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

Abstract

In this article we consider the following family of nonlinear elliptic problems, [equation presented] We will analyze the interaction between the Hardy-Leray potential and the gradient term getting existence and nonexistence results in bounded domains Ώ = RN, N ≥ 3, containing the pole of the potential. Recall that [equation presented] 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 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 λ. We prove that there exists solution if and only if 2λ ≤ ΛN 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 [equation presented] < -1 and 1 < q = 2, there exists positive solutions for any f L 1(Ω).