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I’m interested in solving nonlinear elliptic boundary value problems of the type $$ -a\Delta u + f(u) = 0, $$ $$ u|_\Gamma = u_0 $$ by Newton’s method when its convergence is global and monotonic. Could you advice some references concerning this problem, containing proofs of global convergence?

Newton's method takes the form $$ -a\Delta u + f(\widetilde u) + f'(\widetilde u)(u - \widetilde u) = 0 $$ where $\widetilde u$ is the previous approximation for the solution.

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