Elliptic problems with nonlinear terms depending on the gradient and singular on the boundary: Interaction with a Hardy-Leray potential

In this article we consider the following family of nonlinear elliptic problems, -Delta(u(m)) - lambda u(m)/vertical bar x vertical bar(2) = vertical bar Du vertical bar(q) + cf(x). We will analyze the interaction between the Hardy-Leray potential and the gradient term getting existence and nonexistence results in bounded domains Omega subset of R-N, N >= 3, containing the pole of the potential. Recall that Lambda(N) = (N - 2/2)(2) is the optimal constant in the Hardy-Leray inequality. 1. For 0 < m <= 2 we prove the existence of a critical exponent q(+) <= 2 such that for q > q(+), the above equation has no positive distributional solution. If q < q(+) we find solutions by using different alternative arguments. Moreover if q = q(+) > 1 we get the following alternative results. (a) If m < 2 and q = q(+) there is no solution. (b) If m = 2, then q(+) = 2 for all lambda. We prove that there exists solution if and only if 2 lambda <= A(N) and, moreover, we find infinitely many positive solutions. 2. If m > 2 we obtain some partial results on existence and nonexistence. We emphasize that if q(1m-1) < -1 and 1 < q <= 2, there exists positive solutions for any f is an element of L-1 (Omega)