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1.
We consider the nonlinear Sturm–Liouville problem
(1)
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   

2.
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.
This paper studies the existence of solutions to the singular boundary value problem
, where g: (0, 1) × (0, ∞) → ℝ and h: (0, 1) × [0, ∞) → [0, ∞) are continuous. So our nonlinearity may be singular at t = 0, 1 and u = 0 and, moreover, may change sign. The approach is based on an approximation method together with the theory of upper and lower solutions. The research is supported by NNSF of China(10301033).  相似文献   

4.
Positive solutions and eigenvalue intervals for nonlinear systems   总被引：1，自引：0，他引：1  
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   

5.
We investigate the large time behavior of positive solutions with finite mass for the viscous Hamilton-Jacobi equationu t = Δu + |Δu| p ,t>0,x ∈ ℝ N , wherep≥1 andu(0,.)=u 0≥0,u 0≢0,u 0∈L 1. DenotingI ∞=lim t→∞‖u(t)‖1≤∞, we show that the asymptotic behavior of the mass can be classified along three cases as follows:
–  • ifp≤(N+2)/(N+1), thenI ∞=∞ for allu 0;
–  • if (N+2)/(N+1)<p<2, then bothI ∞=∞ andI ∞<∞ occur;
–  • ifp≥2, thenI ∞<∞ for allu 0.
We also consider a similar question for the equationu t=Δu+u p .  相似文献   

6.
The problem (where B is a unit ball in R n )
, with , is known to have a curve of positive solutions bifurcating from infinity at λ = λ1, the principal eigenvalue. It turns out that a similar situation may occur, when g(u) is oscillatory for large u, instead of being small. In case n = 1, we can also prove existence of infinitely many solutions at λ = λ1 on this curve. Similarly, we consider oscillatory bifurcation from zero.   相似文献   

7.
Given (M,g) a smooth compact Riemannian manifold of dimension n ≥ 5, we consider equations like
where is a Paneitz-Branson type operator with constant coefficients α and aα, u is required to be positive, and is critical from the Sobolev viewpoint. We define the energy function Em as the infimum of over the u’s which are solutions of the above equation. We prove that Em (α ) →+∞ as α →+∞ . In particular, for any Λ > 0, there exists α0 > 0 such that for α ≥ α0, the above equation does not have a solution of energy less than or equal to Λ.  相似文献   

8.
In this paper we establish some oscillation or nonoscillation criteria for the second order half-linear differential equation
where (i) r,c ∈ C([t 0, ∞), ℝ := (− ∞, ∞)) and r(t) > 0 on [t 0, ∞) for some t 0 ⩾ 0; (ii) Φ(u) = |u|p−2 u for some fixed number p > 1. We also generalize some results of Hille-Wintner, Leighton and Willet.  相似文献   

9.
Consider the space C0(Ω) endowed with a Banach lattice-norm ‖ · ‖ that is not assumed to be the usual spectral norm ‖ · ‖∞ of the supremum over Ω. A recent extension of the classical Banach-Stone theorem establishes that each surjective linear isometry U of the Banach lattice (C 0(Ω), ‖ · ‖) induces a partition Π of Ω into a family of finite subsets S ⊂ Ω along with a bijection T: Π → Π which preserves cardinality, and a family [u(S): S ∈ Π] of surjective linear maps u(S): C(T(S)) → C(S) of the finite-dimensional C*-algebras C(S) such that
$ (Uf)|_{T(S)} = u(S)(f|_s ) \forall f \in \mathcal{C}_0 (\Omega ) \forall S \in \prod . $ (Uf)|_{T(S)} = u(S)(f|_s ) \forall f \in \mathcal{C}_0 (\Omega ) \forall S \in \prod .   相似文献   

10.
This paper is concerned with nonoscillatory solutions of the fourth order quasilinear differential equation
where α > 0, β > 0 and p(t) and q(t) are continuous functions on an infinite interval [a,∞) satisfying p(t) > 0 and q(t) > 0 (t≥a). The growth bounds near t = ∞ of nonoscillatory solutions are obtained, and necessary and sufficient integral conditions are established for the existence of nonoscillatory solutions having specific asymptotic growths as t→∞. This revised version was published online in June 2006 with corrections to the Cover Date.  相似文献   

