共查询到20条相似文献,搜索用时 15 毫秒
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We study the nonlinear Schröodinger equation
with critical exponent
2*= 2
N/(
N-2),
N 4,
where
a 0,
has a potential well. Using variational methods we
establish existence and multiplicity of positive solutions which
localize near the potential well for small and
large. 相似文献
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We consider the multiple existence of positive solutions of the following nonlinear Schrödinger equation: where if N3 and p(1, ) if N=1,2, and a(x), b(x) are continuous functions. We assume that a(x) is nonnegative and has a potential well := int a–1(0) consisting of k components and the first eigenvalues of –+b(x) on j under Dirichlet boundary condition are positive for all . Under these conditions we show that (PM) has at least 2k–1 positive solutions for large . More precisely we show that for any given non-empty subset , (P) has a positive solutions u(x) for large . In addition for any sequence n we can extract a subsequence ni along which uni converges strongly in H1(RN). Moreover the limit function u(x)=limiuni satisfies (i) For jJ the restriction u|j of u(x) to j is a least energy solution of –v+b(x)v=vp in j and v=0 on j. (ii) u(x)=0 for .Mathematics Subject Classifications (2000):35Q55, 35J20 相似文献
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We combine the F-expansion method with the homogeneous balance principle to build a strategy to find analytical solitonic and periodic wave solutions to a generalized nonlinear Schrödinger equation with distributed coefficients, linear gain/loss, and nonlinear gain/absorption. In the case of a dimensionless effective Gross–Pitaevskii equation which describes the evolution of the wave function of a quasi-one-dimensional cigar-shaped Bose–Einstein condensate, the building strategy is applied to generate analytical solutions. 相似文献
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In this paper we consider a class of semilinear Schrödinger equation which terms are asymptotically periodic at infinity. Under a weaker superquadratic condition on the nonlinearity, the existence of a ground state solution is established. The main tools employed here to overcome the new difficulties are the concentration-compactness principle and the Local Mountain Pass Theorem. 相似文献
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General soliton solutions to a reverse-time nonlocal nonlinear Schrödinger (NLS) equation are discussed via a matrix version of binary Darboux transformation. With this technique, searching for solutions of the Lax pair is transferred to find vector solutions of the associated linear differential equation system. From vanishing and nonvanishing seed solutions, general vector solutions of such linear differential equation system in terms of the canonical forms of the spectral matrix can be constructed by means of triangular Toeplitz matrices. Several explicit one-soliton solutions and two-soliton solutions are provided corresponding to different forms of the spectral matrix. Furthermore, dynamics and interactions of bright solitons, degenerate solitons, breathers, rogue waves, and dark solitons are also explored graphically. 相似文献
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Mónica Clapp Yanheng Ding 《Zeitschrift für Angewandte Mathematik und Physik (ZAMP)》2004,25(2):592-605
We study the nonlinear Schröodinger equation
-Du+la(x)u=mu+u2*-1, u ? \mathbbRN,-\Delta u+\lambda a(x)u=\mu u+u^{2^{\ast }-1},{ \ }u\in \mathbb{R}^{N},
with critical exponent
2*= 2
N/(
N-2),
N 4,
where
a 0,
has a potential well. Using variational methods we
establish existence and multiplicity of positive solutions which
localize near the potential well for small and
large. 相似文献
12.
Mónica Clapp Andrzej Szulkin 《NoDEA : Nonlinear Differential Equations and Applications》2010,17(2):229-248
We consider the magnetic nonlinear Schrödinger equations $\begin{array}{ll}{\left(-i\nabla + sA\right)^{2} u + u \, = \, |u|^{p-2}\, u, \quad p \in (2, 6),} \\ \quad \quad {\left(-i\nabla + sA\right) ^{2}u \, = \, |u|^{4}\, u,}\end{array}$ in ${\Omega=\mathcal{O}\times \mathbb{R}}We consider the magnetic nonlinear Schr?dinger equations
ll(-i?+ sA)2 u + u = |u|p-2 u, p ? (2, 6), (-i?+ sA) 2u = |u|4 u,\begin{array}{ll}{\left(-i\nabla + sA\right)^{2} u + u \, = \, |u|^{p-2}\, u, \quad p \in (2, 6),} \\ \quad \quad {\left(-i\nabla + sA\right) ^{2}u \, = \, |u|^{4}\, u,}\end{array} 相似文献
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《Applied Mathematics Letters》2003,16(5):663-667
The goal of the paper is to study the structure of the eigenfunctions of the one-dimensional Schrödinger equation from the point of view of the Euler theorem. It turns out that analog of exponent is exponentially increasing solution. Sometimes linear combinations of such solutions cancel each other at infinity and then we obtain an eigenfunction from L2(R1). 相似文献
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Jiaqi Liu Peter A. Perry Catherine Sulem 《Annales de l'Institut Henri Poincaré (C) Analyse Non Linéaire》2018,35(1):217-265
The large-time behavior of solutions to the derivative nonlinear Schrödinger equation is established for initial conditions in some weighted Sobolev spaces under the assumption that the initial conditions do not support solitons. Our approach uses the inverse scattering setting and the nonlinear steepest descent method of Deift and Zhou as recast by Dieng and McLaughlin. 相似文献
16.
Elena Prestini 《Monatshefte für Mathematik》1990,109(2):135-143
Letf be a radial function and setT * f(x)=sup0<t<1 |T t f(x)|, x ∈ ?n, n≥2, where(Tt f)^ (ξ)=e it|ξ|a \(\hat f\) (ξ),a > 1. We show that, ifB is the ball centered at the origin, of radius 100, then \(\int\limits_B {|T^ * f(x)|} dx \leqslant c(\int {|\hat f(\xi )|^2 (l + |\xi |^s )ds} )^{1/2} \) if and only ifs≥1/4. 相似文献
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Using Morse theory, truncation arguments and an abstract critical point theorem, we obtain the existence of at least three or infinitely many nontrivial solutions for the following quasilinear Schrödinger equation in a bounded smooth domain 相似文献
$$\left\{ {\begin{array}{*{20}{c}} { - {\Delta _p}u - \frac{p}{{{2^{p - 1}}}}u{\Delta _p}\left( {{u^2}} \right) = f\left( {x,u} \right)\;in\;\Omega } \\ {u = 0\;on\;\partial \Omega .} \end{array}} \right.$$ (0.1) |