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1.
Let a be a quadratic form associated with a Schrödinger operator L=-∇·(A∇)+V on a domain Ω⊂Rd. If a is nonnegative on , then either there is W>0 such that for all , or there is a sequence and a function ?>0 satisfying L?=0 such that a[?k]→0, ?k? locally uniformly in Ω?{x0}. This dichotomy is equivalent to the dichotomy between L being subcritical resp. critical in Ω. In the latter case, one has an inequality of Poincaré type: there exists W>0 such that for every satisfying there exists a constant C>0 such that for all .  相似文献   

2.
Suppose that β?0 is a constant and that is a continuous function with R+:=(0,∞). This paper investigates N-dimensional singular, quasilinear elliptic equations of the form
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3.
In Rm×Rnm, endowed with coordinates X=(x,y), we consider the PDE
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4.
Let H?1 be a selfadjoint operator in H, let J be a linear and bounded operator from (D(H1/2),∥H1/2·∥) to Haux and for β>0 let be the nonnegative selfadjoint operator in H satisfying
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5.
Let (Mn,g) be a compact riemannian manifold of dimension n?3. Under some assumptions, we prove that there exists a positive function φ solution of the Yamabe equation
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6.
This paper shows the existence and the uniqueness of the positive solution ?(t) of the singular boundary value problem
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7.
Multiplicity of solutions for the plasma problem in two dimensions   总被引:1,自引:0,他引:1  
Let Ω be a bounded domain in R2, u+=u if u?0, u+=0 if u<0, u=u+u. In this paper we study the existence of solutions to the following problem arising in the study of a simple model of a confined plasma
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8.
9.
In this paper we investigate linear three-term recurrence formulae with sequences of integers (T(n))n?0 and (U(n))n?0, which are ultimately periodic modulo m, e.g.
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10.
Let K denote a field, and let V denote a vector space over K with finite positive dimension. By a Leonard pair on V we mean an ordered pair of linear transformations A : V → V and A : V → V that satisfy the following two conditions:
(i)
There exists a basis for V with respect to which the matrix representing A is irreducible tridiagonal and the matrix representing A is diagonal.
(ii)
There exists a basis for V with respect to which the matrix representing A is irreducible tridiagonal and the matrix representing A is diagonal.
Let (respectively v0v1, … , vd) denote a basis for V that satisfies (i) (respectively (ii)). For 0 ? i ? d, let ai denote the coefficient of , when we write as a linear combination of , and let denote the coefficient of vi, when we write Avi as a linear combination of v0v1, … , vd.In this paper we show a0 = ad if and only if . Moreover we show that for d ? 1 the following are equivalent; (i) a0 = ad and a1 = ad−1; (ii) and ; (iii) ai = adi and for 0 ? i ? d. These give a proof of a conjecture by the second author. We say A, A is balanced whenever ai = adi and for 0 ? i ? d. We say A,A is essentially bipartite (respectively essentially dual bipartite) whenever ai (respectively ) is independent of i for 0 ? i ? d. Observe that if A, A is essentially bipartite or dual bipartite, then A, A is balanced. For d ≠ 2, we show that if A, A is balanced then A, A is essentially bipartite or dual bipartite.  相似文献   

11.
Let (n?3) be a ball, and let fC3. We are concerned with the Neumann problem
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12.
Perturbation from Dirichlet problem involving oscillating nonlinearities   总被引:1,自引:0,他引:1  
In this paper we prove that if the potential has a suitable oscillating behavior in any neighborhood of the origin (respectively +∞), then under very mild conditions on the perturbation term g, for every kN there exists bk>0 such that
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13.
In this paper, we prove some optimal uniqueness results for large solutions of a canonical class of semilinear equations under minimal regularity conditions on the weight function in front of the non-linearity and combine these results with the localization method introduced in [López-Gómez, The boundary blow-up rate of large solutions, J. Differential Equations 195 (2003) 25-45] to prove that any large solution L of Δu=a(x)up, p>1, a>0, must satisfy
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14.
If X is a real Banach space, we denote by WX the class of all functionals possessing the following property: if {un} is a sequence in X converging weakly to uX and lim infnΦ(un)≤Φ(u), then {un} has a subsequence converging strongly to u.In this paper, we prove the following result:Let X be a separable and reflexive real Banach space; an interval; a sequentially weakly lower semicontinuous C1 functional, belonging to WX, bounded on each bounded subset of X and whose derivative admits a continuous inverse on X; a C1 functional with compact derivative. Assume that, for each λI, the functional ΦλJ is coercive and has a strict local, not global minimum, say .Then, for each compact interval [a,b]⊆I for which , there exists r>0 with the following property: for every λ∈[a,b] and every C1 functional with compact derivative, there exists δ>0 such that, for each μ∈[0,δ], the equation
Φ(x)=λJ(x)+μΨ(x)  相似文献   

15.
Let ΩRN, N?2, be a bounded domain. We consider the following quasilinear problem depending on a real parameter λ>0:
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16.
In this paper, we consider the Brezis-Nirenberg problem in dimension N?4, in the supercritical case. We prove that if the exponent gets close to and if, simultaneously, the bifurcation parameter tends to zero at the appropriate rate, then there are radial solutions which behave like a superposition of bubbles, namely solutions of the form
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17.
We study the Cauchy problem for a class of p-evolution operators P(t,x,Dt,Dx) in , with less than coefficients with respect to the time variable.According to Lipschitz, log-lipschitz or Hölder regularity we find well-posedness in Sobolev spaces or in Gevrey classes.  相似文献   

18.
L. Hörmander's extension of Ásgeirsson's mean value theorem states that if u is a solution of the inhomogeneous ultrahyperbolic equation (Δx−Δy)u=f, , , then
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19.
A set {a1,…,am} of m distinct positive integers is called a Diophantine m-tuple if aiaj+1 is a perfect square for all i, j with 1?i<j?m. It is conjectured that if {a,b,c,d} is a Diophantine quadruple with a<b<c<d, then d=d+, where d+=a+b+c+2abc+2rst and , , . In this paper, we show that if {a,b,c,d,e} is a Diophantine quintuple with a<b<c<d<e, then d=d+.  相似文献   

20.
We study the degeneration dimension of non-archimedean analytic maps into the complement of hypersurface divisors of smooth projective varieties. We also show that there exist no non-archimedean analytic maps into where Di, 1?i?n, are hypersurfaces of degree at least 2 in general position and intersecting transversally. Moreover, we prove that there exist no non-archimedean analytic maps into when D1, D2 are generic plane curves with degD1+degD2?4.  相似文献   

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