11.
Suppose λ is a positive number. Basic theory of cardinal interpolation ensures the existence of the Gaussian cardinal functionL λ(x)
x∈R, satisfying the interpolatory conditionsL k = δ0k,k∈Z . One objective of this paper is to derive several additional properties ofL λ. For example, it is shown thatL λ possesses the signregularity property sgn[L λ(x)]=sgn[sin(πx)/(πx)],x∈R, and that |L λ (x)|≤2e 8 min {(⌊|x|⌋+1)-1,exp(-λ⌊|x|⌋)},x∈R. The analysis is based on a simple representation formula forL λ and employs some methods from classical function theory. A second consideration in the paper is the Gaussian cardinal-interpolation operatorL λ, defined by the equation (L λy)(x):=
,x∈R, y=(yk)k∈Z. On account of the exponential decay of the cardinal functionL λ,L λ is a well-defined linear map froml ∞ (Z) intoL ∞ (R). Its associated operatornorm ‖L λ‖ is called the Lebesgue constant ofL λ. The latter half of the paper establishes the following estimates for the Lebesgue constant: ‖L λ‖≍1, λ→∞, and ║Lλ║≍log(1/λ), λ→0+. Suitable multidimensional analogues of these results are also given. For Carl de Boor, on the occasion of his sixtieth birthday  相似文献   

12.
In this paper we study the Dirichlet problem in Q T = Ω × (0, T) for degenerate equations of porous medium-type with a lower order term:
The principal part of the operator degenerates in u = 0 according to a nonnegative increasing real function α(u), and the term grows quadratically with respect to the gradient. We prove an existence result for solutions to this problem in the framework of the distributional solutions under the hypotheses that both f and the initial datum u 0 are bounded nonnegative functions. Moreover as further results we get an existence result for the model problem
in the case that the principal part of the operator is of fast-diffusion type, i.e. α(u) = u m , with −1 < m < 0.   相似文献   

13.
This paper summarized recent achievements obtained by the authors about the box dimensions of the Besicovitch functions given byB(t) := ∞∑k=1 λs-2k sin(λkt),where 1 ＜ s ＜ 2, λk ＞ 0 tends to infinity as k →∞ and λk satisfies λk 1/λk ≥λ＞ 1. The results show thatlimk→∞ log λk 1/log λk = 1is a necessary and sufficient condition for Graph(B(t)) to have same upper and lower box dimensions.For the fractional Riemann-Liouville differential operator Du and the fractional integral operator D-v,the results show that if λ is sufficiently large, then a necessary and sufficient condition for box dimension of Graph(D-v(B)),0 ＜ v ＜ s - 1, to be s - v and box dimension of Graph(Du(B)),0 ＜ u ＜ 2 - s, to be s uis also lim k→∞logλk 1/log λk = 1.  相似文献   

14.
A system of linear differential equations with oscillatory decreasing coefficients is considered. The coefficients have the formt -α a(t), α > 0 wherea(t) is a trigonometric polynomial with an arbitrary set of frequencies. The asymptotic behavior of the solutions of this system ast → ∞ is studied. We construct an invertible (for sufficiently larget) change of variables that takes the original system to a system not containing oscillatory coefficients in its principal part. The study of the asymptotic behavior of the solutions of the transformed system is a simpler problem. As an example, the following equation is considered:
, where λ andα, 0 <α ≤ 1, are real numbers. Translated fromMatematicheskie Zametki, Vol. 64, No. 5, pp. 658–666, November, 1998.  相似文献   

15.
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   

16.
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   

17.
For the singular Cauchy problem
where α: (0, τ) → (0, +∞) is a continuous function and , the authors prove the existence of a nonempty set of continuously differentiable solutions x: (0, ρ] → ℝ (ρ ∈ (0, τ) is sufficiently small) with the known asymptotic as t → +0. Translated from Sovremennaya Matematika i Ee Prilozheniya (Contemporary Mathematics and Its Applications), Vol. 57, Suzdal Conference–2006, Part 3, 2008.  相似文献   

