Existence and Regularity for Solution to a Degenerate Problem with Singular Gradient Lower Order Term

We study the existence and regularity results for non-linear elliptic equation with degenerate coercivity and a singular gradient lower order term. The model problems is { -div(b(x)| ∇u |p-2∇u(1+| u |)γ)+| ∇u |p| u |θ=f,in Ω,u=0,on ∂Ω, \left\{ {\matrix{ { - div\left( {b\left( x \right){{{{\left| {\nabla u} \right|}^{p - 2}}\nabla u} \over {\left( {1 + \left| u \right|} \right)\gamma }}} \right) + {{{{\left| {\nabla u} \right|}^p}} \over {{{\left| u \right|}^\theta }}} = f,} \hfill & {in\,\Omega ,} \hfill \cr {u = 0,} \hfill & {on\,\partial \Omega ,} \hfill \cr } } \right. swhere Ω is a bounded open subset in ℝ^N, 1 ≤ θ < 2, p > 2 and γ > 0. We will show that, even if the lower order term is singular, we obtain existence and regularity of positive solution, under various assumptions on the summability of the source f.