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1.
An Application of a Mountain Pass Theorem 总被引:3,自引:0,他引:3
We are concerned with the following Dirichlet problem:
−Δu(x) = f(x, u), x∈Ω, u∈H
1
0(Ω), (P)
where f(x, t) ∈C (×ℝ), f(x, t)/t is nondecreasing in t∈ℝ and tends to an L
∞-function q(x) uniformly in x∈Ω as t→ + ∞ (i.e., f(x, t) is asymptotically linear in t at infinity). In this case, an Ambrosetti-Rabinowitz-type condition, that is, for some θ > 2, M > 0,
0 > θF(x, s) ≤f(x, s)s, for all |s|≥M and x∈Ω, (AR)
is no longer true, where F(x, s) = ∫
s
0
f(x, t)dt. As is well known, (AR) is an important technical condition in applying Mountain Pass Theorem. In this paper, without assuming
(AR) we prove, by using a variant version of Mountain Pass Theorem, that problem (P) has a positive solution under suitable
conditions on f(x, t) and q(x). Our methods also work for the case where f(x, t) is superlinear in t at infinity, i.e., q(x) ≡ +∞.
Received June 24, 1998, Accepted January 14, 2000. 相似文献
2.
Extending the problem of determining Ramsey numbers Erdős and Rogers introduced the following function. For given integers
2 ≤ s < t let f
s,t
(n) = min{max{|S|: S ⊆ V (H) and H[S] contains no K
s
}}, where the minimum is taken over all K
t
-free graphs H of order n. This function attracted a considerable amount of attention but despite that, the gap between the lower and upper bounds
is still fairly wide. For example, when t=s+1, the best bounds have been of the form Ω(n
1/2+o(1)) ≤ f
s,s+1(n) ≤ O(n
1−ɛ(s)), where ɛ(s) tends to zero as s tends to infinity. In this paper we improve the upper bound by showing that f
s,s+1(n) ≤ O(n
2/3). Moreover, we show that for every ɛ > 0 and sufficiently large integers 1 ≪ k ≪ s, Ω(n
1/2−ɛ
) ≤ f
s,s+k
(n) ≤ O(n
1/2+ɛ
. In addition, we also discuss some connections between the function f
s,t
and vertex Folkman numbers. 相似文献
3.
Explosive solutions of elliptic equations with absorption and nonlinear gradient term 总被引:2,自引:0,他引:2
Marius Ghergu Constantin Niculescu Vicenţiu Rădulescu 《Proceedings Mathematical Sciences》2002,112(3):441-451
Letf be a non-decreasing C1-function such that
andF(t)/f
2
a(t)→ 0 ast → ∞, whereF(t)=∫
0
t
f(s) ds anda ∈ (0, 2]. We prove the existence of positive large solutions to the equationΔu +q(x)|Δu|
a
=p(x)f(u) in a smooth bounded domain Ω ⊂RN, provided thatp, q are non-negative continuous functions so that any zero ofp is surrounded by a surface strictly included in Ω on whichp is positive. Under additional hypotheses onp we deduce the existence of solutions if Ω is unbounded. 相似文献
4.
Evangelos A. Latos Dimitrios E. Tzanetis 《NoDEA : Nonlinear Differential Equations and Applications》2010,17(2):137-151
We investigate the behaviour of solution u = u(x, t; λ) at λ = λ* for the non-local porous medium equation ${u_t = (u^n)_{xx} + {\lambda}f(u)/({\int_{-1}^1} f(u){\rm d}x)^2}We investigate the behaviour of solution u = u(x, t; λ) at λ = λ* for the non-local porous medium equation ut = (un)xx + lf(u)/(ò-11 f(u)dx)2{u_t = (u^n)_{xx} + {\lambda}f(u)/({\int_{-1}^1} f(u){\rm d}x)^2} with Dirichlet boundary conditions and positive initial data. The function f satisfies: f(s),−f ′ (s) > 0 for s ≥ 0 and s
n-1
f(s) is integrable at infinity. Due to the conditions on f, there exists a critical value of parameter λ, say λ*, such that for λ > λ* the solution u = u(x, t; λ) blows up globally in finite time, while for λ ≥ λ* the corresponding steady-state problem does not have any solution.
For 0 < λ < λ* there exists a unique steady-state solution w = w(x; λ) while u = u(x, t; λ) is global in time and converges to w as t → ∞. Here we show the global grow-up of critical solution u* = u(x, t; λ*) (u* (x, t) → ∞, as t → ∞ for all x ? (-1,1){x\in(-1,1)}. 相似文献
5.