18.
LetX be a Banach space and letA be the infinitesimal generator of a differentiable semigroup {T(t) |t ≥ 0}, i.e. aC 0-semigroup such thatt ↦T(t)x is differentiable on (0, ∞) for everyx εX. LetB be a bounded linear operator onX and let {S(t) |t ≥ 0} be the semigroup generated byA +B. Renardy recently gave an example which shows that {S(t) |t ≥ 0} need not be differentiable. In this paper we give a condition on the growth of ‖T′(t)‖ ast ↓ 0 which is sufficient to ensure that {S(t) |t ≥ 0} is differentiable. Moreover, we use Renardy’s example to study the optimality of our growth condition. Our results can be summarized roughly as follows:
(i)  If lim sup t→0+t log‖T′(t)‖/log(1/2) = 0 then {S(t) |t ≥ 0} is differentiable.
(ii)  If 0<L=lim sup t→0+t log‖T′(t)‖/log(1/2)<∞ thent ↦S(t ) is differentiable on (L, ∞) in the uniform operator topology, but need not be differentiable near zero
(iii)  For each function α: (0, 1) → (0, ∞) with α(t)/log(1/t) → ∞ ast ↓ 0, Renardy’s example can be adjusted so that limsup t→0+t log‖T′(t)‖/α(t) = 0 andt →S(t) is nowhere differentiable on (0, ∞).
We also show that if lim sup t→0+t p ‖T′(t)‖<∞ for a givenp ε [1, ∞), then lim sup t→0+t p‖S′(t)‖<∞; it was known previously that if limsup t→0+t p‖T′(t)‖<∞, then {S(t) |t ≥ 0} is differentiable and limsup t→0+t 2p–1‖S′(t)‖<∞.  相似文献   

19.
A group of necessary and sufficient conditions for the nonoscillation of a second order linear delayequation with impulses(r(t)u')'=-p(t)u(t-τ)are obtained in this paper,where p(t)=sum from ∞to n=1 a_n δ(t-t_n),δ(t) is a Dirac δ-unction,and for each n∈N,a_n>0,t_n→∞as n→∞.Furthermore,the boundedness of the solutions is also investigated if the equationis nonoscillatory.An example is given to illustrate the use of the main theorems.  相似文献   

20.
Let u=u（x,t,uo）represent the global solution of the initial value problem for the one-dimensional fluid dynamics equation ut-εuxxt＋δux＋γHuxx＋βuxxx＋f（u）x=αuxx,u（x,0）=uo（x）, whereα〉0,β〉0,γ〉0,δ〉0 andε〉0 are constants.This equation may be viewed as a one-dimensional reduction of n-dimensional incompressible Navier-Stokes equations. The nonlinear function satisfies the conditions f（0）=0,｜f（u）｜→∞as ｜u｜→∞,and f∈C^1（R）,and there exist the following limits Lo=lim sup/u→o f（u）/u^3 and L∞=lim sup/u→∞ f（u）/u^5 Suppose that the initial function u0∈L^I（R）∩H^2（R）.By using energy estimates,Fourier transform,Plancherel＇s identity,upper limit estimate,lower limit estimate and the results of the linear problem vt-εv（xxt）＋δvx＋γHv（xx）＋βv（xxx）=αv（xx）,v（x,0）=vo（x）, the author justifies the following limits（with sharp rates of decay） lim t→∞[（1＋t）^（m＋1/2）∫｜uxm（x,t）｜^2dx]=1/2π（π/2α）^（1/2）m!!/（4α）^m[∫R uo（x）dx]^2, if∫R uo（x）dx≠0, where 0!!=1,1!!=1 and m!!=1·3…（2m-3）…（2m-1）.Moreover lim t→∞[（1＋t）^（m＋3/2）∫R｜uxm（x,t）｜^2dx]=1/2π（x/2α）^（1/2）（m＋1）!!/（4α）^（m＋1）[∫Rρo（x）dx]^2, if the initial function uo（x）=ρo′（x）,for some functionρo∈C^1（R）∩L^1（R）and∫Rρo（x）dx≠0.  相似文献   

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