A remark on the existence of positive entire solutions of a sublinear elliptic problem

Abstract

By means of a sub–supersolutions argument and a perturbed argument, we show the existence of entire solutions to a semilinear elliptic problem − △ u = b ( x ) g ( u ) , u > 0 , x ∈ R N , lim | x | → ∞ u ( x ) = 0 , where b ∈ C loc α ( R N ) for some α ∈ ( 0 , 1 ) and b ( x ) > 0 , ∀ x ∈ R N , g ∈ C 1 ( ( 0 , ∞ ) , ( 0 , ∞ ) ) which may be singular at 0. No monotonicity condition is imposed on the functions g ( s ) and g ( s ) / s .