ELLIPTIC EQUATIONS HAVING A SINGULAR QUADRATIC GRADIENT TERM AND A CHANGING SIGN DATUM

In this paper we study a singular elliptic problem whose model is { -Delta u = vertical bar del u vertical bar(2)/vertical bar u vertical bar(theta) + f(x), in Omega; u = 0, on partial derivative Omega; where theta is an element of (0, 1) and f is an element of L-m (Omega), with m >= N/2. We do not assume any sign condition on the lower order term, nor assume the datum f has a constant sign. We carefully de fi ne the meaning of solution to this problem giving sense to the gradient term where u = 0, and prove the existence of such a solution. We also discuss related questions as the existence of solutions when the datum f is less regular or the boundedness of the solutions when the datum f is an element of L-m (Omega) with m > N/2.