共查询到20条相似文献,搜索用时 31 毫秒
1.
We obtain Sobolev inequalities for the Shcrödinger operator −Δ−V, where V has critical behaviour V(x)=((N−2)/2)2|x|−2 near the origin. We apply these inequalities to obtain point-wise estimates on the associated heat kernel, improving upon earlier results. 相似文献
2.
Elena Cordero 《Journal of Functional Analysis》2008,254(2):506-534
We study the action of metaplectic operators on Wiener amalgam spaces, giving upper bounds for their norms. As an application, we obtain new fixed-time estimates in these spaces for Schrödinger equations with general quadratic Hamiltonians and Strichartz estimates for the Schrödinger equation with potentials V(x)=±2|x|. 相似文献
3.
Lucas C.F. Ferreira 《Journal of Differential Equations》2011,250(4):2045-2063
We study the equation Δu+u|u|p−1+V(x)u+f(x)=0 in Rn, where n?3 and p>n/(n−2). The forcing term f and the potential V can be singular at zero, change sign and decay polynomially at infinity. We can consider anisotropic potentials of form h(x)|x|−2 where h is not purely angular. We obtain solutions u which blow up at the origin and do not belong to any Lebesgue space Lr. Also, u is positive and radial, in case f and V are. Asymptotic stability properties of solutions, their behavior near the singularity, and decay are addressed. 相似文献
4.
We discuss the regularity of the oscillatory semigroup eitH, where H=-Δ+|x|2 is the n-dimensional Hermite operator. The main result is a Strichartz-type estimate for the oscillatory semigroup eitH in terms of the mixed Lp spaces. The result can be interpreted as the regularity of solution to the Schrödinger equation with potential V(x)=|x|2. 相似文献
5.
Let V(x) be a non-negative, bounded potential in RN, N?3 and p supercritical, . We look for positive solutions of the standing-wave nonlinear Schrödinger equation Δu−V(x)u+up=0 in RN, with u(x)→0 as |x|→+∞. We prove that if V(x)=o(−2|x|) as |x|→+∞, then for N?4 and this problem admits a continuum of solutions. If in addition we have, for instance, V(x)=O(|x|−μ) with μ>N, then this result still holds provided that N?3 and . Other conditions for solvability, involving behavior of V at ∞, are also provided. 相似文献
6.
In this paper the notion of critical tangent cone CT(x|Q) to Q at x is introduced for the case when Q is a convex subset of a normed space X. If Q is closed with nonempty interior, and x∈Q, the nonemptiness of the Dubovitskii–Milyutin set of second-order admissible variations, V(x,d|Q), is then characterized by the condition d∈CT(x|Q). Furthermore, the support function of V(x,d|Q) is shown to be evaluated in terms of that support function of Q. For the cases when Q is the set of continuous or L ∞ selections of a certain set-valued map, the corresponding characterization of the cone CT(x|Q) and the formula for the support function of V(x,d|Q) are obtained in terms of more verifiable conditions. 相似文献
7.
Consider the Schrödinger equation {fx25-1}. The following estimates are proved: (A) IfV≡0 then for any 0≤α<1/2, {fx25-2}, and for α=1/2,s>1/2, {fx25-3} (B) If |V(x)|≤C(1+|x|2)?1?δ, δ>0, then (if 0 is neither an eigenvalue nor a resonance of ?Δ+V), {fx25-4}. 相似文献
8.
B. A. Khudaikuliev 《Mathematical Notes》2012,92(5-6):820-829
This paper deals with the behavior of the nonnegative solutions of the problem $$- \Delta u = V(x)u, \left. u \right|\partial \Omega = \varphi (x)$$ in a conical domain Ω ? ? n , n ≥ 3, where 0 ≤ V (x) ∈ L1(Ω), 0 ≤ ?(x) ∈ L1(?Ω) and ?(x) is continuous on the boundary ?Ω. It is proved that there exists a constant C *(n) = (n ? 2)2/4 such that if V 0(x) = (c + λ 1)|x|?2, then, for 0 ≤ c ≤ C *(n) and V(x) ≤ V 0(x) in the domain Ω, this problem has a nonnegative solution for any nonnegative boundary function ?(x) ∈ L 1(?Ω); for c > C *(n) and V(x) ≥ V 0(x) in Ω, this problem has no nonnegative solutions if ?(x) > 0. 相似文献
9.
