共查询到20条相似文献,搜索用时 31 毫秒
1.
Jorge García-Melián Julio D. Rossi José C. Sabina de Lis 《Calculus of Variations and Partial Differential Equations》2008,31(2):187-204
In this work we consider the behaviour for large values of p of the unique positive weak solution u
p
to Δ
p
u = u
q
in Ω, u = +∞ on , where q > p − 1. We take q = q(p) and analyze the limit of u
p
as p → ∞. We find that when q(p)/p → Q the behaviour strongly depends on Q. If 1 < Q < ∞ then solutions converge uniformly in compacts to a viscosity solution of with u = +∞ on . If Q = 1 then solutions go to ∞ in the whole Ω and when Q = ∞ solutions converge to 1 uniformly in compact subsets of Ω, hence the boundary blow-up is lost in the limit. 相似文献
2.
Juan Dávila Manuel del Pino Monica Musso Juncheng Wei 《Calculus of Variations and Partial Differential Equations》2008,32(4):453-480
We consider the elliptic problem Δu + u
p
= 0, u > 0 in an exterior domain, under zero Dirichlet and vanishing conditions, where is smooth and bounded in , N ≥ 3, and p is supercritical, namely . We prove that this problem has infinitely many solutions with slow decay
at infinity. In addition, a solution with fast decay
O(|x|2-N
) exists if p is close enough from above to the critical exponent. 相似文献
3.
We study the boundary value problem in Ω, u = 0 on ∂Ω, where Ω is a bounded domain in with smooth boundary, λ is a positive real number, and the continuous functions p
1, p
2, and q satisfy 1 < p
2(x) < q(x) < p
1(x) < N and for any . The main result of this paper establishes the existence of two positive constants λ0 and λ1 with λ0 ≤ λ1 such that any is an eigenvalue, while any is not an eigenvalue of the above problem. 相似文献
4.
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
5.
Berardino Sciunzi 《NoDEA : Nonlinear Differential Equations and Applications》2007,14(3-4):315-334
We consider the Dirichlet problem in Ω with zero Dirichlet boundary conditions. We prove local summability properties of and we exploit these results to give geometric characterizations of the critical set . We extend to the case of changing sign nonlinearities some results known in the case f(s) > 0 for s > 0.
Berardino Sciunzi: Supported by MURST, Project “Metodi Variazionali ed Equazioni Differenziali Non Lineari” 相似文献
6.
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
7.
Changshou Lin Liping Wang Juncheng Wei 《Calculus of Variations and Partial Differential Equations》2007,30(2):153-182
We consider the following critical elliptic Neumann problem on , Ω; being a smooth bounded domain in is a large number. We show that at a positive nondegenerate local minimum point Q
0 of the mean curvature (we may assume that Q
0 = 0 and the unit normal at Q
0 is − e
N
) for any fixed integer K ≥ 2, there exists a μ
K
> 0 such that for μ > μ
K
, the above problem has K−bubble solution u
μ concentrating at the same point Q
0. More precisely, we show that u
μ has K local maximum points Q
1μ, ... , Q
K
μ ∈∂Ω with the property that and approach an optimal configuration of the following functional
(*) Find out the optimal configuration that minimizes the following functional: where are two generic constants and φ (Q) = Q
T
G
Q with G = (∇
ij
H(Q
0)).
Research supported in part by an Earmarked Grant from RGC of HK. 相似文献
8.
Kin Ming Hui 《Mathematische Annalen》2007,339(2):395-443
We prove the existence of a unique solution of the following Neumann problem , u > 0, in (a, b) × (0, T), u(x, 0) = u
0(x) ≥ 0 in (a, b), and , where if m < 0, if m = 0, and
m≤ 0, , and the case −1 < m ≤ 0, , for some constant p > 1 − m. We also obtain a similar result in higher dimensions. As a corollary we will give a new proof of a result of A. Rodriguez
and J.L. Vazquez on the existence of infinitely many finite mass solutions of the above equation in for any −1 < m ≤ 0. We also obtain the exact decay rate of the solution at infinity. 相似文献
9.
Jorge García-Melián José C. Sabina De Lis Julio D. Rossi 《NoDEA : Nonlinear Differential Equations and Applications》2007,14(5-6):499-525
We deal with positive solutions of Δu = a(x)u
p
in a bounded smooth domain subject to the boundary condition ∂u/∂v = λu, λ a parameter, p > 1. We prove that this problem has a unique positive solution if and only if 0 < λ < σ1 where, roughly speaking, σ1 is finite if and only if |∂Ω ∩ {a = 0}| > 0 and coincides with the first eigenvalue of an associated eigenvalue problem. Moreover, we find the limit profile
of the solution as λ → σ1.
Supported by DGES and FEDER under grant BFM2001-3894 (J. García-Melián and J. Sabina) and ANPCyT PICT No. 03-05009 (J. D.
Rossi). J.D. Rossi is a member of CONICET. 相似文献
10.
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
11.
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
12.
Xianling Fan Shao-Gao Deng 《NoDEA : Nonlinear Differential Equations and Applications》2009,16(2):255-271
We study the existence and multiplicity of positive solutions for the inhomogeneous Neumann boundary value problems involving
the p(x)-Laplacian of the form
where Ω is a bounded smooth domain in , and p(x) > 1 for with and φ ≢ 0 on ∂Ω. Using 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 = λ*, has at least one positive solution if λ = λ*, and has no positive solution if λ = λ*. To prove the result we establish a special strong comparison principle for the Neumann problems.
The research was supported by the National Natural Science Foundation of China 10371052,10671084). 相似文献
13.
