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

- Appears in Collection
- MA-Journal Papers(저널논문)

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