where p>1, q>1, is a nonnegative continuous function, λ is a real number. The sufficient condition to have positive solutions of the above problem is . However, if , there is no solution which belongs to it. Therefore, our results are optimal.  相似文献   

3.
A nonlocal p-Laplacian evolution equation with Neumann boundary conditions   总被引:1,自引:0,他引:1  
F. Andreu  J.M. Mazn  J.D. Rossi  J. Toledo 《Journal de Mathématiques Pures et Appliquées》2008,90(2):201-227
In this paper we study the nonlocal p-Laplacian type diffusion equation,
If p>1, this is the nonlocal analogous problem to the well-known local p-Laplacian evolution equation ut=div(|u|p−2u) with homogeneous Neumann boundary conditions. We prove existence and uniqueness of a strong solution, and if the kernel J is rescaled in an appropriate way, we show that the solutions to the corresponding nonlocal problems converge strongly in L(0,T;Lp(Ω)) to the solution of the p-Laplacian with homogeneous Neumann boundary conditions. The extreme case p=1, that is, the nonlocal analogous to the total variation flow, is also analyzed. Finally, we study the asymptotic behavior of the solutions as t goes to infinity, showing the convergence to the mean value of the initial condition.  相似文献   

4.
Analytic approximation of matrix functions in     
L. Baratchart  F.L. Nazarov  V.V. Peller   《Journal of Approximation Theory》2009,158(2):242
We consider the problem of approximation of matrix functions of class Lp on the unit circle by matrix functions analytic in the unit disk in the norm of Lp, 2≤p<. For an m×n matrix function Φ in Lp, we consider the Hankel operator , 1/p+1/q=1/2. It turns out that the space of m×n matrix functions in Lp splits into two subclasses: the set of respectable matrix functions and the set of weird matrix functions. If Φ is respectable, then its distance to the set of analytic matrix functions is equal to the norm of HΦ. For weird matrix functions, to obtain the distance formula, we consider Hankel operators defined on spaces of matrix functions. We also describe the set of p-badly approximable matrix functions in terms of special factorizations and give a parametrization formula for all best analytic approximants in the norm of Lp. Finally, we introduce the notion of p-superoptimal approximation and prove the uniqueness of a p-superoptimal approximant for rational matrix functions.  相似文献   

5.
Nonradial large solutions of sublinear elliptic problems   总被引:1,自引:0,他引:1  
Khalifa El Mabrouk  Wolfhard Hansen 《Journal of Mathematical Analysis and Applications》2007,330(2):1025-1041
Let p be a nonnegative locally bounded function on , N3, and 0<γ<1. Assuming that the oscillation sup|x|=rp(x)−inf|x|=rp(x) tends to zero as r→∞ at a specified rate, it is shown that the equation Δu=p(x)uγ admits a positive solution in satisfying lim|x|→∞u(x)=∞ if and only if
  相似文献   

6.
Existence and estimates of solutions to a singular Dirichlet problem for the Monge–Ampère equation     
Ahmed Mohammed   《Journal of Mathematical Analysis and Applications》2008,340(2):1226-1234
Given a strictly convex, smooth, and bounded domain Ω in we establish the existence of a negative convex solution in with zero boundary value to the singular Monge–Ampère equation det(D2u)=p(x)g(−u). An associated Dirichlet problem will be employed to provide a necessary and sufficient condition for the solvability of the singular boundary value problem. Estimates of solutions will also be given and regularity of solutions will be deduced from the estimates.  相似文献   

7.
A Palais–Smale approach to Lane–Emden equations     
Huei-li Lin  Weichung Wang   《Journal of Mathematical Analysis and Applications》2007,330(2):1220-1237
We consider the unbounded domain problems −Δu+u=|u|p−2u in Ω, u>0 in Ω, and u=0 on ∂Ω, where Ω is an unbounded domain in , 2<p<2*, for N>2, and 2*=∞ for N=2. The existence of a ground state solution to the problems is greatly affected by the shape of the domain. To determine the existence of the solutions in a general domain remains a challenge task. For the flat interior flask domain that consists a strip and a ball attached to the bottom of the strip, previous results have asserted the existence of a ground state solution when the diameter of the ball is greater than a positive constant. However, the existence of the solutions when the diameter of the ball equals to the width of the strip is still an important open question. This article resolves the open question partially by considering a variation of the flat interior flask domain, which is formed by attaching a stretched ball to the bottom of the strip.  相似文献   

8.
Converse and Smoothness Theorems for Erdo&#x030B;s Weights inL_(0<p?∞)     
S.B. Damelin 《Journal of Approximation Theory》1998,93(3):349-398
We prove converse and smoothness theorems of polynomial approximation in weightedLpspaces with norm ‖fWLp()(0<p?∞) for Erdo&#x030B;s weights on the real line. In particular we prove characterization theorems involving realization functionals and thereby establish some interesting properties of our weighted modulus of continuity.  相似文献   

