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1.
Zhaoli Liu Jiabao Su Zhi-Qiang Wang 《Calculus of Variations and Partial Differential Equations》2009,35(4):463-480
In this paper, we study existence of nontrivial solutions to the elliptic equation
and to the elliptic system
where Ω is a bounded domain in with smooth boundary ∂Ω, , f (x, 0) = 0, with m ≥ 2 and . Nontrivial solutions are obtained in the case in which the nonlinearities have linear growth. That is, for some c > 0, for and , and for and , where I
m
is the m × m identity matrix. In sharp contrast to the existing results in the literature, we do not make any assumptions at infinity
on the asymptotic behaviors of the nonlinearity f and .
Z. Liu was supported by NSFC(10825106, 10831005). J. Su was supported by NSFC(10831005), NSFB(1082004), BJJW-Project(KZ200810028013)
and the Doctoral Programme Foundation of NEM of China (20070028004). 相似文献
2.
Francesca Alessio Piero Montecchiari 《Calculus of Variations and Partial Differential Equations》2007,30(1):51-83
We consider a class of semilinear elliptic equations of the form
where is a periodic, positive function and is modeled on the classical two well Ginzburg-Landau potential . We show, via variational methods, that if the set of solutions to the one dimensional heteroclinic problem
has a discrete structure, then (0.1) has infinitely many solutions periodic in the variable y and verifying the asymptotic conditions as uniformly with respect to .
Supported by MURST Project ‘Metodi Variazionali ed Equazioni Differenziali Non Lineari’. 相似文献
3.
Yehuda Pinchover Kyril Tintarev 《Calculus of Variations and Partial Differential Equations》2007,28(2):179-201
Let Ω be a domain in , d ≥ 2, and 1 < p < ∞. Fix . Consider the functional Q and its Gateaux derivative Q′ given by If Q ≥ 0 on, then either there is a positive continuous function W such that for all, or there is a sequence and a function v > 0 satisfying Q′ (v) = 0, such that Q(u
k
) → 0, and in . In the latter case, v is (up to a multiplicative constant) the unique positive supersolution of the equation Q′ (u) = 0 in Ω, and one has for Q an inequality of Poincaré type: there exists a positive continuous function W such that for every satisfying there exists a constant C > 0 such that . As a consequence, we prove positivity properties for the quasilinear operator Q′ that are known to hold for general subcritical resp. critical second-order linear elliptic operators. 相似文献
4.
We prove a Γ-convergence result for the family of functionals defined on H
1(Ω) by for a given and a parameter . We show that in either of the two cases, p = 2 or , any limit of the minimizers is an optimal lifting. 相似文献
5.
We consider one-dimensional difference Schr?dinger equations with real analytic function V(x). Suppose V(x) is a small perturbation of a trigonometric polynomial V
0(x) of degree k
0, and assume positive Lyapunov exponents and Diophantine ω. We prove that the integrated density of states is H?lder continuous for any k > 0. Moreover, we show that is absolutely continuous for a.e. ω. Our approach is via finite volume bounds. I.e., we study the eigenvalues of the problem
on a finite interval [1, N] with Dirichlet boundary conditions. Then the averaged number of these Dirichlet eigenvalues which fall into an interval
, does not exceed , k > 0. Moreover, for , this averaged number does not exceed exp , for any . For the integrated density of states of the problem this implies that for any . To investigate the distribution of the Dirichlet eigenvalues of on a finite interval [1, N] we study the distribution of the zeros of the characteristic determinants with complexified phase x, and frozen ω, E. We prove equidistribution of these zeros in some annulus and show also that no more than 2k
0 of them fall into any disk of radius exp. In addition, we obtain the lower bound (with δ > 0 arbitrary) for the separation of the eigenvalues of the Dirichlet eigenvalues over the interval [0, N]. This necessarily requires the removal of a small set of energies.
Received: February 2006, Accepted: December 2007 相似文献
6.
Juan A. Aledo José M. Espinar José A. Gálvez 《Calculus of Variations and Partial Differential Equations》2007,29(3):347-363
We study isometric immersions of surfaces of constant curvature into the homogeneous spaces and . In particular, we prove that there exists a unique isometric immersion from the standard 2-sphere of constant curvature
c > 0 into and a unique one into when c > 1, up to isometries of the ambient space. Moreover, we show that the hyperbolic plane of constant curvature c < −1 cannot be isometrically immersed into or .
