Letf be a non-decreasing C1-function such that$$f > 0 on (0,\infty ), f(0) = 0, \int_1^\infty {1/\sqrt {F(t)} dt< \infty } $$ andF(t)/f2a(t)→ 0 ast → ∞, whereF(t)=∫0tf(s) ds anda ∈ (0, 2]. We prove the existence of positive large solutions to the equationΔu +q(x)|Δu|a=p(x)f(u) in a smooth bounded domain Ω ⊂RN, provided thatp, q are non-negative continuous functions so that any zero ofp is surrounded by a surface strictly included in Ω on whichp is positive. Under additional hypotheses onp we deduce the existence of solutions if Ω is unbounded.
