Lane–Emden–Fowler equations with convection and singular potential

Abstract

We are concerned with singular elliptic problems of the form − Δ u ± p ( d ( x ) ) g ( u ) = λ f ( x , u ) + μ | ∇ u | a in Ω, where Ω is a smooth bounded domain in R N , d ( x ) = dist ( x , ∂ Ω ) , λ > 0 , μ ∈ R , 0 < a ⩽ 2 , and f is a nondecreasing function. We assume that p ( d ( x ) ) is a positive weight with possible singular behavior on the boundary of Ω and that the nonlinearity g is unbounded around the origin. Taking into account the competition between the anisotropic potential p ( d ( x ) ) , the convection term | ∇ u | a , and the singular nonlinearity g, we establish various existence and nonexistence results.