Solutions concentrating on higher dimensional subsets for singularly perturbed elliptic equations II
We construct spike layered solutions for the semilinear elliptic equation −ε2Δu+V(x)u=K(x)up−1 on a domain Ω⊂RN which may be bounded or unbounded. The solutions concentrate simultaneously on a finite number of m-dimensional spheres in Ω. These spheres accumulate as ε→0 at a prescribed sphere in Ω whose location is determined by the potential functions V,K. 相似文献
10.
The existence of solutions is obtained for a class of the non-periodic Schrödinger equation −Δu + V(x)u = f(x, u), x ∈ RN, by the generalized mountain pass theorem, where V is large at infinity and f is superlinear as |u| → ∞. 相似文献
11.
Lucas C. F. Ferreira Everaldo S. Medeiros Marcelo Montenegro 《Annali di Matematica Pura ed Applicata》2013,192(4):539-552
The equation Δu + V(x)u + b(x)u|u| ρ -1 + h(x) = 0 in ${\mathbb{R}^{n}}$ is studied in anisotropic Lebesgue spaces. We assume ${\frac{n-\theta}{n-2} < \rho < \infty}$ , with n ≥ 3 and 0 ≤ θ < 2, which covers the supercritical range. Our approach relies on estimates of the Riesz potential and allows us to consider a wide class of potentials V, including anisotropic ones. The symmetry and antisymmetry of the solutions are also addressed. 相似文献
12.
Ting Cheng 《Journal of Differential Equations》2008,244(4):766-802
The blow-up rate estimate for the solution to a semilinear parabolic equation ut=Δu+V(x)|u|p−1u in Ω×(0,T) with 0-Dirichlet boundary condition is obtained. As an application, it is shown that the asymptotic behavior of blow-up time and blow-up set of the problem with nonnegative initial data u(x,0)=Mφ(x) as M goes to infinity, which have been found in [C. Cortazar, M. Elgueta, J.D. Rossi, The blow-up problem for a semilinear parabolic equation with a potential, preprint, arXiv: math.AP/0607055, July 2006], is improved under some reasonable and weaker conditions compared with [C. Cortazar, M. Elgueta, J.D. Rossi, The blow-up problem for a semilinear parabolic equation with a potential, preprint, arXiv: math.AP/0607055, July 2006]. 相似文献
13.
We study the collision of two solitons for the nonlinear Schrödinger equation iψt=−ψxx+F(2|ψ|)ψ, F(ξ)=−2ξ+O(ξ2) as ξ→0, in the case where one soliton is small with respect to the other. We show that in general, the two soliton structure is not preserved after the collision: while the large soliton survives, the small one splits into two outgoing waves that for sufficiently long times can be controlled by the cubic NLS: iψt=−ψxx−22|ψ|ψ. 相似文献
14.
Sebastián Lorca 《Journal of Mathematical Analysis and Applications》2004,295(1):276-286
We study the existence of positive solutions to the elliptic equation ε2Δu(x,y)−V(y)u(x,y)+f(u(x,y))=0 for (x,y) in an unbounded domain subject to the boundary condition u=0 whenever is nonempty. Our potential V depends only on the y variable and is a bounded or unbounded domain which may coincide with . The positive parameter ε is tending to zero and our solutions uε concentrate along minimum points of the unbounded manifold of critical points of V. 相似文献
15.
