共查询到20条相似文献，搜索用时 31 毫秒
1.
Here, we solve non-convex, variational problems given in the form
where u ∈ (W
1,∞(0, 1))
k
and is a non-convex, coercive polynomial. To solve (1) we analyse the convex hull of the integrand at the point a, so that we can find vectors and positive values λ1, . . . , λ
N
satisfying the non-linear equation
Thus, we can calculate minimizers of (1) by following a proposal of Dacorogna in (Direct Methods in the Calculus of Variations.
Springer, Heidelberg, 1989). Indeed, we can solve (2) by using a semidefinite program based on multidimensional moments.
We dedicate this work to our colleague Jesús Bermejo. 相似文献
2.
Tetsutaro Shibata 《Annales Henri Poincare》2008,9(6):1217-1227
We consider the nonlinear eigenvalue problem
,
where f(u) = u
p
+ h(u) (p > 1) and λ > 0 is a parameter. Typical example of h(u) is with 1 < q < (p+ 1)/2. We establish the precise asymptotic formula for L
m
-bifurcation branch λ = λ
m
(α) of positive solutions as α → ∞, where α > 0 is the L
m
-norm of the positive solution associated with .
Submitted: September 27, 2007. Accepted: May 28, 2008. 相似文献
3.
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
4.
Marcello Lucia 《Calculus of Variations and Partial Differential Equations》2006,26(3):313-330
We consider the equation
If Ω is of class C
2, we show that this problem has a non-trivial solution u
λ for each λ ∊ (8 π, λ*). The value λ* depends on the domain and is bounded from below by 2 j
0
2 π, where j
0 is the first zero of the Bessel function of the first kind of order zero (λ*≥ 2 j
0
2 π > 8 π). Moreover, the family of solution u
λ blows-up as λ → 8 π. 相似文献
5.
Steven D. Taliaferro 《Mathematische Annalen》2007,338(3):555-586
We study C
2,1 nonnegative solutions u(x,t) of the nonlinear parabolic inequalities
in a punctured neighborhood of the origin in , when and . We show that a necessary and sufficient condition on λ for such solutions u to satisfy an a priori bound near the origin is , and in this case, the a priori bound on u is
This a priori bound for u can be improved by imposing an upper bound on the initial condition of u. 相似文献
6.
In the present paper, the following Dirichlet problem and Neumann problem involving the p-Laplacian
and
are studied and some new multiplicity results of solutions for systems (1.λ) and (2.λ) are obtained. Moreover, by using the
KKM principle we give also two new existence results of solutions for systems (1.1) and (2.1).
This Work is supported in part by the National Natural Science Foundation of China (10561011). 相似文献
| ((1.λ)) |
| ((2.λ)) |
7.
We consider the following Liouville equation in
For each fixed and a
j
> 0 for 1 ≤ j ≤ k, we construct a solution to the above equation with the following asymptotic behavior:
相似文献
8.
Some remarks on trigonometric sums 总被引：1，自引：1，他引：0
Let
where m
1 < m
2 < … < m
t
≦ , δ
x
→ 0, p runs over the primes p ≧ ≦ 1, |X
p
| ≦ 1. It is assumed that m
v
, , X
p
may depend on x.
Assume that . It is proved that
for almost all irrational α, π(x) = number of primes up to x.
Research supported by the Applied Number Theory Research Group of the Hungarian Academy of Science and by a grant from OTKA
T46993. 相似文献
9.
This paper deals with the existence of positive solutions for the nonlinear system
. This system often arises in the study of positive radial solutions of nonlinear elliptic system. Here u = (u
1, …, u
n) and f
i, i = 1, 2, …, n are continuous and nonnegative functions, p(t), q(t): [0, 1] → (0, ∞) are continuous functions. Moreover, we characterize the eigenvalue intervals for
. The proof is based on a well-known fixed point theorem in cones. 相似文献
10.
Thomas Bartsch Zhi-Qiang Wang Juncheng Wei 《Journal of Fixed Point Theory and Applications》2007,2(2):353-367
We consider the existence of bound states for the coupled elliptic system
where n ≤ 3. Using the fixed point index in cones we prove the existence of a five-dimensional continuum of solutions (λ1, λ2, μ
1, μ
2, β, u
1, u
2) bifurcating from the set of semipositive solutions (where u
1 = 0 or u
2 = 0) and investigate the parameter range covered by .
