共查询到20条相似文献,搜索用时 500 毫秒
1.
We consider the fractional Hartree equation in the -supercritical case, and find a sharp threshold of the scattering versus blow-up dichotomy for radial data: If and , then the solution is globally well-posed and scatters; if and , the solution blows up in finite time. This condition is sharp in the sense that the solitary wave solution is global but not scattering, which satisfies the equality in the above conditions. Here, Q is the ground-state solution for the fractional Hartree equation. 相似文献
2.
《International Journal of Approximate Reasoning》2014,55(9):1830-1842
3.
In this work, we prove the existence of convex solutions to the following k-Hessian equation in the neighborhood of a point , where , is nonnegative near , and . 相似文献
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6.
A.P. Bergamasco P.L. Dattori da Silva R.B. Gonzalez 《Journal of Differential Equations》2018,264(5):3500-3526
Let be a vector field defined on the torus , where , are real-valued functions and belonging to the Gevrey class , , for . We present a complete characterization for the s-global solvability and s-global hypoellipticity of L. Our results are linked to Diophantine properties of the coefficients and, also, connectedness of certain sublevel sets. 相似文献
7.
赵晓军 《数学物理学报(B辑英文版)》2018,38(2):673-680
In this article, we study the nonexistence of solution with finite Morse index for the following Choquard type equation-△u=∫RN|u(y)|p|x-y|αdy|u(x)|p-2u(x) in RN where N ≥ 3, 0 α min{4, N}. Suppose that 2 p (2 N-α)/(N-2),we will show that this problem does not possess nontrivial solution with finite Morse index. While for p=(2 N-α)/(N-2),if i(u) ∞, then we have ∫_RN∫_RN|u(x)p(u)(y)~p/|x-y|~α dxdy ∞ and ∫_RN|▽u|~2 dx=∫_RN∫_RN|u(x)p(u)(y)~p/|x-y|~αdxdy. 相似文献
8.
We study solutions of the focusing energy-critical nonlinear heat equation in . We show that solutions emanating from initial data with energy and -norm below those of the stationary solution W are global and decay to zero, via the “concentration-compactness plus rigidity” strategy of Kenig–Merle [33], [34]. First, global such solutions are shown to dissipate to zero, using a refinement of the small data theory and the -dissipation relation. Finite-time blow-up is then ruled out using the backwards-uniqueness of Escauriaza–Seregin–Sverak [17], [18] in an argument similar to that of Kenig–Koch [32] for the Navier–Stokes equations. 相似文献
9.
In this article, we study the multiplicity and concentration behavior of positive solutions for the p-Laplacian equation of Schrödinger-Kirchhoff type
in
, where Δp is the p-Laplacian operator, 1 < p < N, M:
and V:
are continuous functions, ε is a positive parameter, and f is a continuous function with subcritical growth. We assume that V satisfies the local condition introduced by M. del Pino and P. Felmer. By the variational methods, penalization techniques, and Lyusternik-Schnirelmann theory, we prove the existence, multiplicity, and concentration of solutions for the above equation. 相似文献
10.
Let . Let and be open bounded connected subsets of containing the origin. Let be such that contains the closure of for all . Then, for a fixed we consider a Dirichlet problem for the Laplace operator in the perforated domain . We denote by the corresponding solution. If and , then we know that under suitable regularity assumptions there exist and a real analytic operator from to such that for all . Thus it is natural to ask what happens to the equality for ? negative. We show a general result on continuation properties of some particular real analytic families of harmonic functions in domains with a small hole and we prove that the validity of the equality for ? negative depends on the parity of the dimension n. 相似文献
11.
This paper investigates the orbital stability of periodic traveling wave solutions to the generalized Zakharov equations
First, we prove the existence of a smooth curve of positive traveling wave solutions of dnoidal type with a fixed fundamental period L for the generalized Zakharov equations. Then, by using the classical method proposed by Benjamin, Bona et al., we show that this solution is orbitally stable by perturbations with period L. The results on the orbital stability of periodic traveling wave solutions for the generalized Zakharov equations in this paper can be regarded as a perfect extension of the results of [15, 16, 19]. 相似文献
12.
Yohei Fujishima 《Journal of Differential Equations》2018,264(11):6809-6842
We are concerned with the existence of global in time solution for a semilinear heat equation with exponential nonlinearity
(P)
where is a continuous initial function. In this paper, we consider the case where decays to ?∞ at space infinity, and study the optimal decay bound classifying the existence of global in time solutions and blowing up solutions for (P). In particular, we point out that the optimal decay bound for is related to the decay rate of forward self-similar solutions of . 相似文献
14.
