共查询到20条相似文献,搜索用时 46 毫秒
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.
Let κ be a cardinal which is measurable after generically adding many Cohen subsets to κ and let be the κ-Rado graph. We prove, for 2 ≤ m < ω, that there is a finite value such that the set [κ]
m
can be partitioned into classes such that for any coloring of any of the classes C
i
in fewer than κ colors, there is a copy of in such that is monochromatic. It follows that , that is, for any coloring of with fewer than κ colors there is a copy of such that has at most colors. On the other hand, we show that there are colorings of such that if is any copy of then for all , and hence . We characterize as the cardinality of a certain finite set of types and obtain an upper and a lower bound on its value. In particular, and for m > 2 we have where r
m
is the corresponding number of types for the countable Rado graph.
Research of M. Džamonja and J. A. Larson were partially supported by Engineering and Physical Sciences Research Council and
research of W. J. Mitchell was partly supported by grant number DMS 0400954 from the United States National Science Foundation. 相似文献
3.
New variational principles based on the concept of anti-selfdual (ASD) Lagrangians were recently introduced in “AIHP-Analyse
non linéaire, 2006”. We continue here the program of using such Lagrangians to provide variational formulations and resolutions
to various basic equations and evolutions which do not normally fit in the Euler-Lagrange framework. In particular, we consider
stationary boundary value problems of the form as well ass dissipative initial value evolutions of the form where is a convex potential on an infinite dimensional space, A is a linear operator and is any scalar. The framework developed in the above mentioned paper reformulates these problems as and respectively, where is an “ASD” vector field derived from a suitable Lagrangian L. In this paper, we extend the domain of application of this approach by establishing existence and regularity results under
much less restrictive boundedness conditions on the anti-selfdual Lagrangian L so as to cover equations involving unbounded operators. Our main applications deal with various nonlinear boundary value
problems and parabolic initial value equations governed by transport operators with or without a diffusion term.
Nassif Ghoussoub research was partially supported by a grant from the Natural Sciences and Engineering Research Council of
Canada. The author gratefully acknowledges the hospitality and support of the Centre de Recherches Mathématiques in Montréal
where this work was initiated.
Leo Tzou’s research was partially supported by a doctoral postgraduate scholarship from the Natural Science and Engineering
Research Council of Canada. 相似文献
4.
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
5.
Consider a smooth bounded domain , and the Navier–Stokes system in with initial value and external force f = div F, where , are so-called Serrin exponents. It is an important question what is the optimal (weakest possible) initial value condition
in order to obtain a unique strong solution in some initial interval [0, T), . Up to now several sufficient conditions on u
0 are known which need not be necessary. Our main result, see Theorem 1.1, shows that the condition , A denotes the Stokes operator, is sufficient and necessary for the existence of such a strong solution u. In particular, if , , then any weak solution u in the usual sense does not satisfy Serrin’s condition for each 0 < T ≤ ∞.
相似文献
6.
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
7.
Mirjana Stojanović 《Acta Appl Math》2006,92(1):1-14
We consider several types of nonlinear parabolic equations with singular like potential and initial data. To prove the existence-uniqueness theorems we employ regularized derivatives. As a framework we use Colombeau space
and Colombeau vector space
相似文献
8.
Pairs of numerically satisfactory solutions as for the three-term recurrence relations satisfied by the families of functions , , are given. It is proved that minimal solutions always exist, except when and z is in the positive or negative real axis, and that is minimal as whenever . The minimal solution is identified for any recurrence direction, that is, for any integer values of and . When the confluent limit , with fixed, is the main tool for identifying minimal solutions together with a connection formula; for , is the main tool to be considered. 相似文献
9.
Elena I. Kaikina Leonardo Guardado-Zavala Hector F. Ruiz-Paredes Jesus A. Mendez Navarro 《NoDEA : Nonlinear Differential Equations and Applications》2009,16(1):63-77
We study nonlinear nonlocal equations on a half-line in the critical case
where . The linear operator is a pseudodifferential operator defined by the inverse Laplace transform with dissipative symbol , the number . The aim of this paper is to prove the global existence of solutions to the inital-boundary value problem (0.1) and to find
the main term of the large time asymptotic representation of solutions in the critical case.
相似文献
10.
Nicola Garofalo 《manuscripta mathematica》2008,126(3):353-373
We prove some new a priori estimates for H
2-convex functions which are zero on the boundary of a bounded smooth domain Ω in a Carnot group . Such estimates are global and are geometric in nature as they involve the horizontal mean curvature of ∂Ω. As a consequence of our bounds we show that if has step two, then for any smooth H
2-convex function in vanishing on ∂Ω one has
.
Supported in part by NSF Grant DMS-07010001. 相似文献
11.
We consider the problem
where Ω is a bounded smooth domain in , 1 < p< + ∞ if N = 2, if N ≥ 3 and ε is a parameter. We show that if the mean curvature of ∂Ω is not constant then, for ε small enough, such a problem
has always a nodal solution u
ε with one positive peak and one negative peak on the boundary. Moreover, and converge to and , respectively, as ε goes to zero. Here, H denotes the mean curvature of ∂Ω.