Dedicated to Albrecht Dold and Edward Fadell 相似文献
11.
Luisa Fattorusso 《Czechoslovak Mathematical Journal》2008,58(1):113-129
Let Ω be a bounded open subset of ℝ
n
, n > 2. In Ω we deduce the global differentiability result
for the solutions u ∈ H
1 (Ω, ℝ
n
) of the Dirichlet problem
with controlled growth and nonlinearity q = 2.
The result was obtained by first extending the interior differentiability result near the boundary and then proving the global
differentiability result making use of a covering procedure. 相似文献
12.
We consider the following semilinear elliptic equation with singular nonlinearity:
where
and Ω is an open subset in
. Let u be a non-negative finite energy stationary solution and
be the rupture set of u. We show that the Hausdorff dimension of Σ is less than or equal to [(n−2) α+(n+2)]/(α +1). 相似文献
13.
Mean-value theorems and extensions of the Elliott-Daboussi theorem on additive arithmetic semigroups
Wen-Bin Zhang 《The Ramanujan Journal》2008,15(1):47-75
We present more general forms of the mean-value theorems established before for multiplicative functions on additive arithmetic
semigroups and prove, on the basis of these new theorems, extensions of the Elliott-Daboussi theorem. Let
be an additive arithmetic semigroup with a generating set ℘ of primes p. Assume that the number G(m) of elements a in
with “degree” ∂(a)=m satisfies
with constants q>1, ρ
1<ρ
2<⋅⋅⋅<ρ
r
=ρ, ρ≥1, γ>1+ρ. For the main result, let α,τ,η be positive constants such that α>1,τ
ρ≥1, and τ
α
ρ≥1. Then for a multiplicative function f(a) on
the following two conditions (A) and (B) are equivalent. These are (A) All four series
converge and
and (B) The order τ
ρ mean-value
exists with m
f
≠0 and the limit
exists with M
v
(α)>0.
相似文献
14.
Let L(x, v) be a Lagrangian which is convex and superlinear in the velocity variable v, and let H(x, p) be the associated Hamiltonian. Conditions are obtained under which every viscosity solution
of the Hamilton-Jacobi equation
is an action function in the large, i.e.,
for all
Received: 13 June 2003 相似文献
15.
Noriko Mizoguchi 《Mathematische Annalen》2007,339(4):839-877
A solution u of a Cauchy problem for a semilinear heat equation
is said to undergo Type II blowup at t = T if lim sup Let be the radially symmetric singular steady state. Suppose that is a radially symmetric function such that and (u
0)
t
change sign at most finitely many times. We determine the exact blowup rate of Type II blowup solution with initial data
u
0 in the case of p > p
L
, where p
L
is the Lepin exponent. 相似文献
16.
T. Shibata 《Annali di Matematica Pura ed Applicata》2007,186(3):525-537
We consider the nonlinear Sturm–Liouville problem
where λ > 0 is an eigenvalue parameter. To understand well the global behavior of the bifurcation branch in R
+ × L
2(I), we establish the precise asymptotic formula for λ(α), which is associated with eigenfunction u
α with ‖ u
α ‖2 = α, as α → ∞. It is shown that if for some constant p > 1 the function h(u) ≔ f(u)/u
p
satisfies adequate assumptions, including a slow growth at ∞, then λ(α) ∼ α
p−1
h(α) as α → ∞ and the second term of λ(α) as α → ∞ is determined by lim
u → ∞
uh′(u).
Mathematics Subject Classification (2000) 34B15 相似文献
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17.
18.
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
19.
20.
The gradient method for the symmetric positive definite linear system
is as follows
where
is the residual of the system at xk and αk is the stepsize. The stepsize
is optimal in the sense that it minimizes the modulus
, where λ1 and λn are the minimal and maximal eigenvalues of A respectively. Since λ1 and λn are unknown to users, it is usual that the gradient method with the optimal stepsize is only mentioned in theory. In this
paper, we will propose a new stepsize formula which tends to the optimal stepsize as
. At the same time, the minimal and maximal eigenvalues, λ1 and λn, of A and their corresponding eigenvectors can be obtained.
This research was initiated while the first author was visiting The Hong Kong Polytechnic University.
This author was supported by the Chinese NSF grants (No. 40233029 and 101071104) and an innovation fund of Chinese Academy
of Sciences.
This author was supported by a grant from the Research Committee of the Hong Kong Polytechnic University (A-PC36). 相似文献
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