In this article, we investigate the initial value problem(IVP) associated with the defocusing nonlinear wave equation on ?2 as follows:
where the initial data (u0, u1) ? Hs(?2) × Hs?1(?2). It is shown that the IVP is global well-posedness in Hs(?2) × Hs?1(?2) for any 1 > s > 2/5. The proof relies upon the almost conserved quantity in using multilinear correction term. The main difficulty is to control the growth of the variation of the almost conserved quantity. Finally, we utilize linear-nonlinear decomposition benefited from the ideas of Roy [1]. 相似文献
15.
Changlin Xiang 《数学物理学报(B辑英文版)》2017,37(1):58
This note is a continuation of the work[17].We study the following quasilinear elliptic equations(■)where 1 p N,0 ≤μ ((N-p)/p)~p and Q ∈ L~∞(R~N).Optimal asymptotic estimates on the gradient of solutions are obtained both at the origin and at the infinity. 相似文献
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朱新才 《数学物理学报(B辑英文版)》2018,38(2):733-744
In this article,we study constrained minimizers of the following variational problem e(p):=inf{u∈H1(R3),||u||22=p}E(u),p〉0,where E(u)is the Schrdinger-Poisson-Slater(SPS)energy functional E(u):=1/2∫R3︱▽u(x)︱2dx-1/4∫R3∫R3u2(y)u2(x)/︱x-y︱dydx-1/p∫R3︱u(x)︱pdx in R3 and p∈(2,6).We prove the existence of minimizers for the cases 2p10/3,ρ0,and p=10/3,0ρρ~*,and show that e(ρ)=-∞for the other cases,whereρ~*=||φ||_2~2 andφ(x)is the unique(up to translations)positive radially symmetric solution of-△u+u=u~(7/3)in R~3.Moreover,when e(ρ~*)=-∞,the blow-up behavior of minimizers asρ↗ρ~*is also analyzed rigorously. 相似文献
18.
Huyuan Chen Patricio Felmer Jianfu Yang 《Annales de l'Institut Henri Poincaré (C) Analyse Non Linéaire》2018,35(3):729-750
In this paper, we study the elliptic problem with Dirac mass
(1)
where , , , is the Dirac mass at the origin and the potential V is locally Lipchitz continuous in , with non-empty support and satisfying with , and . We obtain two positive solutions of (1) with additional conditions for parameters on , p and k. The first solution is a minimal positive solution and the second solution is constructed via Mountain Pass Theorem. 相似文献
19.
2 Abstract For the 2-D quasilinear wave equation(?_t~2-?_x)u+2∑ij=0g~(ij)(?u)?_(ij)u = 0 satisfying i,j=0 null condition or both null conditions, a blowup or global existence result has been shown by Alinhac. In this paper, we consider a more general 2-D quasilinear wave equation(?_t~2-?_x)_u+2∑ij=0g~(ij)(?u)?_(ij)u = 0 satisfying null conditions with small initial data and the coefficients i,j=0 depending simultaneously on u and ?u. Through construction of an approximate solution,combined with weighted energy integral method, a quasi-global or global existence solution are established by continuous induction. 相似文献
20.
Andrew Lorent Guanying Peng 《Annales de l'Institut Henri Poincaré (C) Analyse Non Linéaire》2018,35(2):481-516
Let be a bounded simply-connected domain. The Eikonal equation for a function has very little regularity, examples with singularities of the gradient existing on a set of positive measure are trivial to construct. With the mild additional condition of two vanishing entropies we show ?u is locally Lipschitz outside a locally finite set. Our condition is motivated by a well known problem in Calculus of Variations known as the Aviles–Giga problem. The two entropies we consider were introduced by Jin, Kohn [26], Ambrosio, DeLellis, Mantegazza [2] to study the Γ-limit of the Aviles–Giga functional. Formally if u satisfies the Eikonal equation and if
(1)
where and are the entropies introduced by Jin, Kohn [26], and Ambrosio, DeLellis, Mantegazza [2], then ?u is locally Lipschitz continuous outside a locally finite set.Condition (1) is motivated by the zero energy states of the Aviles–Giga functional. The zero energy states of the Aviles–Giga functional have been characterized by Jabin, Otto, Perthame [25]. Among other results they showed that if for some sequence and then ?u is Lipschitz continuous outside a finite set. This is essentially a corollary to their theorem that if u is a solution to the Eikonal equation a.e. and if for every “entropy” Φ (in the sense of [18], Definition 1) function u satisfies distributionally in Ω then ?u is locally Lipschitz continuous outside a locally finite set. In this paper we generalize this result in that we require only two entropies to vanish.The method of proof is to transform any solution of the Eikonal equation satisfying (1) into a differential inclusion where is a connected compact set of matrices without Rank-1 connections. Equivalently this differential inclusion can be written as a constrained non-linear Beltrami equation. The set K is also non-elliptic in the sense of Sverak [32]. By use of this transformation and by utilizing ideas from the work on regularity of solutions of the Eikonal equation in fractional Sobolev space by Ignat [23], DeLellis, Ignat [15] as well as methods of Sverak [32], regularity is established. 相似文献