Existence and non-existence phenomena for nonlinear elliptic equations with L1 data and singular reactions

We study existence and non-existence of solutions for singular elliptic boundary value problems as (1) { − Δ p u + a ( x ) u γ = μ f ( x ) in Ω , u > 0 in Ω , u = 0 on ∂ Ω , where Ω is a smooth bounded open subset of R N ( N ≥ 2 ), Δ p u is the p -Laplacian with p > 1 , 0 < γ ≤ 1 , and a ≥ 0 is bounded and non-trivial. For any positive f ∈ L 1 ( Ω ) we show that problem ( 1 ) is solvable for any μ > μ 0 > 0 , for some μ 0 large enough. As a reciprocal outcome we also show that no finite energy solution exists if 0 < μ < μ 0 ⁎ , for some small μ 0 ⁎ . This paper extends the celebrated one of J. I. Diaz, J. M. Morel and L. Oswald ( [16] ) to the case p ≠ 2 . Our result is also new for p = 2 provided the singular term has a critical growth near zero (i.e. γ = 1 ).