Daniele Castorina Pierpaolo Esposito Berardino Sciunzi 《Calculus of Variations and Partial Differential Equations》2009,34(3):279-306
The behavior of the “minimal branch” is investigated for quasilinear eigenvalue problems involving the p-Laplace operator, considered in a smooth bounded domain of , and compactness holds below a critical dimension N
#. The nonlinearity f(u) lies in a very general class and the results we present are new even for p = 2. Due to the degeneracy of p-Laplace operator, for p ≠ 2 it is crucial to define a suitable notion of semi-stability: the functional space we introduce in the paper seems to
be the natural one and yields to a spectral theory for the linearized operator. For the case p = 2, compactness is also established along unstable branches satisfying suitable spectral information. The analysis is based
on a blow-up argument and stronger assumptions on the nonlinearity f(u) are required.
Authors are partially supported by MIUR, project “Variational methods and nonlinear differential equations”. 相似文献
14.
Piotr Kot 《Czechoslovak Mathematical Journal》2009,59(2):371-379
We solve the following Dirichlet problem on the bounded balanced domain with some additional properties: For p > 0 and a positive lower semi-continuous function u on ∂Ω with u(z) = u(λ z) for |λ| = 1, z ∈ ∂Ω we construct a holomorphic function f ∈ (Ω) such that for z ∈ ∂Ω, where = {λ ∈ ℂ: |λ| < 1}.
相似文献
15.
Nicolas Th. Varopoulos 《Milan Journal of Mathematics》2007,75(1):1-60
In this paper I consider a class of non-standard singular integrals motivated by potential theoretic and probabilistic considerations.
The probabilistic applications, which are by far the most interesting part of this circle of ideas, are only outlined in Section
1.5: They give the best approximation of the solution of the classical Dirichlet problem in a Lipschitz domain by the corresponding solution by finite differences.
The potential theoretic estimate needed for this gives rise to a natural duality between the L
p
functions on the boundary ∂Ω and a class of functions A on Ω that was first considered by Dahlberg. The actual duality is given by ∫Ω
S f(x)A(x)dx = (f, A) where S f(x) = ∫∂Ω |x − y|1−n
f(y)dy is the Newtonian potential.
We can identify the upper half Lipschitz space with in the obvious way and express for an appropriate kernel K. It is the boundedness properties of the above (for , ) that is the essential part of this work. This relates with more classical (but still “rough”) singular integrals that have
been considered by Christ and Journé.
Lecture held in the Seminario Matematico e Fisico on March 14, 2005
Received: April 2007 相似文献
16.
Mihai Mihăilescu 《Czechoslovak Mathematical Journal》2008,58(1):155-172
We study the boundary value problem in Ω, u = 0 on ∂Ω, where Ω is a smooth bounded domain in ℝ
N
. Our attention is focused on two cases when , where m(x) = max{p
1(x), p
2(x)} for any x ∈ or m(x) < q(x) < N · m(x)/(N − m(x)) for any x ∈ . In the former case we show the existence of infinitely many weak solutions for any λ > 0. In the latter we prove that if λ is large enough then there exists a nontrivial weak solution. Our approach relies on the variable exponent theory of generalized
Lebesgue-Sobolev spaces, combined with a ℤ2-symmetric version for even functionals of the Mountain Pass Theorem and some adequate variational methods. 相似文献
17.
Tiziana Giorgi Robert Smits 《Zeitschrift für Angewandte Mathematik und Physik (ZAMP)》2008,59(4):600-618
We consider the principal eigenvalue λ
1Ω(α) corresponding to Δu = λ (α) u in on ∂Ω, with α a fixed real, and a C
0,1 bounded domain. If α > 0 and small, we derive bounds for λ
1Ω(α) in terms of a Stekloff-type eigenvalue; while for α > 0 large we study the behavior of its growth in terms of maximum curvature.
We analyze how domain monotonicity of the principal eigenvalue depends on the geometry of the domain, and prove that domains
which exhibit domain monotonicity for every α are calibrable. We conjecture that a domain has the domain monotonicity property for some α if and only if it is calibrable.
Robert Smits: This author was partially supported by a grant of the National Security Agency, grant #H98230-05-1-0060. 相似文献
18.
Marc Briane Juan Casado-Díaz 《Calculus of Variations and Partial Differential Equations》2007,29(4):455-479
In this paper, we study the asymptotic behaviour of a given equicoercive sequence of diffusion energies F
n
, , defined in L
2(Ω), for a bounded open subset Ω of . We prove that, contrary to the dimension three (or greater), the Γ-limit of any convergent subsequence of F
n
is still a diffusion energy. We also provide an explicit representation formula of the Γ-limit when its domains contains
the regular functions with compact support in Ω. This compactness result is based on the uniform convergence satisfied by
some minimizers of the equicoercive sequence F
n
, which is specific to the dimension two. The compactness result is applied to the period framework, when the energy density
is a highly oscillating sequence of equicoercive matrix-valued functions. So, we give a definitive answer to the question
of the asymptotic behaviour of periodic conduction problems under the only assumption of equicoerciveness for the two-dimensional
conductivity. 相似文献
19.
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
20.
Jorge García-Melián 《Zeitschrift für Angewandte Mathematik und Physik (ZAMP)》2009,60(4):594-607
In this paper we consider the boundary blow-up problem Δpu = a(x)uq in a smooth bounded domain Ω of , with u = +∞ on ∂Ω. Here is the well-known p-Laplacian operator with p > 1, q > p − 1, and a(x) is a nonnegative weight function which can be singular on ∂Ω. Our results include existence, uniqueness and exact boundary
behavior of positive solutions.
相似文献