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This paper treats nonlinear elliptic boundary value problems of the form (1) L[u] = p(x,u) in $\Omega \subset {ℝ}^n$, $u=Du=...=D^{m-1}u$ on ∂Ω in the Sobolev space $W_0^{m,2}\left(\Omega \right)$, where L is any selfadjoint strongly elliptic linear differential operator of order 2m. Using both topological degree arguments and minimax methods we obtain existence and multiplicity results for the above problem.