Moreover, if Ω is a ball and , we prove that for ε small enough the problem has nodal solutions with two positive peaks on the boundary and arbitrarily
many negative peaks on the boundary.
The authors are supported by the M.I.U.R. National Project “Metodi variazionali e topologici nello studio di fenomeni non
lineari”. 相似文献
12.
Arrigo Cellina Mihai Vornicescu 《Calculus of Variations and Partial Differential Equations》2009,35(2):263-270
In this paper we establish an existence and regularity result for solutions to the problem
for boundary data that are constant on each connected component of the boundary of Ω. The Lagrangean L belongs to a class that contains both extended valued Lagrangeans and Lagrangeans with linear growth. Regularity means that
the solution is Lipschitz continuous and that, in addition, is bounded. 相似文献
13.
14.
Jean Bourgain 《Geometric And Functional Analysis》2009,18(5):1477-1502
The main result of this paper is an exponential sum bound in prime fields for multilinear expressions of the type under nearly optimal conditions on . It provides the expected generalization of the well-known inequality for r = 2. We also establish a new result on Gauss sums for multiplicative subgroups H of , obtaining a nontrivial estimate provided . This is a further improvement on [BGK].
Received: May 2007, Revision: October 2007, Accepted: October 2007 相似文献
15.
In this paper, we prove that if is a radially symmetric, sign-changing stationary solution of the nonlinear heat equation
in the unit ball of , N ≥ 3, with Dirichlet boundary conditions, then the solution of (NLH) with initial value blows up in finite time if |λ − 1| > 0 is sufficiently small and if α is subcritical and sufficiently close to 4/(N − 2).
F. Dickstein was partially supported by CNPq (Brazil). 相似文献
16.
We study the semiflow defined by a semilinear parabolic equation with a singular square potential . It is known that the Hardy-Poincaré inequality and its improved versions, have a prominent role on the definition of the
natural phase space. Our study concerns the case 0 < μ ≤ μ*, where μ* is the optimal constant for the Hardy-Poincaré inequality. On a bounded domain of , we justify the global bifurcation of nontrivial equilibrium solutions for a reaction term f(s) = λs − |s|2γ
s, with λ as a bifurcation parameter. We remark some qualitative differences of the branches in the subcritical case μ < μ* and the critical case μ = μ*. The global bifurcation result is used to show that any solution , initiating form initial data tends to the unique nonnegative equilibrium. 相似文献
17.
We prove regularity results for solutions to a class of quasilinear elliptic equations in divergence form in the Heisenberg
group . The model case is the non-degenerate p-Laplacean operator where , and p is not too far from 2. 相似文献
18.
Marcelo M. Cavalcanti Valéria N. Domingos Cavalcanti Ryuichi Fukuoka Daniel Toundykov 《Journal of Evolution Equations》2009,9(1):143-169
This paper is devoted to the study of uniform energy decay rates of solutions to the wave equation with Cauchy–Ventcel boundary
conditions:
where Ω is a bounded domain of (n ≥ 2) having a smooth boundary , such that with , being closed and disjoint. It is known that if a(x) = 0 then the uniform exponential stability never holds even if a linear frictional feedback is applied to the entire boundary of the domain [see, for instance, Hemmina (ESAIM, Control Optim Calc Var 5:591–622, 2000, Thm. 3.1)]. Let be a smooth function; define ω
1 to be a neighbourhood of , and subdivide the boundary into two parts: and . Now, let ω
0 be a neighbourhood of . We prove that if a(x) ≥ a
0 > 0 on the open subset and if g is a monotone increasing function satisfying k|s| ≤ |g(s)| ≤ K|s| for all |s| ≥ 1, then the energy of the system decays uniformly at the rate quantified by the solution to a certain nonlinear ODE dependent
on the damping [as in Lasiecka and Tataru (Differ Integral Equ 6:507–533, 1993)].
Research of Marcelo M. Cavalcanti was partially supported by the CNPq Grant 300631/2003-0.
Research of Valéria N. Domingos Cavalcanti was partially supported by the CNPq Grant 304895/2003-2. 相似文献
19.
Consider the instationary Navier–Stokes system in a smooth bounded domain with vanishing force and initial value . Since the work of Kiselev and Ladyzhenskaya (Am. Math. Soc. Transl. Ser. 2 24:79–106, 1963) there have been found several
conditions on u
0 to prove the existence of a unique strong solution with u(0) = u
0 in some time interval [0, T), 0 < T ≤ ∞, where the exponents 2 < s < ∞, 3 < q < ∞ satisfy . Indeed, such conditions could be weakened step by step, thus enlarging the corresponding solution classes. Our aim is to
prove the following optimal result with the weakest possible initial value condition and the largest possible solution class:
Given u
0, q, s as above and the Stokes operator A
2, we prove that the condition is necessary and sufficient for the existence of such a local strong solution u. The proof rests on arguments from the recently developed theory of very weak solutions. 相似文献
20.
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