Elliptic problems with a Hardy potential and critical growth in the gradient: Non-resonance and blow-up results

In this article we analyze existence and nonexistence of positive solutions to problem(P±)−Δu±|∇u|2=λu|x|2+fin Ω,u=0on ∂Ω. The main results are the following:(i)If the quadratic term in the gradient appears in the equation as a reaction term (−|∇u|2−|∇u|2) and λ>0λ>0, then there is no solution to problem (P−)(P−) (even in a very weak sense).(ii)If the quadratic term in the gradient appears in the equation as an absorption term (+|∇u|2+|∇u|2), then there exists a positive solution to (P+)(P+) for all λ>0λ>0 and f∈L1(Ω)f∈L1(Ω).