J.A. Aledo was partially supported by Ministerio de Education y Ciencia Grant No. MTM2004-02746 and Junta de Comunidades de
Castilla-La Mancha, grant no. PAI-05-034.
J.M. Espinar and J.A. Gálvez were partially supported by Ministerio de Education y Ciencia grant no. MTM2004-02746 and Junta
de Andalucía Grant No. FQM325. 相似文献
7.
Alexander V. Kolesnikov 《Probability Theory and Related Fields》2008,140(1-2):1-17
Let γ be a Gaussian measure on a Suslin space X, H be the corresponding Cameron–Martin space and {e
i
} ⊂ H be an orthonormal basis of H. Suppose that μ
n
= ρ
n
· γ is a sequence of probability measures which converges weakly to a probability measure μ = ρ · γ Consider a sequence of Dirichlet forms , where and . We prove some sufficient conditions for Mosco convergence where . In particular, if X is a Hilbert space, and can be uniformly approximated by finite dimensional conditional expectations for every fixed e
i
, then under broad assumptions Mosco and the distributions of the associated stochastic processes converge weakly. 相似文献
8.
Juncheng Wei Dong Ye Feng Zhou 《Calculus of Variations and Partial Differential Equations》2007,28(2):217-247
We consider the following anisotropic Emden–Fowler equation where is a bounded smooth domain and a(x) is a positive smooth function. We investigate the effect of anisotropic coefficient a(x) on the existence of bubbling solutions. We show that at given local maximum points of a(x), there exists arbitrarily many bubbles. As a consequence, the quantity can approach to
as . These results show a striking difference with the isotropic case [ Constant]. 相似文献
9.
Miguel Ramos Hugo Tavares 《Calculus of Variations and Partial Differential Equations》2008,31(1):1-25
We consider a system of the form , in an open domain of , with Dirichlet conditions at the boundary (if any). We suppose that f and g are power-type non-linearities, having superlinear and subcritical growth at infinity. We prove the existence of positive
solutions and which concentrate, as , at a prescribed finite number of local minimum points of V(x), possibly degenerate. 相似文献
10.
Olivier Druet Emmanuel Hebey 《Calculus of Variations and Partial Differential Equations》2008,31(2):205-230
Let (M, g) be a smooth compact Riemannian n-manifold, n ≥ 3. Let also p ≥ 1 be an integer, and be the vector space of symmetrical p × p real matrix. We consider critical elliptic systems of equations which we write in condensed form as
where , is a p-map, is the Laplace–Beltrami operator acting on p-maps, and 2* is the critical Sobolev exponent. We fully answer the question of getting sharp asymptotics for local minimal
type solutions of such systems. As an application, we prove compactness of minimal type solutions and prove that the result
is sharp by constructing explicit examples where blow-up occurs when the compactness assumptions are not fulfilled. 相似文献
11.
Annalisa Malusa Luigi Orsina 《Calculus of Variations and Partial Differential Equations》2006,27(2):179-202
We study the limit as n goes to +∞ of the renormalized solutions u
n
to the nonlinear elliptic problems
where Ω is a bounded open set of ℝ
N
, N≥ 2, and μ is a Radon measure with bounded variation in Ω. Under the assumption of G-convergence of the operators , defined for , to the operator , we shall prove that the sequence (u
n
) admits a subsequence converging almost everywhere in Ω to a function u which is a renormalized solution to the problem
相似文献
12.
Thomas Bartsch Shuangjie Peng Zhitao Zhang 《Calculus of Variations and Partial Differential Equations》2007,30(1):113-136
We investigate elliptic equations related to the Caffarelli–Kohn–Nirenberg inequalities: and such that . For various parameters α, β and various domains Ω, we establish some existence and non-existence results of solutions in
rather general, possibly degenerate or singular settings. 相似文献
13.
Nikolaos Bournaveas Hua Wang 《NoDEA : Nonlinear Differential Equations and Applications》2009,16(1):131-142
We show how the methods of [6–8] can be used to prove velocity averaging lemmas in hyperbolic Sobolev spaces for the kinetic
transport equation . Here v is allowed to vary in the whole space and the velocity field lies on the unit sphere. We work in dimensions and, in contrast with [6, 8], we allow right-hand sides with velocity derivatives in any direction and not necessarily tangential
to the sphere.
