where λR+:=[0,∞), and satisfies the conditions
We prove a strong maximum principle for the linear operator defined by the left-hand side of (1), and use this to show that for every solution (λ,u) of (1)–(2), u is positive on Ta,b . In addition, we show that there exists λmax>0 (possibly λmax=∞), such that, if 0λ<λmax then (1)–(2) has a unique solution u(λ), while if λλmax then (1)–(2) has no solution. The value of λmax is characterised as the principal eigenvalue of an associated weighted eigenvalue problem (in this regard, we prove a general existence result for such eigenvalues for problems with general, nonnegative weights).  相似文献   

2.
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   

3.
Eigenvalue problems of nonhomogeneous semilinear elliptic equations in Esteban–Lions domains with holes     
Tsing-San Hsu  Huei-Li Lin 《Journal of Mathematical Analysis and Applications》2007,330(2):1273-1292
In this article, we consider the following eigenvalue problems
('∗
λ' render=n">
where λ>0, N2 and is the upper semi-strip domain with a hole in . Under some suitable conditions on f and h, we show that there exists a positive constant λ* such that Eq. (*)λ has at least two solutions if λ(0,λ*), a unique positive solution if λ=λ*, and no positive solution if λ>λ*. We also obtain some further properties of the positive solutions of (*)λ.  相似文献   

4.
Multiplicity of positive solutions for a class of quasilinear nonhomogeneous Neumann problems   总被引:1,自引:0,他引:1  
Emerson A.M. Abreu  Joo Marcos do   Everaldo S. Medeiros 《Nonlinear Analysis: Theory, Methods & Applications》2005,60(8):1443-1471
In this paper we study the existence, nonexistence and multiplicity of positive solutions for nonhomogeneous Neumann boundary value problem of the type
where Ω is a bounded domain in with smooth boundary, 1<p<n,Δpu=div(|u|p-2u) is the p-Laplacian operator, , , (x)0 and λ is a real parameter. The proofs of our main results rely on different methods: lower and upper solutions and variational approach.  相似文献   

5.
Weighted Hardy inequalities     
D.E. Edmunds  R. Hurri-Syrjnen 《Journal of Mathematical Analysis and Applications》2005,310(2):424-435
For bounded Lipschitz domains D in it is known that if 1<p<∞, then for all β[0,β0), where β0=p−1>0, there is a constant c<∞ with
for all . We show that if D is merely assumed to be a bounded domain in that satisfies a Whitney cube-counting condition with exponent λ and has plump complement, then the same inequality holds with β0 now taken to be . Further, we extend the known results (see [H. Brezis, M. Marcus, Hardy's inequalities revisited, Dedicated to Ennio De Giorgi, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 25 (1997–1998) 217–237; M. Hoffmann-Ostenhof, T. Hoffmann-Ostenhof, A. Laptev, A geometrical version of Hardy's inequality, J. Funct. Anal. 189 (2002) 537–548; J. Tidblom, A geometrical version of Hardy's inequality for W1,p(Ω), Proc. Amer. Math. Soc. 132 (2004) 2265–2271]) concerning the improved Hardy inequality
c=c(n,p), by showing that the class of domains for which the inequality holds is larger than that of all bounded convex domains.  相似文献   

6.
Characterizations of spaces in the unit ball of     
Songxiao Li  Hasi Wulan   《Journal of Mathematical Analysis and Applications》2009,360(2):689-696
Let B denote the unit ball of . For 0<p<∞, the holomorphic function spaces Qp and Qp,0 on the unit ball of are defined as
and
In this paper, we give some derivative-free, mixture and oscillation characterizations for Qp and Qp,0 spaces in the unit ball of .  相似文献   

7.
Markov-Type Inequalities for Products of Müntz Polynomials     
Tams Erdlyi 《Journal of Approximation Theory》2001,112(2):171
Let Λ(λj)j=0 be a sequence of distinct real numbers. The span of {xλ0xλ1, …, xλn} over is denoted by Mn(Λ)span{xλ0xλ1, …, xλn}. Elements of Mn(Λ) are called Müntz polynomials. The principal result of this paper is the following Markov-type inequality for products of Müntz polynomials. T 2.1. LetΛ(λj)j=0andΓ(γj)j=0be increasing sequences of nonnegative real numbers. Let