Yuan Gong Sun 《Journal of Mathematical Analysis and Applications》2004,298(1):114-119
In the case of oscillatory potentials, we establish an oscillation theorem for the forced sublinear differential equation x(n)+q(t)λ|x|sgnx=e(t), t∈[t0,∞). No restriction is imposed on the forcing term e(t) to be the nth derivative of an oscillatory function. In particular, we show that all solutions of the equation x″+tαsintλ|x|sgnx=mtβcost, t?0, 0<λ<1 are oscillatory for all m≠0 if β>(α+2)/(1−λ). This provides an analogue of a result of Nasr [Proc. Amer. Math. Soc. 126 (1998) 123] for the forced superlinear equation and answers a question raised in an earlier paper [J.S.W. Wong, SIAM J. Math. Anal. 19 (1988) 673]. 相似文献
16.
《Nonlinear Analysis: Theory, Methods & Applications》2003,52(5):1441-1453
We consider the Dirichlet problem for the equation −Δu=αu+m(x)u|u|q−2+g(x,u), where q∈(1,2) and m changes sign. We prove that the Morse critical groups at zero of the energy functional of the problem are trivial. As a consequence, existence and bifurcation of nontrivial solutions of the problem are established. 相似文献
17.
In this paper, we consider one-dimensional nonlinear Schrödinger equation iut−uxx+V(x)u+f(2|u|)u=0 on [0,π]×R under the boundary conditions a1u(t,0)−b1ux(t,0)=0, a2u(t,π)+b2ux(t,π)=0, , for i=1,2. It is proved that for a prescribed and analytic positive potential V(x), the above equation admits small-amplitude quasi-periodic solutions corresponding to d-dimensional invariant tori of the associated infinite-dimensional dynamical system. 相似文献
18.
Anders Yeo 《Journal of Graph Theory》1999,32(2):123-136
The local irregularity of a digraph D is defined as il(D) = max {|d+ (x) − d− (x)| : x ϵ V(D)}. Let T be a tournament, let Γ = {V1, V2, …, Vc} be a partition of V(T) such that |V1| ≥ |V2| ≥ … ≥ |Vc|, and let D be the multipartite tournament obtained by deleting all the arcs with both end points in the same set in Γ. We prove that, if |V(T)| ≥ max{2il(T) + 2|V1| + 2|V2| − 2, il(T) + 3|V1| − 1}, then D is Hamiltonian. Furthermore, if T is regular (i.e., il(T) = 0), then we state slightly better lower bounds for |V(T)| such that we still can guarantee that D is Hamiltonian. Finally, we show that our results are best possible. © 1999 John Wiley & Sons, Inc. J Graph Theory 32: 123–136, 1999 相似文献
19.
Xiaofei Guan Mingxia Li Wenming He Zhengwu Jiang 《Central European Journal of Mathematics》2014,12(11):1733-1747
In this paper, some superconvergence results of high-degree finite element method are obtained for solving a second order elliptic equation with variable coefficients on the inner locally symmetric mesh with respect to a point x 0 for triangular meshes. By using of the weak estimates and local symmetric technique, we obtain improved discretization errors of O(h p+1 |ln h|2) and O(h p+2 |ln h|2) when p (≥ 3) is odd and p (≥ 4) is even, respectively. Meanwhile, the results show that the combination of the weak estimates and local symmetric technique is also effective for superconvergence analysis of the second order elliptic equation with variable coefficients. 相似文献
20.
Leonardo P. Bonorino Eduardo H.M. Brietzke Carlos A. Taschetto 《Journal of Differential Equations》2005,214(1):156-175
In this work we study the period function T of solutions to the conservative equation x″(t)+f(x(t))=0. We present conditions on f that imply the monotonicity and convexity of T. As a consequence we obtain the criterium established by C. Chicone and find conditions easier to apply. We also get a condition obtained by Cima, Gasull and Mañosas about monotonicity and, following some of their calculations, present results on the period function of Hamiltonian systems where H(x,y)=F(x)+n-1|y|n. Using the monotonicity of T, we count the homogeneous solutions to the semilinear elliptic equation Δu=γuγ-1 in two dimensions. 相似文献