Positive entire stable solutions of inhomogeneous semilinear elliptic equations

Abstract

For n ≥ 3 and p > 1 , the elliptic equation Δ u + K ( x ) u p + μ f ( x ) = 0 in R n possesses a continuum of positive entire solutions, provided that (i) locally Hölder continuous functions K and f vanish rapidly, for instance, K ( x ) , f ( x ) = O ( | x | l ) near ∞ for some l < − 2 and (ii) μ ≥ 0 is sufficiently small. Especially, in the radial case with K ( x ) = k ( | x | ) and f ( x ) = g ( | x | ) for some appropriate functions k , g on [ 0 , ∞ ) , there exist two intervals I μ , 1 , I μ , 2 such that for each α ∈ I μ , 1 the equation has a positive entire solution u α with u α ( 0 ) = α which converges to l ∈ I μ , 2 at ∞ , and u α 1 < u α 2 for any α 1 < α 2 in I μ , 1 . Moreover, the map α to l is one-to-one and onto from I μ , 1 to I μ , 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 also weakly asymptotically stable in the weighted uniform norm with weight function | x | n − 2 .