9.
Nonexistence of backward self-similar blowup solutions to a supercritical semilinear heat equation     
Noriko Mizoguchi   《Journal of Functional Analysis》2009,257(9):2911-2937
We consider a Cauchy problem for a semilinear heat equation
with p>pS where pS is the Sobolev exponent. If u(x,t)=(Tt)−1/(p−1)φ((Tt)−1/2x) for xRN and t[0,T), where φ is a regular positive solution of
(P)
then u is called a backward self-similar blowup solution. It is immediate that (P) has a trivial positive solution κ≡(p−1)−1/(p−1) for all p>1. Let pL be the Lepin exponent. Lepin obtained a radial regular positive solution of (P) except κ for pS<p<pL. We show that there exist no radial regular positive solutions of (P) which are spatially inhomogeneous for p>pL.  相似文献   

10.
Positive solutions for Robin problem involving the -Laplacian     
Shao-Gao Deng   《Journal of Mathematical Analysis and Applications》2009,360(2):548-560
Consider Robin problem involving the p(x)-Laplacian on a smooth bounded domain Ω as follows
Applying the sub-supersolution method and the variational method, under appropriate assumptions on f, we prove that there exists λ*>0 such that the problem has at least two positive solutions if λ(0,λ*), has at least one positive solution if λ=λ*<+∞ and has no positive solution if λ>λ*. To prove the results, we prove a norm on W1,p(x)(Ω) without the part of ||Lp(x)(Ω) which is equivalent to usual one and establish a special strong comparison principle for Robin problem.  相似文献   

11.
Blow-up of solutions for semilinear heat equation with nonlinear nonlocal boundary condition     
Alexander Gladkov  Kwang Ik Kim 《Journal of Mathematical Analysis and Applications》2008,338(1):264-273
In this paper, we consider a semilinear heat equation utu+c(x,t)up for (x,t)∈Ω×(0,∞) with nonlinear and nonlocal boundary condition and nonnegative initial data where p>0 and l>0. We prove global existence theorem for max(p,l)?1. Some criteria on this problem which determine whether the solutions blow up in a finite time for sufficiently large or for all nontrivial initial data or the solutions exist for all time with sufficiently small or with any initial data are also given.  相似文献   

12.
Global existence and blow-up of solutionsfor a higher-order kirchhoff-type equation with nonlinear dissipation     
Fucai Li 《Applied Mathematics Letters》2004,17(12):686
This paper deals with the higher-order Kirchhoff-type equation with nonlinear dissipationutt+(Ω׀Dmu׀2dx)q(−Δ)mu+ut׀ut׀ru׀pu,xΩ,t>0,in a bounded domain, where m < 1 is a positive integer, q, p, r < 0 arepositive constants. We obtain that the solution exists globally if pr, while ifp > max r, 2q , then for any initial data with negative initial energy, the solution blowsup at finite time in Lp+2 norm.  相似文献   

13.
Well-posedness of the Cauchy problem for the fractional power dissipative equation in critical Besov spaces     
Gang Wu  Jia Yuan   《Journal of Mathematical Analysis and Applications》2008,340(2):1326-1335
In this paper we study the Cauchy problem for the semilinear fractional power dissipative equation ut+(−Δ)αu=F(u) for the initial data u0 in critical Besov spaces with , where α>0, F(u)=P(D)ub+1 with P(D) being a homogeneous pseudo-differential operator of order d[0,2α) and b>0 being an integer. Making use of some estimates of the corresponding linear equation in the frame of mixed time–space spaces, the so-called “mono-norm method” which is different from the Kato's “double-norm method,” Fourier localization technique and Littlewood–Paley theory, we get the well-posedness result in the case .  相似文献   

14.
Asymptotic behavior of large solution to elliptic equation of Bieberbach–Rademacher type with convection terms     
Shuibo Huang  Qiaoyu Tian  Chunlai Mu 《Applied mathematics and computation》2009,210(2):284-293
We analyze the asymptotic behavior of solutions to nonlinear elliptic equation Δu±|u|q=b(x)f(u) in Ω, subject to the singular boundary condition u(x)= as , where Ω is a smooth bounded domain in RN, for some , and . Our approach employs Karamata regular variation theory combined with the method of lower and supper solution.  相似文献   

15.
Refined blowup criteria and nonsymmetric blowup of an aggregation equation     
Dong Li  Jose L. Rodrigo   《Advances in Mathematics》2009,220(6):1717-1738
We consider an aggregation equation in , d2, with fractional dissipation: ut+(uK*u)=−νΛγu, where ν0, 0<γ<1, and K(x)=e−|x|. We prove a refined blowup criteria by which the global existence of solutions is controlled by its norm, for any . We prove the finite time blowup of solutions for a general class of nonsymmetric initial data. The argument presented works for both the inviscid case ν=0 and the supercritical case ν>0 and 0<γ<1. Additionally, we present new proofs of blowup which does not use free energy arguments.  相似文献   