Alessandra Lunardi 《Israel Journal of Mathematics》1987,60(3):281-314
We find a new construction of the evolution operatorG(t, s) associated to a family {A(t), 0≦t≦T} of generators of analytic semigroups in a Banach spaceX. We study the dependence ofG (t, s) ont ands, and we give regularity results for the solution of the i.v.p.u′(t)=A(t)u(t)+f(t),u(0)=x. 相似文献
6.
In the paper, we obtain the existence of symmetric or monotone positive solutions and establish a corresponding iterative
scheme for the equation (ϕ
p
(u′))′+q(t)f(u) = 0, 0 < t < 1, where ϕ
p
(s):= |s|
p−2
s, p > 1, subject to nonlinear boundary condition. The main tool is the monotone iterative technique. Here, the coefficient q(t) may be singular at t = 0; 1. 相似文献
7.
Letf(s, t; k) be the largest value ofm such that it is possible tok-color the edges of the complete graphK
m
so that everyK
s
⊆K
m
has exactlyt colors occuring on its edges. The main object of this paper is to describe the behavior of the functionf(s,t;k), usually thinking ofs andt fixed, and lettingk become large.
Dedicated to Paul Erdős on his seventieth birthday 相似文献
8.
Der-Shin Chang Yuang-Chin Chiang 《Annals of the Institute of Statistical Mathematics》1980,32(1):275-281
Consider a realization of the process
on the intervalT=[0,1] for functionsf
1(t),f
2(t),...,f
n
(t) inH(R), the reproducing kernel Hilbert space with reproducing kernelR(s,t) onT×T, whereR(s,t)=E[ξ(s)ξt)] is assumed to be continuous and known. Problems of the selection of functions {f
k
(t)}
k=1
n
to be ϕ-optimal design are given, and an unified approach to the solutions ofD-,A-,E- andD
s-optimal design problems are discussed. 相似文献
9.
Given a function f : ℕ→ℝ, call an n-vertex graph f-connected if separating off k vertices requires the deletion of at least f(k) vertices whenever k≤(n−f(k))/2. This is a common generalization of vertex connectivity (when f is constant) and expansion (when f is linear). We show that an f-connected graph contains a cycle of length linear in n if f is any linear function, contains a 1-factor and a 2-factor if f(k)≥2k+1, and contains a Hamilton cycle if f(k)≥2(k+1)2. We conjecture that linear growth of f suffices to imply hamiltonicity. 相似文献
10.
Letf
t
be aC
2 Axiom A dynamical system on a compact manifold satisfying the transversality condition. We prove that ifB
x
(ε,t)=[y: dist (f
s
x,f
s
y)≤ε for all 0≤s≤t], then volB
x
(ε,t) has the order exp(∫
0
t
φ (f
s
x)ds) in the continuous time case and exp (Σ
s
t−1
φ (f
s
x)) in the discrete time case, whereφ is a Holder continuous extension from basic hyperbolic sets of the negative of the differential expansion coefficient in
the unstable direction. An application to the theory of large deviations is given.
Partially supported by US-Israel BSF.
Partially supported by a Darpa grant. 相似文献
11.
Antonio Tineo 《Annali di Matematica Pura ed Applicata》2003,182(2):113-128
In this paper, we prove a result of Ambrosetti–Prodi type for the problem x′=f(t,x)+λx, where f(t,x) is T-periodic in t, f(t,0)≡0 and f(t,x) has “cubic nonlinearities”.
Received: February 4, 2000?Published online: April 14, 2003
RID="*"
ID="*"This paper was partially supported by CDCHT, Universidad de los Andes. 相似文献
12.
Peter Müller 《Israel Journal of Mathematics》1999,109(1):319-337
Letf (X, t)εℚ[X, t] be an irreducible polynomial. Hilbert’s irreducibility theorem asserts that there are infinitely manyt
0εℤ such thatf (X, t
0) is still irreducible. We say thatf (X, t) isgeneral if the Galois group off (X, t) over ℚ(t) is the symmetric group in its natural action. We show that if the degree off with respect toX is a prime ≠ 5 or iff is general of degree ≠ 5, thenf (X, t
0) is irreducible for all but finitely manyt
0εℤ unless the curve given byf (X, t)=0 has infinitely many points (x
0,t
0) withx
0εℚ,t
0εℤ. The proof makes use of Siegel’s theorem about integral points on algebraic curves, and classical results about finite
groups, going back to Burnside, Schur, Wielandt, and others.