Then

18(n+m+1)(λnm).In particular ,

Under some necessary extra assumptions, an analog of the above Markov-type inequality is extended to the cases when the factor x is dropped, and when the interval [0, 1] is replaced by [ab](0, ∞).  相似文献   

8.
Existence of multiple positive solutions for nonlinear m-point boundary-value problems     
Chuan-zhi Bai  Jin-xuan Fang   《Applied mathematics and computation》2003,140(2-3):297-305
In this paper, we afford some sufficient conditions to guarantee the existence of multiple positive solutions for the nonlinear m-point boundary-value problem for the one-dimensional p-Laplacian
p(u))+f(t,u)=0, t(0,1),
  相似文献   

9.
Free resolutions in multivariable operator theory     
Devin C. V. Greene 《Journal of Functional Analysis》2003,200(2):429-450
Let be the complex polynomial ring in d variables. A contractive -module is Hilbert space equipped with an action such that for any ,
||z1ξ1+z2ξ++zdξd||2||ξ1||2+||ξ2||2++||ξd||2.
Such objects have been shown to be useful for modeling d-tuples of mutually commuting operators acting on a Hilbert space. There is a subclass of the category of contractive modules whose members play the role of free objects. Given a contractive -module, one can construct a free resolution, i.e. an exact sequence of partial isometries of the following form:
(*)
where is a free module for each i0. The notion of a localization of a free resolution will be defined, in which for each λBd there is a vector space complex of linear maps derived from (*):
We shall show that the homology of this complex is isomorphic to the homology of the Koszul complex of the d-tuple (1,2,…,d), of where i is the ith coordinate function of a Möbius transform on Bd such that (λ)=0.  相似文献   

10.
11.
Entropy numbers of Sobolev embeddings of radial Besov spaces     
Thomas Kühn  Hans-Gerd Leopold  Winfried Sickel  Leszek Skrzypczak   《Journal of Approximation Theory》2003,121(2):244-268
Let be the radial subspace of the Besov space . We prove the independence of the asymptotic behavior of the entropy numbers
from the difference s0s1 as long as the embedding itself is compact. In fact, we shall show that
This is in a certain contrast to earlier results on entropy numbers in the context of Besov spaces Bp,qs(Ω) on bounded domains Ω.  相似文献   

12.
Second-order boundary value problems with nonhomogeneous boundary conditions (II)     
Lingju Kong  Qingkai Kong   《Journal of Mathematical Analysis and Applications》2007,330(2):1393-1411
Sufficient conditions are given for the existence of solutions of the following nonlinear boundary value problem with nonhomogeneous multi-point boundary condition
We prove that the whole plane is divided by a “continuous decreasing curve” Γ into two disjoint connected regions ΛE and ΛN such that the above problem has at least one solution for (λ1,λ2)Γ, has at least two solutions for (λ1,λ2)ΛEΓ, and has no solution for (λ1,λ2)ΛN. We also find explicit subregions of ΛE where the above problem has at least two solutions and two positive solutions, respectively.  相似文献   

13.
Critical points of the regular part of the harmonic Green function with Robin boundary condition     
Juan Dvila  Micha&#x; Kowalczyk  Marcelo Montenegro 《Journal of Functional Analysis》2008,255(5):1057-1101
In this paper we consider the Green function for the Laplacian in a smooth bounded domain with Robin boundary condition
and its regular part Sλ(x,y), where b>0 is smooth. We show that in general, as λ→∞, the Robin function Rλ(x)=Sλ(x,x) has at least 3 critical points. Moreover, in the case bconst we prove that Rλ has critical points near non-degenerate critical points of the mean curvature of the boundary, and when bconst there are critical points of Rλ near non-degenerate critical points of b.  相似文献   

14.
Blow-up analysis for a system of heat equations with nonlinear flux which obey different laws   总被引:1,自引:0,他引:1  
Xianfa Song   《Nonlinear Analysis: Theory, Methods & Applications》2008,69(7):1971-1980
We consider a system of heat equations ut=Δu and vt=Δv in Ω×(0,T) completely coupled by nonlinear boundary conditions
We prove that the solutions always blow up in finite time for non-zero and non-negative initial values. Also, the blow-up only occurs on Ω with
for p,q>0, 0≤α<1 and 0≤β<p.  相似文献   