16.
Best simultaneous approximation in     
Jos Mendoza  Tijani Pakhrou 《Journal of Approximation Theory》2007,145(2):212-220
Let X be a Banach space, (Ω,Σ,μ) a finite measure space, and L1(μ,X) the Banach space of X-valued Bochner μ-integrable functions defined on Ω endowed with its usual norm. Let us suppose that Σ0 is a sub-σ-algebra of Σ, and let μ0 be the restriction of μ to Σ0. Given a natural number n, let N be a monotonous norm in . It is shown that if X is reflexive then L1(μ0,X) is N-simultaneously proximinal in L1(μ,X) in the sense of Fathi et al. [Best simultaneous approximation in Lp(I,E), J. Approx. Theory 116 (2002), 369–379]. Some examples and remarks related with N-simultaneous proximinality are also given.  相似文献   

17.
Convex-transitivity and function spaces     
Jarno Talponen   《Journal of Mathematical Analysis and Applications》2009,350(2):537-549
It is shown that if X is a convex-transitive Banach space and 1p<∞, then Lp([0,1],X) and are convex-transitive. Here is the closed linear span of the simple functions in the Bochner space L([0,1],X). If H is an infinite-dimensional Hilbert space and C0(L) is convex-transitive, then C0(L,H) is convex-transitive. Some new fairly concrete examples of convex-transitive spaces are provided.  相似文献   

18.
Boundedness of generalized higher commutators of Marcinkiewicz integrals   总被引:1,自引:0,他引:1  
默会霞  陆善镇 《数学物理学报(B辑英文版)》2007,27(4):852-866
Let (b) = (b1,…,bm) be a finite family of locally integrable functions. Then,we introduce generalized higher commutator of Marcinkiwicz integral as follows:μ(b)Ω=(∫∞o|F(b)Ω,t(f)(x)|2et/t)1/2,whereF(b)Ω(f)(x)=1/t∫|x-y|≤tΩ(x-y)/|x-y|n-1m∏j=1(bj(x)-bj(y))f(y)dy.When bj ∈(A)βj, 1≤j≤m, 0<βj<1,m∑j=1βj =β<n, and Ω is homogeneous of degree zero and satisfies the cancelation condition, we prove that μ(b)Ω is bounded from Lp(Rn)to Ls(Rn), where 1 < p < n/β and 1/s = 1/p -β/n. Moreover, if Ω also satisfies some Lq-Dini condition, then μ(b)Ω is bounded from Lp(Rn) to (F)β,∞p(Rn) and on certain Hardy spaces. The article extends some known results.  相似文献   

19.
A regularity criterion for the solutions of 3D Navier-Stokes equations     
Xicheng Zhang 《Journal of Mathematical Analysis and Applications》2008,346(1):336-339
In terms of two partial derivatives of any two components of velocity fields, we give a new criterion for the regularity of solutions of the Navier-Stokes equation in R3. More precisely, let u=(u1,u2,u3) be a weak solution in (0,TR3. Then u becomes a classical solution if any two functions of 1u1, 2u2 and 3u3 belong to Lθ(0,T;Lr(R3)) provided with , .  相似文献   

20.
Sub-criticality of non-local Schrödinger systems with antisymmetric potentials and applications to half-harmonic maps     
Francesca Da Lio  Tristan Rivière 《Advances in Mathematics》2011,(3):1862
We consider non-local linear Schrödinger-type critical systems of the type(1) where Ω is antisymmetric potential in L2(R,so(m)), v is an Rm valued map and Ωv denotes the matrix multiplication. We show that every solution vL2(R,Rm) of (1) is in fact in , for every 2?p<+∞, in other words, we prove that the system (1) which is a-priori only critical in L2 happens to have a subcritical behavior for antisymmetric potentials. As an application we obtain the regularity of weak 1/2-harmonic maps into C2 compact sub-manifolds without boundary.  相似文献   

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1.
M. Ramos  H. Tavares  W. Zou   《Advances in Mathematics》2009,222(6):2173-2195
In 1988, A. Bahri and P.L. Lions [A. Bahri, P.L. Lions, Morse-index of some min–max critical points. I. Application to multiplicity results, Comm. Pure Appl. Math. 41 (1988) 1027–1037] studied the following elliptic problem:
where Ω is a bounded smooth domain of , 2<p<(2N−2)/(N−2) and f(x,u) is not assumed to be odd in u. They proved the existence of infinitely many solutions under an appropriate growth restriction on f. In the present paper, we improve this result by showing that under the same growth assumption on f the problem admits in fact infinitely many sign-changing solutions. In addition we derive an estimate on the number of their nodal domains. We also deal with the corresponding fourth order equation Δ2u=|u|p−2u+f(x,u) with both Dirichlet and Navier boundary conditions, as well as with strongly coupled elliptic systems.  相似文献   

2.
The aim of this paper is to discuss the positive solutions of the p-Laplace problem
−div(|u|p−2u)+g(u)|u|p=λuq,
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