EXISTENCE AND BLOW-UP BEHAVIOR OF CONSTRAINED MINIMIZERS FOR SCHRDINGER-POISSON-SLATER SYSTEM

In this article,we study constrained minimizers of the following variational problem e(p):=inf{u∈H1(R3),||u||22=p}E(u),p〉0,where E(u)is the Schrdinger-Poisson-Slater(SPS)energy functional E(u):=1/2∫R3︱▽u(x)︱2dx-1/4∫R3∫R3u2(y)u2(x)/︱x-y︱dydx-1/p∫R3︱u(x)︱pdx in R3 and p∈(2,6).We prove the existence of minimizers for the cases 2p10/3,ρ0,and p=10/3,0ρρ~*,and show that e(ρ)=-∞for the other cases,whereρ~*=||φ||_2~2 andφ(x)is the unique(up to translations)positive radially symmetric solution of-△u+u=u~(7/3)in R~3.Moreover,when e(ρ~*)=-∞,the blow-up behavior of minimizers asρ↗ρ~*is also analyzed rigorously.     

Existence of Nonnegative Solutions for a Class of Systems Involving Fractional(p,q)-Laplacian Operators

The authors study the following Dirichlet problem of a system involving fractional(p, q)-Laplacian operators:{(-△)_p~su=λa(x)|u|+~(p-2)u+λb(x)|u|~(α-2)|u|~βu+μ(x)/αδ|u|~(γ-2)|v|~δu in Ω,(-△)_p~su=λc(x)|v|+~(q-2)v+λb(x)|u|~α|u|~(β-2)v+μ(x)/βγ|u|~γ|v|~(δ-2)v in Ω,u=v=0 on R~N\Ω where λ 0 is a real parameter, ? is a bounded domain in RN, with boundary ?? Lipschitz continuous, s ∈(0, 1), 1 p ≤ q ∞, sq N, while(-?)s pu is the fractional p-Laplacian operator of u and, similarly,(-?)s qv is the fractional q-Laplacian operator of v. Since possibly p = q, the classical definitions of the Nehari manifold for systems and of the Fibering mapping are not suitable. In this paper, the authors modify these definitions to solve the Dirichlet problem above. Then, by virtue of the properties of the first eigenvalueλ_1 for a related system, they prove that there exists a positive solution for the problem when λ λ_1 by the modified definitions. Moreover, the authors obtain the bifurcation property when λ→λ_1~-. Finally, thanks to the Picone identity, a nonexistence result is also obtained when λ≥λ_1.     

Multiplicity of solutions for quasilinear elliptic systems with singularity

In this paper, we study the existence of multiple solutions for the following quasilinear elliptic system:p*(t)|u-2β- △pu1-μ|-2u up1= α1u + β1-2|xp||xt|vβ2||u|u, x∈,|q*β- △qv-μ2 |v|q-2v αv(s)-2|2x|q=|x|sv + β2|uβ1||v2 |-2v, x∈,u(x) = v(x) = 0, x∈ .Multiplicity of solutions for the quasilinear problem is obtained via variational method.     

EXISTENCE RESULT FOR A CLASS OF N-LAPLACIAN EQUATIONS INVOLVING CRITICAL GROWTH

In this paper, we consider a class of N-Laplacian equations involving critical growth{-?_N u = λ|u|~(N-2) u + f(x, u), x ∈ ?,u ∈ W_0~(1,N)(?), u(x) ≥ 0, x ∈ ?,where ? is a bounded domain with smooth boundary in R~N(N 2), f(x, u) is of critical growth. Based on the Trudinger-Moser inequality and a nonstandard linking theorem introduced by Degiovanni and Lancelotti, we prove the existence of a nontrivial solution for any λ λ_1, λ = λ_?(? = 2, 3, ···), and λ_? is the eigenvalues of the operator(-?_N, W_0~(1,N)(?)),which is defined by the Z_2-cohomological index.     

THE GLOBAL SMOOTH SOLUTIONS OF SECOND ORDER QUASILINEAR HYPERBOLIC EQUATIONS WITH DISSIPATIVE BOUNDARY CONDITIONS

The paper deals with the following boundary problem of the second order quasilinear hyperbolic equation with a dissipative boundary condition on a part of the boundary:u_(tt)-sum from i,j=1 to n a_(ij)(Du)u_(x_ix_j)=0, in (0, ∞)×Ω,u|Γ_0=0,sum from i,j=1 to n, a_(ij)(Du)n_ju_x_i+b(Du)u_t|Γ_1=0,u|t=0=φ(x), u_t|t=0=ψ(x), in Ω, where Ω=Γ_0∪Γ_1, b(Du)≥b_0>0. Under some assumptions on the equation and domain, the author proves that there exists a global smooth solution for above problem with small data.     