Supported by the DFG. 相似文献
13.
Given a∈L
1(ℝ) and A the generator of an L
1-integrable family of bounded and linear operators defined on a Banach space X, we prove the existence of almost automorphic solution to the semilinear integral equation u(t)=∫
−∞
t
a(t−s)[Au(s)+f(s,u(s))]ds for each f:ℝ×X→X almost automorphic in t, uniformly in x∈X, and satisfying diverse Lipschitz type conditions. In the scalar case, we prove that a∈L
1(ℝ) positive, nonincreasing and log-convex is already sufficient. 相似文献
14.
Josepii Weier 《Annali dell'Universita di Ferrara》1959,9(1):123-148
Riassunto Sianos, t dei campi tensoriali antisi metrici sopra unan-varietà riamanniana orientata. Siano, rispettivamente,a eb i gradi dis et. Allora rot(s·t)=±(a+1)(grads)·(dual
n−(b−a)−1
dual
b−a
t) ±s·(dual
n−(b−a)−1
div dual
b−a
t), dove dual
i
sono delle modificazioni dell’operatore ben noto dual. Cons⋎t=(duals)·t, il prodottos⋎t possiede delle proprità, sotto certi aspetti duali a quelle dei prodotto esterno,s⋏t. Discutendo il prodottos⋏t, si vede: l'operatore div ed il prodotto ⋎ corrispondono all’operatore rot e al prodotto ⋏.
Résumé Soients, t des champs tensoriels antisy métriques sur unen-variété riemannienne orientée. Soient, respectivement,a etb les degrés des ett. Alors rot(s·t)=±(a+1)(grads)·(dual n−(b−a)−1 dual b−a t) ±s·(dual n−(b−a)−1 div dual b−a t), où dual i sont des modifications de l'opérateur connu dual. Avecs⋎t=(duali)·t, le produits⋎t possède des propriétés à certains égards duales à ceux du produit extérieur,s⋏t. En discutant le produits⋎t, l'on voit de plus: l'opérateur div et le produit ⋎ correspondent à l'opérateur rot et au produit ⋏.相似文献
15.
For positive integersr ands, f′(r, s) is defined as the smallest positive integerp such that every connected (ordinary) graph of orderp contains eitherr mutually adjacent lines ors mutually disjoint lines. It is found thatf’(r,s) =(r−1) (s−1)+2 unlessr=2 and s ≠ 1, in which casef′(2,s)=3.
Definitions not given here can be found in [7, 8]. 相似文献
16.
Antonio Tineo 《Annali di Matematica Pura ed Applicata》2003,182(2):129-141
In this paper we study the scalar equation x′=f(t,x), where f(t,x) has cubic non-linearities in x and we prove that this equation has at most three bounded separate solutions. We say that λ∈ℝ is a critical value of the
equation x′=f(t,x)+λx if this equation has a degenerate bounded solution and we exhibit two classes of functions f such that the above equation has a unique critical value.
Received: February 4, 2000; in final form: March 19, 2002?Published online: April 14, 2003
RID="*"
ID="*"This paper was partially supported by CDCHT, Universidad de los Andes. 相似文献
17.
Jiao Li 《印度理论与应用数学杂志》2010,41(3):425-442
Let (B
t
+ f(t))
t∈[0,+∞) be a Brownian motion with polynomial drift f(t), where f(t) is a polynomial. Some Limit Results for Lower tail and large deviation probabilities estimates, and Level crossing probabilities
estimates of (B
t
+ f(t))
t∈[0,+∞) are given in this paper. 相似文献
18.
19.
Yu. B. Koval' 《Ukrainian Mathematical Journal》1994,46(6):832-836
We prove that integral functionals, whose integrands are bounded functions of a Wiener process on a cylinder, weakly converge
to the processw
1(τ(t)), τ(t) = β1
t + (β2 − β1)mes {s:w
2(s)≥0,s<t}, wherew
1(t andw
2(t) are independent one-dimensional Wiener processes, β1 and β2 are nonrandom values, and β2≥β1≥0.
Kiev University, Kiev. Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 46, No. 6, pp. 765–768, June, 1994. 相似文献
20.
The existence of positive solutions of the Fredholm nonlinear equation y(t) = h(t) + ∫T0k(t, s)[f(y(s)) + g(y(s))] ds is discussed. It is assumed that f is a continuous, nondecreasing function and g is continuous, nonincreasing, and possibly singular. 相似文献