The existence and concentration of weak solutions to a class of p-Laplacian type problems in unbounded domains

In this paper,we study the existence and concentration of weak solutions to the p-Laplacian type elliptic problem-εp△pu+V(z)|u|p-2u-f(u)=0 in Ω,u=0 on ■Ω,u0 in Ω,Np2,where Ω is a domain in RN,possibly unbounded,with empty or smooth boundary,εis a small positive parameter,f∈C1(R+,R)is of subcritical and V:RN→R is a locally Hlder continuous function which is bounded from below,away from zero,such that infΛVmin ■ΛV for some open bounded subset Λ of Ω.We prove that there is anε00 such that for anyε∈(0,ε0],the above mentioned problem possesses a weak solution uεwith exponential decay.Moreover,uεconcentrates around a minimum point of the potential V inΛ.Our result generalizes a similar result by del Pino and Felmer(1996)for semilinear elliptic equations to the p-Laplacian type problem.     

Non-Nehari manifold method for asymptotically periodic Schrdinger equations

We consider the semilinear Schrdinger equation-△u + V(x)u = f(x, u), x ∈ RN,u ∈ H 1(RN),where f is a superlinear, subcritical nonlinearity. We mainly study the case where V(x) = V0(x) + V1(x),V0∈ C(RN), V0(x) is 1-periodic in each of x1, x2,..., x N and sup[σ(-△ + V0) ∩(-∞, 0)] 0 inf[σ(-△ +V0)∩(0, ∞)], V1∈ C(RN) and lim|x|→∞V1(x) = 0. Inspired by previous work of Li et al.(2006), Pankov(2005)and Szulkin and Weth(2009), we develop a more direct approach to generalize the main result of Szulkin and Weth(2009) by removing the "strictly increasing" condition in the Nehari type assumption on f(x, t)/|t|. Unlike the Nahari manifold method, the main idea of our approach lies on finding a minimizing Cerami sequence for the energy functional outside the Nehari-Pankov manifold N0 by using the diagonal method.     

Singularly perturbed Neumann problem for fractional Schrdinger equations

This paper is concerned with a Neumann type problem for singularly perturbed fractional nonlinear Schrdinger equations with subcritical exponent. For some smooth bounded domain ?  R~n, our boundary condition is given by∫_?u(x)-u(y)/|x-y|~(n+2s)dy = 0 for x ∈ R~n\?.We establish existence of non-negative small energy solutions, and also investigate the integrability of the solutions on Rn.     

素环上的导子在Lie理想上的协交换子的零化子

Let R be a prime ring of characteristic different from 2,d and g twoderivations of R at least one of which is nonzero,L a non-central Lie ideal of R,anda∈R.We prove that if a(d(u)u-ug(u))=0 for any u∈L,then either a=0,or R is an s_4-ring,d(x)=[p,x],and g(x)=-d(x)for some p in the Martindalequotient ring of R.     

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.     

A reverse Denjoy theorem Ⅲ

Suppose that C 1 and C 2 are two simple curves joining 0 to ∞, non-intersecting in the finite plane except at 0 and enclosing a domain D which is such that, for all large r, the set {θ : re iθ∈ D} has measure at most 2α, where 0 α π. Suppose also that u is a non-constant subharmonic function in the plane such that u(z) = Φ(|z|) for all large z ∈ C 1 ∪ C 2 ∪～D, where Φ(|z|) is a convex, non-decreasing function of |z| and ～D is the complement of D. Let A D (r, u) = inf{u(z) : z ∈ D and |z| = r}. It is shown that if A D (r, u) = O(1) then lim inf r→∞ B(r, u)/r π/(2α) 0.     

Blow-up of p-Laplacian evolution equations with variable source power

We study the blow-up and/or global existence of the following p-Laplacian evolution equation with variable source power ut(x,t)=div(|?u|~(p-2)?u)+u~(q(x)) in?×(0,T),where ? is either a bounded domain or the whole space R~N,and q(x) is a positive and continuous function defined in ? with 0q_-=inf q(x)=q(x)=sup q(x)=q_+∞.It is demonstrated that the equation with variable source power has much richer dynamics with interesting phenomena which depends on the interplay of q(x) and the structure of spatial domain ?,compared with the case of constant source power.For the case that ? is a bounded domain,the exponent p-1 plays a crucial role.If q_+p-1,there exist blow-up solutions,while if q_+p-1,all the solutions are global.If q_-p-1,there exist global solutions,while for given q_-p-1q_+,there exist some function q(x) and ? such that all nontrivial solutions will blow up,which is called the Fujita phenomenon.For the case ?=R~N,the Fujita phenomenon occurs if 1q_-=q_+=p-1+p/N,while if q_-p-1+p/N,there exist global solutions.     

