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In this paper we consider the steady water wave problem for waves thatpossess a merely $L_r-$integrable vorticity, with $r\in(1,\infty)$ beingarbitrary. We first establish the equivalence of the three formulations--thevelocity formulation, the stream function formulation, and the height functionformulation-- in the setting of strong solutions, regardless of the value of$r$. Based upon this result and using a suitable notion of weak solution forthe height function formulation, we then establish, by means of localbifurcation theory, the existence of small amplitude capillary andcapillary-gravity water waves with a $L_r-$integrable vorticity.

Steady periodic water waves with unbounded vorticity: equivalent formulations and existence results