A note on a critical problem with natural growth in the gradient

Abstract

The paper analyzes the influence on the meaning of natural growth in the gradient of a perturbation by a Hardy potential in some elliptic equations. Indeed, in the case of the Laplacian the natural problem becomes -Δu - ΛN-u/|x|2= |▽u + N - 2/2 u/|x| 2x|2 |x|(N-2)/2 + λf(x) in Ω, u=0 on ∂Ω, ΛN = ((N - 2)/2)2. This problem is a particular case of problem (2). Notice that (N - 2)/2 is optimal as coefficient and exponent on the right hand side. © European Mathematical Society 2006.