Uniqueness of solutions for elliptic systems and fourth order equations involving a parameter

Craig  Cowan

doi:10.3934/cpaa.2016.15.519

Article Contents

2016, Volume 15, Issue 2: 519-533. Doi: 10.3934/cpaa.2016.15.519

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Uniqueness of solutions for elliptic systems and fourth order equations involving a parameter

Craig Cowan^1,

1.
Department of Mathematics, University of Manitoba, Winnipeg, MB R3T 2N2

Received: March 31, 2015

Revised: August 31, 2015

Published: February 2016

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Abstract

We examine the equation $$ \Delta^2 u = \lambda f(u) \qquad \Omega, $$ with either Navier or Dirichlet boundary conditions. We show some uniqueness results under certain constraints on the parameter $ \lambda$. We obtain similar results for the sytem \begin{eqnarray} &-\Delta u = \lambda f(v) \qquad \Omega, \\ &-\Delta v = \gamma g(u) \qquad \Omega, \\ &u= v = 0 \qquad \partial \Omega. \end{eqnarray}

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Mathematics Subject Classification: 35J55, 35B05, 35K50, 35K55.

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