Positive solutions to a class of quasilinear elliptic equations on R

Abstract. We discuss the existence of positive solutions of perturbation to a classof quasilinear elliptic equations on R.1. Introduction. This Note is concerned with solutions of perturbations to thefollowing problem on the real line R(−u 00 +u−k(u 2 ) 00 u = u p ,u ∈ H 1 (R) u > 0,(1)where k ∈ R and p > 1. Consider(−u 00 +(1+ea(x))u−k(1+eb(x))(u 2 ) 00 u = (1+ec(x))u p ,u ∈ H 1 (R) u > 0,(2)where a,b,c are assumed to be real valued functions belonging to the class S,S = {h(x) = h 1 (x)+h 2 (x) : h 1 ∈ L r (R)∩L ∞ (R),for some r ∈ [1,∞), h 2 ∈ L ∞ (R), lim |x|→∞ h 2 (x) = 0}.Our main existence result is the followingTheorem 1.1. There is k 0 > 0 such that for k > −k 0 and a,b,c ∈ S, equation (2)has a solution provided |e| is sufficiently small.Solutions of (2) will be found as critical points of a functional I e of the formI e (u) = I 0 (u)+eG(u), u ∈ H 1 (R), (3)whereI 0 (u) =12Z R |u 0 | 2 +u 2 dx+kZ R u 2 |u 0 | 2 dx−1p+1Z R |u| p+1 dx (4)and G is the perturbationG(u) =12Z