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

Abstract

This work discusses the limit as p goes to 1 of solutions to problem P-Δpu=λ|u|p-2u-|u|q-2u,x∈Ω,u=0x∈∂Ω,\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\begin{aligned} {\\left\\{ \\begin{array}{ll} -\\varDelta _p u = \\lambda |{u}|^{{p}-2}{u} -|{u}|^{{q}-2}{u},&{} \\qquad x\\in \\varOmega ,\\\\ \\ u=0 &{}\\qquad x\\in \\partial \\varOmega , \\end{array}\\right. } \\end{aligned}$$\\end{document}where varOmega is a bounded smooth domain of {mathbb {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 prightarrow 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.