15.
Asymptotic behavior of Solutions for Hénon systems with nearly critical exponent     
Haiyang He  Jianfu Yang   《Journal of Mathematical Analysis and Applications》2008,347(2):459-471
We consider in this paper the problem
(0.1)
where Ω is the unit ball in centered at the origin, 0α<pN, β>0, N8, p>1, qε>1. Suppose qεq>1 as ε→0+ and qε,q satisfy respectively
we investigate the asymptotic behavior of the ground state solutions (uε,vε) of (0.1) as ε→0+. We show that the ground state solutions concentrate at a point, which is located at the boundary. In addition, the ground state solution is non-radial provided that ε>0 is small.  相似文献   

16.
The asymptotic behavior of the solutions of the Cauchy problem generated by -accretive operators     
Jesús García-Falset 《Journal of Mathematical Analysis and Applications》2005,310(2):594-608
The purpose of this paper is to study the asymptotic behavior of the solutions of certain type of differential inclusions posed in Banach spaces. In particular, we obtain the abstract result on the asymptotic behavior of the solution of the boundary value problem
where Ω is a bounded open domain in with smooth boundary ∂Ω, f(t,x) is a given L1-function on ]0,∞[×Ω, γ1 and 1p<∞. Δp represents the p-Laplacian operator, is the associated Neumann boundary operator and β a maximal monotone graph in with 0β(0).  相似文献   

17.
Dissecting the Stanley partition function     
Alexander Berkovich  Frank G. Garvan 《Journal of Combinatorial Theory, Series A》2005,112(2):277-291
Let p(n) denote the number of unrestricted partitions of n. For i=0, 2, let pi(n) denote the number of partitions π of n such that . Here denotes the number of odd parts of the partition π and π is the conjugate of π. Stanley [Amer. Math. Monthly 109 (2002) 760; Adv. Appl. Math., to appear] derived an infinite product representation for the generating function of p0(n)-p2(n). Recently, Swisher [The Andrews–Stanley partition function and p(n), preprint, submitted for publication] employed the circle method to show that
(i)
and that for sufficiently large n
(ii)
In this paper we study the even/odd dissection of the Stanley product, and show how to use it to prove (i) and (ii) with no restriction on n. Moreover, we establish the following new result:
Two proofs of this surprising inequality are given. The first one uses the Göllnitz–Gordon partition theorem. The second one is an immediate corollary of a new partition inequality, which we prove in a combinatorial manner. Our methods are elementary. We use only Jacobi's triple product identity and some naive upper bound estimates.  相似文献   

18.
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   

19.
Lower bounds for the merit factors of trigonometric polynomials from Littlewood classes     
Peter Borwein  Tams Erdlyi 《Journal of Approximation Theory》2003,125(2):190-197
With the notation ,
we prove the following result.Theorem 1. Assume that p is a trigonometric polynomial of degree at most n with real coefficients that satisfies
||p||L2(K)An1/2 and ||p′||L2(K)Bn3/2.
Then
M4(p)−M2(p)M2(p)
with
We also prove that
and
M2(p)−M1(p)10−31M2(p)
for every , where denotes the collection of all trigonometric polynomials of the form
  相似文献   

20.
MULTIPLE SOLUTIONS FOR SCHR(O)DINGER EQUATIONS WITH MAGNETIC FIELD     
彭超权  杨健夫 《数学物理学报(B辑英文版)》2009,29(5):1323-1340
The authors consider the semilinear SchrSdinger equation
-△Au+Vλ(x)u= Q(x)|u|γ-2u in R^N,
where 1 〈 γ 〈 2* and γ≠ 2, Vλ= V^+ -λV^-. Exploiting the relation between the Nehari manifold and fibrering maps, the existence of nontrivial solutions for the problem is discussed.  相似文献   

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Let TR be a time-scale, with a=infT, b=supT. We consider the nonlinear boundary value problem
(2)
(4)
u(a)=u(b)=0,
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