GLOBAL EXISTENCE OF THE SOLUTIONS TO NONLINEAR HYPERBOLIC EQUATIONS IN EXTERIOR DOMAINS

This paper deals with the following IBV problem of nonlinear hyperbolic equations u_(tt)- sum from i, j=1 to n a_(jj)(u, Du)u_(x_ix_j)=b(u, Du), t>0, x∈Ω, u(O, x) =u~0(x), u_t(O, x) =u~1(v), x∈Ω, u(t, x)=O t>O, x∈()Ω,where Ωis the exterior domain of a compact set in R~n, and |a_(ij)(y)-δ_(ij)|= O(|y|~k), |b(y)|=O(|y|~(k+1)), near y=O. It is proved that under suitable assumptions on the smoothness,compatibility conditions and the shape of Ω, the above problem has a unique global smoothsolution for small initial data, in the case that k=1 add n≥7 or that k=2 and n≥4.Moreover, the solution ham some decay properties as t→ + ∞.     

GLOBAL EXISTENCE AND BLOW-UP OF SOLUTIONS FOR GENERALIZED POCHHAMMER-CHREE EQUATIONS

In this article,we study the initial boundary value problem of generalized Pochhammer-Chree equation u_(tt)-u_(xx)-u_(xxt)-u_(xxtt)=f(u) xx,x ∈Ω,t 0,u(x,0) = u0(x),u t(x,0)=u1(x),x ∈Ω,u(0,t) = u(1,t) = 0,t≥0,where Ω=(0,1).First,we obtain the existence of local W k,p solutions.Then,we prove that,if f(s) ∈ΩC k+1(R) is nondecreasing,f(0) = 0 and |f(u)|≤C1|u| u 0 f(s)ds+C2,u 0(x),u 1(x) ∈ΩW k,p(Ω) ∩ W 1,p 0(Ω),k ≥ 1,1 p ≤∞,then for any T 0 the problem admits a unique solution u(x,t) ∈ W 2,∞(0,T;W k,p(Ω) ∩ W 1,p 0(Ω)).Finally,the finite time blow-up of solutions and global W k,p solution of generalized IMBq equations are discussed.     

Scattering Versus Blowup Beyond the Mass-Energy Threshold for the Davey–Stewartson Equation in R~3

We study the Cauchy problem for the Davey–Stewartson equation i?_tu + Δu + |u|~2 u + E_1(|u|~2)u = 0,(t, x) ∈ R × R~3.The dichotomy between scattering and finite time blow-up shall be proved for initial data with finite variance and with mass-energy M(u_0)E(u_0) above the ground state threshold M(Q)E(Q).     

ALMOST CONSERVATION LAWS AND GLOBAL ROUGH SOLUTIONS OF THE DEFOCUSING NONLINEAR WAVE EQUATION ON R~2

In this article, we investigate the initial value problem(IVP) associated with the defocusing nonlinear wave equation on R~2 as follows:{?_(ttu)-△u =-u~3,u(0, x) = u_0(x), ?_(tu)(0, x) = u_1(x),where the initial data(u_0, u_1) ∈ H~s(R~2) × H~(s-1)(R~2). It is shown that the IVP is global well-posedness in H~s(R~2) × H~(s-1)(R~2) for any 1 s 2/5. The proof relies upon the almost conserved quantity in using multilinear correction term. The main difficulty is to control the growth of the variation of the almost conserved quantity. Finally, we utilize linear-nonlinear decomposition benefited from the ideas of Roy [1].     

THE EXISTENCE OF GLOBAL SOLUTION AND THE BLOWUP PROBLEM FOR SOME p-LAPLACE HEAT EQUATIONS

This article consider, for the following heat equation ut/|x|s-△pu=uq,(x,t)∈Ω×(0,T), u(x,t)=0,(x,t)∈(?)Ω×(0,T), u(x,0)=u0(x),u0(x)≥0,u0(x)(?)0 the existence of global solution under some conditions and give two sufficient conditions for the blow up of local solution in finite time, whereΩis a smooth bounded domain in RN(N>p),0∈Ω,△pu=div(|▽u|p-2▽u),0≤s≤2,p≥2,p-1相似文献   

A Remark on the Existence of Entire Large and Bounded Solutions to a(k_1, k_2)-Hessian System with Gradient Term

In this paper, we study the existence of positive entire large and bounded radial positive solutions for the following nonlinear system{S_k_1(λ(D_(u1)~2)) + a_1(|x|) |▽_(u_1) |~(k_1)= p_1(|x|) f_1(u_2) for x ∈ R~N,S_k_2(λ(D_(u_2)~2)) + a_2(|x|) |▽_(u2) |~(k_2)= p_2(|x|) f_2(u_1) for x ∈ R~N.Here S_k_i(λ(D_(u_i)~2) is the k_i-Hessian operator, a_1, p_1, f_1, a_2, p_2 and f_2 are continuous functions.     

