TITLE

A local mountain pass type result for a system of nonlinear Schr�dinger equations

AUTHOR(S)
Ikoma, Norihisa; Tanaka, Kazunaga
PUB. DATE
March 2011
SOURCE
Calculus of Variations & Partial Differential Equations;Mar2011, Vol. 40 Issue 3/4, p449
SOURCE TYPE
Academic Journal
DOC. TYPE
Article
ABSTRACT
We consider a singular perturbation problem for a system of nonlinear Schr�dinger equations: where N = 2, 3, �, �, � > 0 and V( x), V( x): R ? (0, 8) are positive continuous functions. We consider the case where the interaction � > 0 is relatively small and we define for $${P\in{\bf R}^N}$$ the least energy level m( P) for non-trivial vector solutions of the rescaled 'limit' problem:We assume that there exists an open bounded set $${\Lambda\subset{\bf R}^N}$$ satisfyingWe show that (*) possesses a family of non-trivial vector positive solutions $${\{(v_{1\varepsilon}(x), v_{2\varepsilon} (x))\}_{\varepsilon\in (0,\varepsilon_0]}}$$ which concentrates-after extracting a subsequence e ? 0-to a point $${P_0\in\Lambda}$$ with $${m(P_0)={\rm inf}_{P\in\Lambda}m(P)}$$. Moreover ( v( x), v( x)) converges to a least energy non-trivial vector solution of (**) after a suitable rescaling.
ACCESSION #
57389810

 

Related Articles

  • Existence of solutions to a class of quasilinear Schrödinger systems involving the fractional p-Laplacian. Mingqi Xiang; Binlin Zhang; Zhe Wei // Electronic Journal of Qualitative Theory of Differential Equatio;2016, Issue 1-122, p1 

    The purpose of this paper is to investigate the existence of solutions to the following quasilinear Schrödinger type system driven by the fractional p-Laplacian (-Δ)psu + a(x)|u|p-2u = Hu(x, u, v) in RN, (-Δ)qsv + b(x)|v|q-2v = Hv(x, u, v) in RN, where 1 < q ≤ p, sp < N,...

  • Modulational instability and exact solutions for a three-component system of vector nonlinear Schrödinger equations. Yomba, Emmanuel; Sell, George R. // Journal of Mathematical Physics;May2009, Vol. 50 Issue 5, p053518 

    The modulational instability (MI) of the three-component system of vector nonlinear Schrödinger equations is investigated. It is found that there are a number of possibilities for the MI regions due to the generalized nonlinear dispersion relation, which relates the frequency and the wave...

  • POSITIVE SOLUTIONS FOR QUASILINEAR SCHRÖDINGER EQUATIONS WITH CRITICAL GROWTH AND POTENTIAL VANISHING AT INFINITY. YINBIN DENG; WEI SHUAI; Zhi-qiang Wang // Communications on Pure & Applied Analysis;Nov2014, Vol. 13 Issue 6, p2273 

    This paper is concerned with the existence of positive solutions for a class of quasilinear Schrödinger equations in ⃿N with critical growth and potential vanishing at infinity. By using a change of variables, the quasilinear equations are reduced to semilinear one. Since the potential...

  • Truncation Analysis for the Derivative Schrödinger Equation. Xu Peng Cheng; Chang Qian Shun; Guo Bo Ling // Acta Mathematica Sinica;2002, Vol. 18 Issue 1, p137 

    The truncation equation for the derivative nonlinear Schrödinger equation has been discussed in this paper. The existence of a special heteroclinic orbit has been found by using geometrical singular perturbation theory together with Melnikov's technique.

  • Singular boundary perturbations for some eigenvalue problems. Zhevandrov, P.; Alcántar, E. // Journal of Mathematical Physics;May2000, Vol. 41 Issue 5 

    By means of two examples arising from physics we show that in contrast to a small perturbation of a regular boundary point, a small displacement of a singular boundary is singular in the sense that the expansions of the perturbed eigenvalues contain not only the integer powers of the small...

  • Standing waves of the Schrödinger equation coupled with the Chern-Simons gauge field. Huh, Hyungjin // Journal of Mathematical Physics;Jun2012, Vol. 53 Issue 6, p063702 

    We prove the existence of infinitely many radially symmetric standing-wave solutions of the Chern-Simons-Schrödinger system. Our result is established by applying the mountain-pass theorem to the functional, which is obtained by representing gauge fields Aμ in terms of a scalar field...

  • Quasilinear asymptotically periodic Schrödinger equations with critical growth. Silva, Elves A. B.; Vieira, Gilberto F. // Calculus of Variations & Partial Differential Equations;Sep2010, Vol. 39 Issue 1/2, p1 

    It is established the existence of solutions for a class of asymptotically periodic quasilinear elliptic equations in $${\mathbb{R}^N}$$ with critical growth. Applying a change of variable, the quasilinear equations are reduced to semilinear equations, whose respective associated functionals are...

  • EXISTENCE OF SOLUTIONS FOR SINGULARLY PERTURBED SCHRÖDINGER EQUATIONS WITH NONLOCAL PART. MINBO YANG; YANHENG DING // Communications on Pure & Applied Analysis;Mar2013, Vol. 12 Issue 2, p771 

    In the present paper we study the existence of solutions for a nonlocal Schrödinger equation --ε² Δu + V (x)u = (Δℝ³ |u|p/|x - y|µ dy)|u|p-2u, where 0 < µ < 3 and 6-µ/3 < p < 6 - µ. Under suitable assumptions on the potential V (x), if the parameter ε is...

  • ON THE MOUNTAIN-PASS ALGORITHM FOR THE QUASI-LINEAR SCHRÖDINGER EQUATION. GRUMIAU, CHRISTOPHER; SQUASSINA, MARCO; TROESTLER, CHRISTOPHE // Discrete & Continuous Dynamical Systems - Series B;Jul2013, Vol. 18 Issue 5, p1345 

    We discuss the application of the Mountain Pass Algorithm to the so-called quasi-linear Schrödinger equation, which is naturally associated with a class of nonsmooth functionals so that the classical algorithm cannot directly be used. A change of variable allows us to deal with the lack of...

Share

Read the Article

Courtesy of VIRGINIA BEACH PUBLIC LIBRARY AND SYSTEM

Sign out of this library

Other Topics