustrate the applicability of our results.     

POSITIVE SOLUTIONS FOR WEAKLY COUPLED NONLINEAR ELLIPTIC SYSTEMS

In this article,we consider the existence of positive solutions for weakly cou-pled nonlinear elliptic systems {-△u+u (1+a(x))|u| p-1 u+μ|u| α-2 u|v|β+λv in R~N,-△v+v=(1+bx))|v|p-1v+μ|u|α|v|β-2v+λu in R N.(0.1) To find nontrivial solutions,we first investigate autonomous systems.In this case,results of bifurcation from semi-trivial solutions are obtained by the implicit function theorem.Next,the existence of positive solutions of problem(0.1) is obtained by variational methods.     

On a nonlocal problem modelling Ohmic heating in planar domains

In this paper, we consider the nonlocal problem of the form ut-Δu = (λe-u)/(∫Ωe-udx)2,x ∈Ω, t0 and the associated nonlocal stationary problem -Δv = (λe-v)/(∫Ωe-vdx)2, x ∈Ω,where λ is a positive parameter. For Ω to be an annulus, we prove that the nonlocal stationary problemhas a unique solution if and only if λ 2| Ω| 2 , and for λ = 2|Ω|2, the solution of the nonlocal parabolic problem grows up globally to infinity as t →∞.     

THE EXISTENCE AND LOCAL UNIQUENESS OF MULTI-PEAK SOLUTIONS TO A CLASS OF KIRCHHOFF TYPE EQUATIONS

In this paper, we study the existence and local uniqueness of multi-peak solutions to the Kirchhoff type equations -(ε²a + εb∫_R³|?u|²)?u + V(x)u = u^p, u > 0 in R³,which concentrate at non-degenerate critical points of the potential function V(x), where a, b >0, 1 < p < 5 are constants, and ε > 0 is a parameter. Applying the Lyapunov-Schmidt reduction method and a local Pohozaev type identity, we establish the existe...     

Analysis of a Free Boundary Problem Modeling Tumor Growth

In this paper, we study a free boundary problem arising from the modeling of tumor growth. The problem comprises two unknown functions： R = R（t）, the radius of the tumor, and u = u（r, t）, the concentration of nutrient in the tumor. The function u satisfies a nonlinear reaction diffusion equation in the region 0 〈 r 〈 R（t）, t 〉 0, and the function R satisfies a nonlinear integrodifferential equation containing u. Under some general conditions, we establish global existence of transient solutions, unique existence of a stationary solution, and convergence of transient solutions toward the stationary solution as t →∞.     

Solutions of the Camassa-Holm Equation Near the Soliton

In this paper, solutions of the Camassa-Holm equation near the soliton Q is decomposed by pseudoconformal transformation as follows: λ^1/2(t)u(t, λ(t)y + x(t)) = Q(y) + ε(t, y), and the estimation formula with respect to ε(t, y) is obtained: |ε(t, y)| ≤ Ca₃Te^-θ|y|+ |λ^1/2(t)ε₀|. For the CH equation, we prove that the solution of the Cauchy problem and the soliton Q is sufficiently close as y → ∞, and the approximation degree of the solution an...     

POSITIVE SOLUTIONS TO FOURTH ORDER MULTI-POINT BOUNDARY VALUE PROBLEMS WITH p-LAPLACIAN OPERATOR

In this paper, we consider the existence of positive solutions to a singular fourth order p-Laplacian equation. By the upper and lower solution method and fixed point theorems, the existence of positive solutions to the boundary value problem is obtained under the assumption that the nonlinear term is decreasing.