Global well-posedness and blow-up for the hartree equation

For 2 γ min{4, n}, we consider the focusing Hartree equation iu_t+ △u +(|x|~(-γ)* |u|~2)u = 0, x ∈ R~n.(0.1)Let M [u] and E [u] denote the mass and energy, respectively, of a solution u, and Q be the ground state of-△ Q + Q =(|x|~(-γ)* |Q|~2)Q. Guo and Wang [Z. Angew. Math.Phy.,2014] established a dichotomy for scattering versus blow-up for the Cauchy problem of(0.1) if M [u]~(1-s_c)E [u]~(s_c) M [Q]~(1-s_c)E [Q]~(s_c)(s_c=(γ-2)/2). In this paper, we consider the complementary case M [u]~(1-s_c)E [u]~(s_c)≥ M [Q]~(1-s_c)E [Q]~(s_c) and obtain a criteria on blow-up and global existence for the Hartree equation(0.1).     

Initial Boundary Value Problem for One Class of System of Multi- Dimensional Inhomogeneous GBBM Equations

This paper studies the following initial-boundary value problem for the system of multidimensional inhomogeneous GBBM equations $[\begin{array}{l} {u_r} - \Delta {u_i} + \sum\limits_{i = 1}^n {\frac{\partial }{{\partial {x_i}}}} grad\varphi (u) = f(u),{\rm{ (1}}{\rm{.1)}}\u{|_{t = 0}} = {u_0}(x),x \in \Omega ,{\rm{ (1}}{\rm{.2)}}\u{|_{\partial \Omega }} = 0,t \ge 0,{\rm{ (1}}{\rm{.3)}} \end{array}\]$ The existence and uniqueness of the global solution for the problem(l.l) (1.2) (1.3) are proved. The asymptotic behavior and “blow up” phenomenon of the solution for the problem (1.1) (1.2) (1.3) are investigated under certain conditions.     

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.     

Liouville theorem for Choquard equation with finite Morse indices

In this article, we study the nonexistence of solution with finite Morse index for the following Choquard type equation-△u=∫RN|u(y)|p|x-y|αdy|u(x)|p-2u(x) in RN where N ≥ 3, 0 α min{4, N}. Suppose that 2 p (2 N-α)/(N-2),we will show that this problem does not possess nontrivial solution with finite Morse index. While for p=(2 N-α)/(N-2),if i(u) ∞, then we have ∫_RN∫_RN|u(x)p(u)(y)~p/|x-y|~α dxdy ∞ and ∫_RN|▽u|~2 dx=∫_RN∫_RN|u(x)p(u)(y)~p/|x-y|~αdxdy.     

Global Existence and Blow-Up Results for a Classical Semilinear Parabolic Equation

The author studies the boundary value problem of the classical semilinear parabolic equations ut-△u = |u|p-1u inΩ×(0, T), and u = 0 on the boundary × [0, T) and u = φ at t = 0, where Rnis a compact C1domain, 1 < p ≤ p S is a fixed constant, and φ∈ C1 0(Ω) is a given smooth function. Introducing a new idea, it is shown that there are two sets W and Z, such that for φ∈ W, there is a global positive solution u(t) ∈ W with H1omega limit 0 and for φ∈ Z, the solution blows up at finite time.     

GLOBAL WELL-POSEDNESS IN ENERGY SPACE OF SMALL AMPLITUDE SOLUTIONS FOR KLEIN-GORDON-ZAKHAROV EQUATION IN THREE SPACE DIMENSION

The Cauchy problem of the Klein-Gordon-Zakharov equation in three dimensional space {utt-?u + u =-nu,(x, t) ∈ R~3× R_+,ntt-?n= ?|u|~2,(x, t) ∈ R~3× R_+,u(x, 0) = u_0(x), ?_tu(x, 0) = u_1(x),n(x, 0) = n_0(x), ?_tn(x,0) =n_1(x),(0.1) is considered. It is shown that it is globally well-posed in energy space H~1× L~2× L~2× H~(-1) if small initial data(u_0(x), u_1(x), n_0(x), n_1(x)) ∈(H~1× L~2× L~2× H~(-1)). It answers an open problem: Is it globally well-posed in energy space H~1× L~2× L~2× H~(-1) for 3D Klein-GordonZakharov equation with small initial data [1, 2]? The method in this article combines the linear property of the equation( dispersive property) with nonlinear property of the equation(energy inequalities). We mainly extend the spaces F~s and N~s in one dimension [3] to higher dimension.     