Existence of Semiclassical States for a Quasilinear Schr?dinger Equation on R~N with Exponential Critical Growth

We study a quasilinear Schr?dinger equation{-ε~NΔNu+V(x)|u|~(N-2)u=Q(x)f(u) in R~N,0u∈W~(1,N)(R~N),u(x)|x|→∞→0,where V,Q are two continuous real functions on R~N and ε0 is a real parameter.Assume that the nonlinearity f is of exponential critical growth in the sense of Trudinger–Moser inequality,we are able to establish the existence and concentration of the semiclassical solutions by variational methods.     

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.     

Beurling's theorem and invariant subspaces for the shift on Hardy spaces

Let G be a bounded open subset in the complex plane and let H~2(G) denote the Hardy space on G. We call a bounded simply connected domain W perfectly connected if the boundary value function of the inverse of the Riemann map from W onto the unit disk D is almost 1-1 with respect to the Lebesgue measure on D and if the Riemann map belongs to the weak-star closure of the polynomials in H~∞(W). Our main theorem states: in order that for each M∈Lat (M_z), there exist u∈H~∞(G) such that M=∨{uH~2(G)}, it is necessary and sufficient that the following hold: (1) each component of G is a perfectly connected domain; (2) the harmonic measures of the components of G are mutually singular; (3) the set of polynomials is weak-star dense in H~∞(G). Moreover, if G satisfies these conditions, then every M∈Lat (M_z) is of the form uH~2(G), where u∈H~∞(G) and the restriction of u to each of the components of G is either an inner function or zero.     

THE BOUNDEDNESS FOR COMMUTATORS OF ANISOTROPIC CALDERóN-ZYGMUND OPERATORS

Let T be an anisotropic Calderón-Zygmund operator and φ:R~n×[0,∞)→[0,∞) be an anisotropic Musielak-Orlicz function with φ(x,·) being an Orlicz function andφ(·,t) being a Muckenhoupt A_∞(A) weight.In this paper,our goal is to study two boundedness theorems for commutators of anisotropic Calderon-Zygmund operators.Precisely,when b∈BMO_w(R~n,A)(a proper subspace of anisotropic bounded mean oscillation space BMO(R~n,A)),the commutator [b,T] is bounded from anisotropic weighted Hardy space H_ω~1(R~n,A) to weighted Lebesgue space L_ω~1(R~n) and when b∈BMO(R~n)(bounded mean oscillation space),the commutator [b,T] is bounded on Musielak-Orlicz space L~φ(R~n),which are extensions of the isotropic setting.     

带有临界Sobolev指数的奇异双调和问题

LetΩR~N be a smooth bounded domain such that 0∈Ω,N≥5,2~*:=(2N)/(N-4) is the critical Sobolev exponent,and f(x) is a given function.By using the variational methods, the paper proves the existence of solutions for the singular critical in the homogeneous problemΔ~u-μu/(|x|~4)=|u|~(2~*-2)u f(x) with Dirichlet boundary condition on Ωunder some assumptions on f(x) andμ.     

SOLUTIONS TO NONLINEAR ELLIPTIC EQUATIONS WITH A GRADIENT

In this article, we consider existence and nonexistence of solutions to problem{ -?p u+g(x, u)|▽u|p=fin ?,u= 0 on??(0.1)with 1 p ∞, wherefis a positive measurable function which is bounded away from 0in ?, and the domain ? is a smooth bounded open set in RN(N≥ 2). Especially, under the condition thatg(x, s) = 1/|s|α(α 0) is singular ats= 0, we obtain thatα pis necessary and sufficient for the existence of solutions inW1,p0(?) to problem(0.1) whenfis sufficiently regular.     

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.     

Quasi-periodic solutions of a semilinear Li(?)nard equation at resonance

We are concerned with the existence of quasi-periodic solutions for the follow- ing equation x″ F_x(x,t)x′ ω~2x φ(x,t)=0, where F and φare smooth functions and 2π-periodic in t,ω>0 is a constant.Under some assumptions on the parities of F and φ,we show that the Dancer's function,which is used to study the existence of periodic solutions,also plays a role for the existence of quasi-periodic solutions and the Lagrangian stability (i.e.all solutions are bounded).     

