Semiclassical stationary states for nonlinear Schr�dinger equations with fast decaying potentials

Moroz, Vitaly; Van Schaftingen, Jean
January 2010
Calculus of Variations & Partial Differential Equations;Jan2010, Vol. 37 Issue 1/2, p1
Academic Journal
We study the existence of positive solutions for a class of nonlinear Schr�dinger equations of the type where N = 3, p > 1 is subcritical and V is a nonnegative continuous potential. Amongst other results, we prove that if V has a positive local minimum, and $${\frac{N}{N-2} < p < \frac{N+2}{N-2}}$$ , then for small e the problem admits positive solutions which concentrate as e ? 0 around the local minimum point of V. The novelty is that no restriction is imposed on the rate of decay of V. In particular, we cover the case where V is compactly supported.


Related Articles

  • HOMOGENIZATION OF INCOMPRESSIBLE EULER EQUATIONS. Hou, Thomas Y.; Dan-ping Yang; Ke Wang // Journal of Computational Mathematics;Mar2004, Vol. 22 Issue 2, p220 

    In this paper, we perform a nonlinear multiscale analysis for incompressible Euler equations with rapidly oscillating initial data. The initial condition for velocity field is assumed to have two scales. The fast scale velocity component is periodic and is of order one. One of the important...

  • On the integration of some classes of weakly deformed nonlinear Schrödinger equations. Zenchuk, A. I. // JETP Letters;8/10/97, Vol. 66 Issue 3, p222 

    A method is proposed for constructing the solutions of a nonlinear Schrödinger equation with small corrections arising as a result of the introduction of arbitrary functions of the time and coordinates into the operator that dresses the kernel of a local &∂macr; problem.

  • On a class of homogeneous nonlinear Schrödinger equations. Auberson, G.; Sabatier, P. C. // Journal of Mathematical Physics;Aug94, Vol. 35 Issue 8, p4028 

    A class of homogeneous, norm conserving, nonlinear wave equations of the Schrödinger type is studied. It is shown that those equations which derive from a Lagrangian can be linearized, but have no regular confined solutions, whereas the equations which cannot be obtained from a local...

  • Multiparameter Family of Collapsing Solutions to the Critical Nonlinear Schrödinger Equation in Dimension D = 2. Ovchinnikov, Yu. N.; Sigal, I. M. // Journal of Experimental & Theoretical Physics;Jul2003, Vol. 97 Issue 1, p194 

    We consider the critical nonlinear Schrödinger equation in dimension D = 2 and obtain a system consisting of three equations describing the collapse of solutions. The system admits a five-parameter family of solutions. Almost everywhere, except for an exponentially narrow region near the...

  • Meixner–Pollaczek polynomials and the Heisenberg algebra. Koornwinder, Tom H. // Journal of Mathematical Physics;Apr89, Vol. 30 Issue 4, p767 

    An alternative proof is given for the connection between a system of continuous Hahn polynomials and identities for symmetric elements in the Heisenberg algebra, which was first observed by Bender, Mead, and Pinsky [Phys. Rev. Lett. 56, 2445 (1986); J. Math. Phys. 28, 509 (1987)]. The continuous...

  • Non-Gaussian Lagrangian Feynman-Kac formulas. Sakbaev, V.; Smolyanov, O.; Shamarov, N. // Doklady Mathematics;Aug2014, Vol. 90 Issue 1, p416 

    The article discusses topics such as Feynman-Kac formulas, Lagrangian and Lebesgue measure. Schrodinger equation, evolution equations and Banach space are also presented. Other topics include operator measures and complex-valued measures. Wiener measure, Feynman pseudomeasure and class of semi...

  • New spectral collocation algorithms for one- and two-dimensional Schrödinger equations with a Kerr law nonlinearity. Bhrawy, Ali; Mallawi, Fouad; Abdelkawy, Mohamed // Advances in Difference Equations;1/25/2016, Vol. 2016 Issue 1, p1 

    A shifted Jacobi collocation method in two stages is constructed and used to numerically solve nonlinear Schrödinger equations (NLSEs) with a Kerr law nonlinearity, subject to initial-boundary conditions. An expansion in a series of spatial shifted Jacobi polynomials with temporal...

  • EXISTENCE AND CONCENTRATION OF POSITIVE SOLUTIONS FOR A QUASILINEAR ELLIPTIC EQUATION IN ℝ. GLOSS, ELISANDRA // Electronic Journal of Differential Equations;2010, Vol. 2010, Special section p1 

    We study the existence and concentration of positive solutions for the quasilinear elliptic equation -ε²u" - ε²(u²)"u + V (x)u = h(u) in ℝ as ε → 0, where the potential V : ℝ → ℝ has a positive infimum and inf∂Ω V > infΩ V for some...

  • Collapse in the Nonlinear Schr�dinger Equation of Critical Dimension {s = 1, D = 2}. Ovchinnikov, Yu. N.; Sigal, I. M. // JETP Letters;4/10/2002, Vol. 75 Issue 7, p357 

    Collapsing solutions to the nonlinear Schr�dinger equation of critical dimension {s = 1, D = 2} are analyzed in the adiabatic approximation. A three-parameter set of solutions is obtained for the scale factor ?(t). It is shown that the Talanov solution lies on the separatrix between the...


Read the Article


Sign out of this library

Other Topics