Existence results for nonlinear elliptic equations with degenerate coercivity

with Ω a bounded open subset of R , N ≥ 2, p > 1, θ ≥ 0, and f a measurable function on whose summability we will make different assumptions. It is clear from the above example that the differential operator is defined on W 0 (Ω), but that it may not be coercive on the same space as u becomes large. Due to this lack of coercivity, standard existence theorems for solutions of nonlinear elliptic equations cannot be applied. In this paper, we will prove several existence and regularity results (depending on the summability of the datum f ) for the solutions of (1.1). Let us give the precise assumptions on the problems that we will study. Let Ω be a bounded open subset of RN , N ≥ 2. Let 1 < p < N, and let a : Ω × R × RN → RN be a Carathéodory function (that is, a(·, t, ξ) is measurable on Ω for every (t, ξ) in R × RN , and a(x, ·, ·) is continuous on R×RN for almost every x in Ω), such that the following assumptions hold: a(x, t, ξ) · ξ ≥ b(|t|) |ξ|p , (1.2)