Optimal Transportation for Generalized Lagrangian

This paper deals with the optimal transportation for generalized Lagrangian L = L(x, u, t), and considers the following cost function: c(x, y) = inf x(0)=x x(1)=y u∈U∫_0~1 L(x(s), u(x(s), s), s)ds, where U is a control set, and x satisfies the ordinary equation x(s) = f(x(s), u(x(s), s)).It is proved that under the condition that the initial measure μ0 is absolutely continuous w.r.t. the Lebesgue measure, the Monge problem has a solution, and the optimal transport map just walks along the characteristic curves of the corresponding Hamilton-Jacobi equation:V_t(t, x) + sup u∈UV_x(t, x), f(x, u(x(t), t), t)-L(x(t), u(x(t), t), t) = 0,V(0, x) = Φ0(x).     

Entropy and renormalized solutions for nonlinear elliptic problem involving variable exponent and measure data

We give an existence result of entropy and renormalized solutions for strongly nonlinear elliptic equations in the framework of Sobolev spaces with variable exponents of the type:-div(a(x, u, u) + φ(u)) + g(x, u, u) = μ, where the right-hand side belongs to L1(Ω) + W-1,p(x)(Ω),-div(a(x, u, u)) is a Leray–Lions oper- ator defined from W-1,p(x)(Ω) into its dual and φ∈ C0(R, RN). The function g(x, u, u) is a non linear lower order term with natural growth with respect to |u| satisfying the sign condition, that is,g(x, u, u)u ≥ 0.     

A concentration behavior for semilinear elliptic systems with indefinite weight

Consider the Schrdinger system{-Δu+V1,nu=αQn(x)︱u︱α-2u︱v︱β,-Δv+V2,nv=βQn(x)︱u︱α︱v︱β-2v,u,v∈H10(Ω) where ΩR~N,α,β 1,α + β 2* and the spectrum σ(-△ + V_(i,n))(0,+∞),i = 1,2;Q_n is a bounded function and is positive in a region contained in Ω and negative outside.Moreover,the sets{Q_n 0} shrink to a point x_0∈Ω as n→+∞.We obtain the concentration phenomenon.Precisely,we first show that the system has a nontrivial solution(u_n,v_n) corresponding to Q_n,then we prove that the sequences(u_n) and(v_n) concentrate at x_0 with respect to the H~1-norm.Moreover,if the sets {Q_n 0} shrink to finite points and(u_n,v_n) is a ground state solution,then we must have that both u_n and v_n concentrate at exactly one of these points.Surprisingly,the concentration of u_n and v_n occurs at the same point.Hence,we generalize the results due to Ackermann and Szulkin.     

Jacobian-determinant equation in the critical Besov spaces

Let Ω be a bounded domain in R~n with smooth boundary. Here we consider the following Jacobian-determinant equation det u(x)=f(x),x∈Ω;u(x)=x,x∈?Ω where f is a function on Ω with min_Ω f = δ 0 and Ωf(x)dx = |Ω|. We prove that if f ∈B_(p1)~(np)(Ω) for some p∈(n,∞), then there exists a solution u ∈ B_(p1)~(np+1)(Ω)C~1(Ω) to this equation. On the other hand, we give a simple example such that u ∈ C_0~1(R~2, R~2) while detu does not lie in B_(p1)~(2p)(R~2) for any p∞.     

A critical Trudinger-Moser inequality involving a degenerate potential and nonlinear Schrödinger equations

The classical critical Trudinger-Moser inequality in R~2 under the constraint ∫_(R_2)(|▽u|~2+|u|~2)dx≤1 was established through the technique of blow-up analysis or the rearrangement-free argument:for any τ 0,it holds that ■ and 4π is sharp.However,if we consider the less restrictive constraint ∫_(R_2)(|▽u|~2+|u|~2)dx≤1,where V(x) is nonnegative and vanishes on an open set in R~2,it is unknown whether the sharp constant of the Trudinger-Moser inequality is still 4π.The loss of a positive lower bound of the potential V(x) makes this problem become fairly nontrivial.The main purpose of this paper is twofold.We will first establish the Trudinger-Moser inequality ■ when V is nonnegative and vanishes on an open set in R~2.As an application,we also prove the existence of ground state solutions to the following Sciridinger equations with critical exponeitial growth:-Δu+V(x)u=f u) in R~2,(0.1)where V(x)≥0 and vanishes on an open set of R~2 and f has critical exponential growth.Having a positive constant lower bound for the potential V(x)(e.g.,the Rabinowitz type potential) has been the standard assumption when one deals with the existence of solutions to the above Schr?dinger equations when the nonlinear term has the exponential growth.Our existence result seems to be the first one without this standard assumption.