The logistic problem with nonlinear directed advection

Abstract

We are concerned with the study of positive solutions to the logistic model −Δu+a(x)⋅∇up=λu−b(x)f(u)in Ω,u=0on ∂Ω, where Ω is a smooth and bounded domain of the Euclidean space RN (N≥2), 0<p<1, a∈C(Ω¯)N, b∈C(Ω¯), b≥0. It is assumed that λ>0 is a real parameter and f∈C1[0,∞) is nonnegative and satisfies some growth conditions at 0 and ∞. The two main difficulties in the study of the above problem are the indefinite sign of the advection term a(x)⋅∇up and the fact that ∇up=pup−1∇u may be singular around the boundary of Ω. We discuss the existence of a positive solution as well as its C1-regularity at ∂Ω. Our results extend those recently obtained in Cintra et al. (2022) for the case p>1.