p-Laplacian diffusion coupled to logistic reaction: asymptotic behavior as p goes to 1

This work discusses the limit as p goes to 1 of solutions to problem {-Delta(p)u = lambda vertical bar u vertical bar(p-2)u - vertical bar u vertical bar(q-2)u, x is an element of Omega, (P) u = 0 x is an element of partial derivative Omega, where Omega is a bounded smooth domain of R-N, lambda > 0 is a parameter and the exponents p, q satisfy 1 < p < q. Our interest is focused on the radially symmetric case. We prove in this radial setting that solutions u(p) to (P) converge to a limit u as p -> 1+. Moreover, the limit function u defines a solution to the natural 'limit problem' which involves the 1-Laplacian operator. In addition, a precise description of the structure of the set of all possible solutions to such a problem is achieved. This is accomplished by means of the introduction of a suitable energy condition. Furthermore, a detailed analysis of the profiles of all these solutions is also performed.