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Minimization Methods for Quasi-Linear Problems with an Application to Periodic Water Waves

Penalization and minimization methods are used to give an abstract semiglobal result on the existence of nontrivial solutions of parameter-dependent quasi-linear differential equations in variational form. A consequence is a proof of existence, by infinite-dimensional variational means, of bifurcation points for quasi-linear equations which have a line of trivial solutions. The approach is to penalize the functional twice. Minimization gives the existence of critical points of the resulting problem, and a priori estimates show that the critical points lie in a region unaffected by the leading penalization. The other penalization contributes to the value of the parameter. As applications we prove the existence of periodic water waves, with and without surface tension.