• Title/Summary/Keyword: biharmonic elliptic system

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MULTIPLICITY OF SOLUTIONS FOR BIHARMONIC ELLIPTIC SYSTEMS INVOLVING CRITICAL NONLINEARITY

  • Lu, Dengfeng;Xiao, Jianhai
    • Bulletin of the Korean Mathematical Society
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    • v.50 no.5
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    • pp.1693-1710
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    • 2013
  • In this paper, we consider the biharmonic elliptic systems of the form $$\{{\Delta}^2u=F_u(u,v)+{\lambda}{\mid}u{\mid}^{q-2}u,\;x{\in}{\Omega},\\{\Delta}^2v=F_v(u,v)+{\delta}{\mid}v{\mid}^{q-2}v,\;x{\in}{\Omega},\\u=\frac{{\partial}u}{{\partial}n}=0,\; v=\frac{{\partial}v}{{\partial}n}=0,\;x{\in}{\partial}{\Omega},$$, where ${\Omega}{\subset}\mathbb{R}^N$ is a bounded domain with smooth boundary ${\partial}{\Omega}$, ${\Delta}^2$ is the biharmonic operator, $N{\geq}5$, $2{\leq}q$ < $2^*$, $2^*=\frac{2N}{N-4}$ denotes the critical Sobolev exponent, $F{\in}C^1(\mathbb{R}^2,\mathbb{R}^+)$ is homogeneous function of degree $2^*$. By using the variational methods and the Ljusternik-Schnirelmann theory, we obtain multiplicity result of nontrivial solutions under certain hypotheses on ${\lambda}$ and ${\delta}$.

FINITE ELEMENT APPROXIMATION OF THE DISCRETE FIRST-ORDER SYSTEM LEAST SQUARES FOR ELLIPTIC PROBLEMS

  • SHIN, Byeong-Chun
    • Communications of the Korean Mathematical Society
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    • v.20 no.3
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    • pp.563-578
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    • 2005
  • In [Z. Cai and B. C. Shin, SIAM J. Numer. Anal. 40 (2002), 307-318], we developed the discrete first-order system least squares method for the second-order elliptic boundary value problem by directly approximating $H(div){\cap}H(curl)-type$ space based on the Helmholtz decomposition. Under general assumptions, error estimates were established in the $L^2\;and\;H^1$ norms for the vector and scalar variables, respectively. Such error estimates are optimal with respect to the required regularity of the solution. In this paper, we study solution methods for solving the system of linear equations arising from the discretization of variational formulation which possesses discrete biharmonic term and focus on numerical results including the performances of multigrid preconditioners and the finite element accuracy.