Semi-classical standing waves for nonlinear Schrodinger systems

Cited 7 time in webofscience Cited 0 time in scopus
  • Hit : 484
  • Download : 0
For N <= 3 and beta > 0, we consider the following singularly perturbed elliptic system { epsilon(2) Delta u(1) - W-1(x)u(1) + mu(1) (u(1))(3) + beta u(1) (u(2))(2) = 0, u(1) > 0 in R-N, epsilon(2) Delta u2 - W-2(x)u(2) + mu(2)(u(2))(3) + beta u(2)(u(1))(2) = 0, u(2) > 0 in R-N. There are an enormous number of results for localized solutions of singularly perturbed scalar problems using variational methods or finite dimensional reduction methods. However, there exist no general existence results of localized solutions for elliptic systems. We present some such results here. In the first, by a mini-max characterization for a limiting problem, for small epsilon > 0, we show the existence of one bump solutions with a common concentration point of u1, u2 in a domain O when certain conditions for the limiting problem are satisfied. Typical examples of potentials W1, W2 satisfying the condition are the following: (1) W-1, W-2 have a common non-degenerate critical point in O which share the same stable, unstable directions; (2) for the outnormal n on partial derivative O, partial derivative W-1/partial derivative n > 0, partial derivative W-2/partial derivative n > 0 or partial derivative W-1/partial derivative n < 0, partial derivative W-2/partial derivative n < 0 on partial derivative O ; (3) max(x is an element of O) W-i(x) >> max(x is an element of partial derivative O) W-i(x) for i = 1, 2. results for some potentials W1, W2, not satisfying these conditions, but each W-1, W-2 having a structurally stable critical point in O.
Publisher
SPRINGER HEIDELBERG
Issue Date
2015-10
Language
English
Article Type
Article
Keywords

SOLITARY WAVES; BOUND-STATES; R-N; EQUATIONS; POTENTIALS; SYMMETRY; PACKETS; SPIKES

Citation

CALCULUS OF VARIATIONS AND PARTIAL DIFFERENTIAL EQUATIONS, v.54, no.2, pp.2287 - 2340

ISSN
0944-2669
DOI
10.1007/s00526-015-0866-6
URI
http://hdl.handle.net/10203/205296
Appears in Collection
MA-Journal Papers(저널논문)
Files in This Item
There are no files associated with this item.
This item is cited by other documents in WoS
⊙ Detail Information in WoSⓡ Click to see webofscience_button
⊙ Cited 7 items in WoS Click to see citing articles in records_button

qr_code

  • mendeley

    citeulike


rss_1.0 rss_2.0 atom_1.0