연구홍보포털

논문 초록 (Abstract)

For n >= 3 and p > 1, the elliptic equation Delta u + K(x)u(p) + mu f (x) = 0 in R(n) possesses a continuum of positive entire solutions, provided that (i) locally Holder continuous functions K and f vanis h rapidly, for instance, K(x), f (x) = O(vertical bar x vertical bar( l)) near infinity for some l < -2 and (ii) mu >= 0 is sufficientl y small. Especially, in the radialcase with K(x) = k(vertical bar x v ertical bar) and f (x) = g(vertical bar x vertical bar) for some appr opriate functionsk, g on [0, infinity), there exist two intervals I( mu, 1)l, I(mu, 2) such that for each alpha is an element of I(mu, 1) t he equation has a positive entire solution u(alpha) with u(alpha )(0) = alpha which converges to l is an element of I(mu, 2) at infin ity and u(alpha 1) < u(alpha 2) for any alpha(1) < alpha( 2) in I(mu, 1). Moreover, the map a to l is one-to-one and onto f rom I(mu, 1) to I(mu, 2). If K >= 0, each solution regarded as a steady state for the corresponding parabolic equation is stable in the uniform norm; moreover, in the radial case the solutions are al so weakly asymptotically stable in the weighted uniform nor m with weight function vertical bar x vertical bar(n-2). (C) 2011 Elsevi er Ltd. All rights reserved.