相似文献
14.
Anna Maria Candela Giuliana Palmieri 《Calculus of Variations and Partial Differential Equations》2009,34(4):495-530
The aim of this paper is investigating the existence of one or more critical points of a family of functionals which generalizes
the model problem
in the Banach space , being Ω a bounded domain in . In order to use “classical” theorems, a suitable variant of condition (C) is proved and is decomposed according to a “good” sequence of finite dimensional subspaces.
The authors acknowledge the support of M.I.U.R. (research funds ex 40% and 60%). 相似文献
15.
Soogil Seo 《manuscripta mathematica》2008,127(3):381-396
A circular distribution is a Galois equivariant map ψ from the roots of unity μ
∞ to an algebraic closure of such that ψ satisfies product conditions, for ϵ ∈ μ
∞ and , and congruence conditions for each prime number l and with (l, s) = 1, modulo primes over l for all , where μ
l
and μ
s
denote respectively the sets of lth and sth roots of unity. For such ψ, let be the group generated over by and let be , where U
s
denotes the global units of . We give formulas for the indices and of and inside the circular numbers P
s
and units C
s
of Sinnott over .
This work was supported by the SRC Program of Korea Science and Engineering Foundation (KOSEF) grant funded by the Korea government
(MOST) (No. R11-2007-035-01001-0). This work was supported by the Korea Research Foundation Grant funded by the Korean Government
(MOEHRD, Basic Research Promotion Fund) (KRF-2006-312-C00455). 相似文献
16.
Liselott Flodén Anders Holmbom Marianne Olsson Nils Svanstedt 《Annali dell'Universita di Ferrara》2007,53(2):217-232
Reiterated homogenization is studied for divergence structure parabolic problems of the form . It is shown that under standard assumptions on the function a(y
1,y
2,t,ξ) the sequence of solutions converges weakly in to the solution u of the homogenized problem .
相似文献
17.
Robert Černý 《Calculus of Variations and Partial Differential Equations》2007,28(2):203-216
We compute the relaxation
where
for sequences of functions from
converging strongly in the
-norm to
. 相似文献
18.
Tobias Lamm 《Calculus of Variations and Partial Differential Equations》2006,27(2):125-157
Let be a compact Riemannian surface and let be a compact Riemannian manifold, both without boundary, and assume that N is isometrically embedded into some ℝ
l
. We consider a sequence of critical points of the functional with uniformly bounded energy. We show that this sequence converges weakly in and strongly away from finitely many points to a smooth harmonic map. One can perform a blow-up to show that there separate at most finitely many non-trivial harmonic two-spheres at these finitely many points. Finally we prove the so called energy identity for this approximation in the case that ↪ ℝ
l
.
Mathematics Subject Classification (2000) 58E20, 35J60, 53C43 相似文献
19.
Pigong Han Zhaoxia Liu 《Calculus of Variations and Partial Differential Equations》2007,30(3):315-352
Let Ω be an open bounded domain in with smooth boundary . We are concerned with the critical Neumann problem
where and Q(x) is a positive continuous function on . Using Moser iteration, we give an asymptotic characterization of solutions for (*) at the origin. Under some conditions
on Q, μ, we, by means of a variational method, prove that there exists such that for every , problem (*) has a positive solution and a pair of sign-changing solutions. 相似文献
20.
Jérôme Droniou Juan-Luis Vázquez 《Calculus of Variations and Partial Differential Equations》2009,34(4):413-434
We study the existence and uniqueness of solutions of the convective–diffusive elliptic equation
posed in a bounded domain , with pure Neumann boundary conditions
Under the assumption that with p = N if N ≥ 3 (resp. p > 2 if N = 2), we prove that the problem has a solution if ∫Ω
f
dx = 0, and also that the kernel is generated by a function , unique up to a multiplicative constant, which satisfies a.e. on Ω. We also prove that the equation
has a unique solution for all ν > 0 and the map is an isomorphism of the respective spaces. The study is made in parallel with the dual problem, with equation
The dependence on the data is also examined, and we give applications to solutions of nonlinear elliptic PDE with measure
data and to parabolic problems. 相似文献