Asymptotic behavior of positive solutions for quasilinear elliptic equations

摘要

We systematically inquire positive radial solutions of the quasilinear elliptic equation -Delta(p)phi = vertical bar x vertical bar(sigma)phi(q) with Delta(p)phi := div(vertical bar del phi vertical bar(p-2)del phi) is the p-Laplace operator, q > p-1, sigma > -p and 1 < p < N. It is known that the radial solutions of this equation can be classified into three different types: the M-solutions (singular at r = 0), the E-solutions (regular at r = 0) and the F-solutions (whose existence begins away from r = 0). For these radial solutions, we find a substitution which can transfer this equation into an autonomous Lotka-Volterra system, such that the F-, E- and M-solutions can be comprehensively characterized. In particular, the M-solution has extremely plentiful properties, and its asymptotic expansions are more complicated. For the subcritical case of q, by virtue of a priori estimates, we employ an iterative method which can improve the accuracy step by step, to derive their precise asymptotic expansions. For the supercritical case of q, with the help of the autonomous Lotka-Volterra system and an inverse transformation, we obtain the asymptotic expansions of the M-solutions near the origin. In the same manner, we could acquire more accurate asymptotic expansions of the E-solutions via the iterative method whereas the F-solutions can be described by the autonomous Lotka-Volterra system.