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

We examine the equation $Δ^2 u = λf(u) \qquad Ω,$ with either Navier or Dirichlet boundary conditions. We show some uniqueness results under certain constraints on the parameter $λ$. We obtain similar results for the sytem {equation*} \{{array}{rrl} -Δu &=& λf(v) \qquad Ω, -Δv &=& γg(u) \qquad Ω, u&=& v = 0 \qquad \partial Omega. {array}. {equation*}

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