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2.
In this paper we are concerned with multi-lump bound states of the nonlinear Schr?dinger equation
for sufficiently small , where for and for . V is bounded on . For any finite collection of nondegenerate critical points of V, we show the uniqueness of solutions of the form for , where u is positive on and is a small perturbation of a sum of one-lump solutions concentrated near , respectively for sufficiently small .
Received: 30 October 2001; in final form: 10 June 2002 /Published online: 2 December 2002
RID="*"
ID="*" Research supported by Alexander von Humboldt Foundation in Germany and NSFC in China 相似文献
3.
We prove the existence of solutions of nonlinear elliptic equations with first-order terms having “natural growth” with respect
to the gradient. The assumptions on the source terms lead to the existence of possibly unbounded solutions (though with exponential
integrability). The domain Ω is allowed to have infinite Lebesgue measure.
Received: April 13, 2001; in final form: September 29, 2001?Published online: July 9, 2002 相似文献
4.
In this paper, we establish rectifiability of the jump set of an S
1–valued conservation law in two space–dimensions. This conservation law is a reformulation of the eikonal equation and is
motivated by the singular limit of a class of variational problems. The only assumption on the weak solutions is that the
entropy productions are (signed) Radon measures, an assumption which is justified by the variational origin. The methods are
a combination of Geometric Measure Theory and elementary geometric arguments used to classify blow–ups.?The merit of our approach
is that we obtain the structure as if the solutions were in BV, without using the BV–control, which is not available in these
variationally motivated problems.
Received June 24, 2002 / final version received November 12, 2002?Published online February 7, 2003 相似文献
5.
We establish a general existence theory for the Cauchy problem associated with a scalar conservation law in one-space dimension. The flux-function is assumed to be nonconvex and we consider nonclassical entropy solutions selected by a kinetic relation. To solve the Cauchy problem, we construct a sequence of approximate solutions using a wave-front tracking scheme. The main difficulty is deriving a uniform estimate on the total variation of the approximate solutions. This is achieved here by introducing a generalized total variation functional, which is decreasing in time and, additionally, reduces to the standard total variation functional when the solutions contain only classical shocks. This functional seems sufficiently robust to be useful for systems as well.Received: June 3, 2002; revised: November 12, 2002 相似文献
6.
The equation with boundary Dirichlet zero data is considered in a bounded domain . Under the assumption that concentrates, as , round a manifold and that f is a superlinear function, satisfying suitable growth assumptions, the existence of multiple distinct positive solutions
is proved.
Received: 19 December 2000 / Accepted: 8 May 2001 / Published online: 5 September 2002 相似文献
7.
Raúl Ferreira Arturo de Pablo Fernando Quirós Julio D. Rossi 《Zeitschrift für Angewandte Mathematik und Physik (ZAMP)》2006,57(4):586-594
We study the solutions of a parabolic system of heat equations coupled at the boundary through a nonlinear flux. We characterize
in terms of the parameters involved when non-simultaneous quenching may appear. Moreover, if quenching is non-simultaneous
we find the quenching rate, which surprisingly depends on the flux associated to the other component.
Partially supported by project BFM2002-04572 (Spain).
Partially supported by UBA grant EX046, CONICET and Fundación Antorchas (Argentina).
Received: February 17, 2004; revised: July 5, 2004 相似文献
8.
Giovanni Cimatti 《Annali di Matematica Pura ed Applicata》2002,181(2):181-212
The stability of the stationary solution of the thermistor as a circuit element is studied using a Liapunov functional and
the Hale–LaSalle invariance principle. The asymptotic stability of a class of periodic solutions is also considered.
Received: November 22, 1999; in final form: May 23, 2001?Published online: May 29, 2002 相似文献
9.
Xianfa Song Sining Zheng Zhaoxin Jiang 《Zeitschrift für Angewandte Mathematik und Physik (ZAMP)》2005,56(1):1-10
This paper deals with a nonlinear diffusion system coupled via nonlinear reaction terms of power type. As results of interactions among the multi-nonlinearities in the system described by six exponents, global boundedness and blow-up criteria of positive solutions are determined.Supported by the National Natural Science Foundation of China.Received: December 12, 2001; revised: May 6, December 3, 2002 相似文献
11.
