Browsing by Author "Huang, Hsin-Yuan"

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Bubbling solutions for a skew-symmetric Chern-Simons system in a torus
(2017-08-15)
Han, Xiaosen
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Huang, Hsin-Yuan
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Lin, Chang-Shou
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應用數學系
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Department of Applied Mathematics
We establish the existence of bubbling solutions for the following skew-symmetric Chern Simons system Delta u(1) + 1/epsilon(2) e(u2) (1 - e(u1)) = 4 pi Sigma N-1 i=1 delta p(1/i) { Delta u(2) + 1/epsilon(2) e(u1) (1 - e(u2)) = 4 pi Sigma N-2 i=1 delta p(2/i) over a parallelogram Omega with doubly periodic boundary condition, where epsilon > 0 is a coupling parameter, and delta(p) denotes the Dirac measure concentrated at p. We obtain that if (N-1 - 1)(N-2 - 1) > 1, there exists an epsilon(o) > 0 such that, for any epsilon is an element of(0, epsilon(o)), the above system admits a solution (u(1),(epsilon), u(2),(epsilon)) satisfying u1,(epsilon) and u(2),(epsilon) blow up simultaneously at the point p*, and 1/epsilon(2) e(uj,k) (1 - e(ui,epsilon)) -> 4 pi N-i delta(p*) , 1 <= i, j <= 2 , i not equal j as epsilon -> 0, where the location of the point p* defined by (1.12) satisfies the condition (1.13). (C) 2017 Elsevier: Inc. All rights reserved.
Existence of bubbling solutions for the Liouville system in a torus
(2019-06-01)
Huang, Hsin-Yuan
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應用數學系
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Department of Applied Mathematics
We consider the following Liouville system on a parallelogram Omega in R-2: Delta u(i) + Sigma(n)(j=1) a(ij)rho(j) (h(j)e(uj)/integral(Omega)h(j)e(u)j - 1/vertical bar Omega vertical bar) = 0, i is an element of I = {1,..., n}, (0.1) where h(i) (x) is an element of C-3(Omega), h(i) (x) > 0, ui is doubly periodic on partial derivative Omega (i is an element of I), and A = (a(ij)) nxn is a non-negative constant matrix. We prove that if q is a non-degenerate critical point of Sigma n i= 1. * i log hi (x) and A satisfies certain conditions stated in Theorem 1.1, (0.1) has a sequence of fully bubbling solutions which blow up at p, as. = (.1,...,.n).. * = (. * 1,...,. * n), where. * satisfies 8p Sigma n i= 1. * i = Sigma n i= 1 Sigma nj = 1 ai j. * i. * j and Sigma n i= 1 ai j. * i. * j > 6p for j is an element of I.