Multi-peak nodal solutions for a two-dimensional elliptic problem with large exponent in weighted nonlinearity

We consider the boundary value problem u +p|x2α||u|-1u = 0,-1 < α = 0, in the unit ball B with the homogeneous Dirichlet boundary condition, when p is a large exponent. By a constructive way, we prove that for any positive integer m, there exists a multi-peak nodal solution up whose maxima and minima are2 located alternately near the origin and the other m points qll=(λ cosπ(l-1)2, λ-1)msinπ(m), l = 2,, m + 1,such that as p goes to +∞,m+1pα|x2||upp|-1up 8πe(1 + α)δ0 + 8πe(-1)l-1δl=2re λ∈(0, 1), m is an odd number with(1 + α)(m + 2)-1 > 0, or m is an even,ql whe number. The same techniques lead also to a more general result on general domains.     

GLOBAL EXISTENCE OF THE SOLUTION OF NONLINEAR SCHRDINGER EQUATIONS IN EXTERIOR DOMAINS

In this paper,we discuss the problem for the nonlinear Schr(?)dinger equation(?)where Ω is the exterior domain of a compact set in B~n,a_j(u)=O(|u|),b_j(u)=O(|u|)(1≤j≤n),c(u)=O(|u|~2)near u=0.If n≥5,some Sobolev norm of u_0(x)is sufficiently small,under suitableassumptions on smoothnessand and compatibility and the shape of Ω,we get that the problem has aunique global solution u(t,x),with the decay estimate‖u(t,·)‖_(L(?)(Ω))=O(t~(-n/4)),‖u(t,·)‖_(L~4(Ω))=O(t~(-n/4)),t→+∞.     

Quasi-Normal Structure and Fixed Point of Nonlinear Mappings

Let (X,|| ||) be a Banach space. For $\Omega \subset X^*$ and $x\in X$ we introduce the following notations (p\geq 1 and n\in N) $|X|_{\Omega _p(n)}=sup{(\sum\limits_{f\in F} |f(x)|^p)^{1/p}:F \subset \Omega,|F|\leq n$ $|X|_{\Omega _\infty}=sup{|f(x)|:f\in \Omega}$ A convex subset E of X is said to have guasi-normal structure whenever there exists a norm 1 | on A which satisfies the following conditions; (i) E has norinal structure relative to the norm ||| |||. (ii) There exist $\Omega \subset X^*$, p\geq 1 and \theta \in (0,1], such that $|x|_{\Omega _p(2) \leq |||x||| \leq ||x||}$ for x\in E and |||x|||<||x|| implies $2^1/p |x|_\Omega_\infty \geq \theta ||x||+(1-\theta)|||x|||$ or (ii)' There exist \Omega \subset X^*,p\geq 1 and \alpha \in [1,4^1/p) such that for all x\in E, |x|_\Omega_\rho(4)\leq |||x|||,||x||=max{|||x|||,\alpha|x|_\Omega_\infty} and for any countable subset w of \Omega $sup{\sum\limits _{\delt\in w |f(x)|^p:x\in E}<+\infty$ We notice that a set with normal steucture must have quasi-normal structure and there exist sets without normal structure which quasi-normal structure. The main result of the present paper is as follows. Theorem. Let (X, || ||) be a Banach space, E a weak compact convex nonempty subset of X with quasi-normal structure. Let T be a mapping of E in to itself. If there exists a sequence {x_n} in any T-invariant convex subset of E such that $lim_{n\rightarrow \infty} ||x_n-x_n+1||=lim_{n\rightarrow \infty}||x_n-Tx_n||=0$ and $lim_{n\rightarrow \infty} ||y-x_n||=\delta(\bar co{x_n}),for y\in \bar co({c_n})$ limll2/-*?ll=3(coK}), for y€co({xa}), then the mapping T has a fixed point in E, In particular, if the mapping T satisfies $||Tx-Ty||\leq max{||x-y||,1/2(||x-Ty||+||y-Tx||)},for x,y\in E$ then the mapping T has a fixed point in E.