J. Chabrowski 《Monatshefte für Mathematik》2002,137(4):261-272
We consider the nonlinear Schr?dinger equation
where W(x) = V(x) − E.
We establish the existence of the least energy solutions. We also formulate conditions guaranteeing the existence of multiple
solutions in terms of the Lusternik–Schnirelemann category.
Received March 30, 2001; in revised form May 29, 2002 相似文献
12.
《偏微分方程通讯》2013,38(3-4):809-845
13.
The finite difference approximation of a nonstationary pseudo-advected
vorticity equation is proved to yield generalized solutions to the
two-dimensional stationary Euler equations with nonvanishing
vorticity. This result is obtained by the simultaneous limiting of
lattice scale and time.Received: 15 May 2002 相似文献
14.
Explicit exact solutions for a new generalized Hamiltonian amplitude equation with nonlinear terms of any order 总被引:1,自引:0,他引:1
Making use of a proper transformation and a generalized ansatz, we consider a new generalized Hamiltonian amplitude equation with nonlinear terms of any order, iux + utt + (|u|p + |u|2p)u + uxt = 0. As a result, many explicit exact solutions, which include kink-shaped soliton solutions, bell-shaped soliton solutions, periodic wave solutions, the combined formal solitary wave solutions and rational solutions, are obtained.Received: April 4, 2002 相似文献
15.
16.
We present here an improved version of the method introduced by the first author to derive
pointwise gradient estimates for the solutions of one-dimensional parabolic problems. After considering
a general qualinear equation in divergence form we apply the method to the case of a nonlinear
diffusion-convection equation. The conclusions are stated first for classical solutions and then for
generalized and mild solutions. In the case of unbounded initial datum we obtain several regularizing
effects for t > 0. Some unilateral pointwise gradient
estimates are also obtained. The case of
the Dirichlet problem is also considered. Finally, we collect, in the last section, several comments
showing the connections among these estimates and the study of the free boundaries
associated to the solutions of the diffusion-convection equation. 相似文献
17.
The purpose of this paper is to study the existence, the uniqueness and the limit in , as of solutions of general initial-boundary-value problems of the form and in a bounded domain with dynamical boundary conditions of the form
Received: 5 December 2000 / Revised version: 20 November 2001 / Published online: 4 April 2002 相似文献
18.
In this work a general class of nonlinear abstract equations satisfying a generalized strong maximum principle is considered in order to study the behavior of the bounded components of positive solutions bifurcating from the curve of trivial states (λ,u)=(λ,0) at a nonlinear eigenvalue λ=λ0 with geometric multiplicity one. Since the unilateral theorems of Rabinowitz (J. Funct. Anal. 7 (1971) 487, Theorems 1.27 and 1.40) are not true as originally stated (cf. the very recent counterexample of Dancer, Bull. London Math. Soc. 34 (2002) 533), in order to get our main results the unilateral theorem of López-Gómez (Spectral Theory and Nonlinear Functional Analysis, Research Notes in Mathematics, vol. 426, CRC Press, Boca Raton, FL, 2001, Theorem 6.4.3) is required. Our analysis fills some serious gaps existing is some published papers that were provoked by a direct use of Rabinowitz's unilateral theory. Actually, the abstract theory developed in this paper cannot be covered with the pioneering results of Rabinowitz (1971), since in Rabinowitz's context any component of positive solutions must be unbounded, by a celebrated result attributable to Dancer (Arch. Rational Mech. Anal. 52 (1973) 181). 相似文献
19.
In this note we prove the uniqueness of weak solutions to a nonlinear hyperbolic system in electrohydrodynamics without the effects of a dissociation–recombination process. It is still open in the presence of a special dissociation–recombination process, although the existence of at least one weak solution was proved via the method of renormalized solutions by Feireisl [E. Feireisl, Weak solutions to a non-linear hyperbolic system arising in the theory of dielectric liquids, Math. Methods Appl. Sci. 18 (1995) 1041–1052] in 1995. 相似文献
20.
We show that there is a class of solutions to the generalized Proudman-Johnson equation which exist globally for all parameters a having the form for n∈N, thereby extending a result of Bressan and Constantin (2005) [2]. Furthermore, we present new proofs of existence of solutions developing spontaneous singularities and compute the corresponding blow-up rates. 相似文献