Asymptotic Behavior of Ground State Radial Solutions for pp-Laplacian Problems

Let pp p p, we take up the existence, the uniqueness and the asymptotic behavior of a positive continuous solution to the following nonlinear problem in (0, +∞), (p/AA)(AAAApp(uu )) ′ + qq(qq)uu = 0, limqqx0AAAApp(uu )(qq) = 0, limqqx+∞uu(qq) = 0, where αα α pp α p, AApp(tt) = tttttt ppαp (tt t t), A is a positive differentiable function in (0, +∞) and q is a positive continuous function in (0, +∞) such that there exists cc p 0 satisfying for each x in (0, +∞), p/cc c qq(qq)(p + qq) exp(α ∫ p (zz(zz)/zz)zzzz) c cc, ββ β pp and zz t zz(zp, +∞)) such that limttx+∞zz(tt) = 0.

In this paper, our main purpose is to obtain the existence of a unique positive solution to the following boundary value problem: and to establish estimates on such solution under an appropriate condition on .e study of this type of ( 1) is motivated by [5,15].Namely, in the special case () =   (),     ( + ) and () =   ,   , the authors in [15] studied (1) and gave some uniqueness results.In this work, we consider a wider class of weights  and we aim to extend the study of (1) in [15], to     .
For the case   , the problem () has been studied in [5].Indeed, the authors of [5] proved the following existence result.eorem 1. e problem () has a unique positive solution   ( )) satisfying for each   ( ) where  is a positive constant and    is the function de�ned on ( ) by We shall improve in this paper the above asymptotic behavior of the solution of problem () and we extend the study of () to  ≤     1. e pure elliptic problem of type has been investigated by several authors with zero Dirichlet boundary value; we refer the reader to [16][17][18][19][20][21][22][23][24] and the references therein.More recently, applying Karamata regular variation theory, Chemmam et al. gave in [17] the asymptotic behavior of solutions of problem ().In this work, we aim to extend the result established in [17] to the radial case associated to problem ().�o simplify our statements, we need to �x some notations and make some assumptions.roughout this paper, we shall use , to denote the set of Karamata As it is mentioned above, our main purpose in this paper is to establish existence and global behavior of a positive solution of problem ().Let us introduce our hypotheses.
Here, the function  is continuous in  ), differentiable and positive in ( ) such that  ()    (8) with     1. e function  is required to satisfy the following hypothesis.
Remark 2. We need to verify condition in hypothesis (), only if  = , (see Lemma 6 below).
As a typical example of function  satisfying (), we quote the following.
e main body of the paper is organized as follows.In Section 2, we establish some estimates and we recall some known results on functions belonging to .eorem 4 is proved in Section 3. e last Section is reserved to some applications.

Key Estimates
In what follows, we are going to give estimates on the functions    and   (   ), where  is a function satisfying () and   is the function given by (12).First, we recall some fundamental properties of functions belonging to the class , taken from [17,22] , Proof.For  ∈ (0, ∞), we have To prove the result, it is sufficient to show that ℎ() ≈ Ψ() for  ∈ (0, ∞).
Since the function ℎ is continuous and positive in [0, ], we have Now, assume that  ∈ [, ∞), then we have Hence, we reach the result by combining (22) with the estimates stated in each case above.is completes the proof.
Proposition 10.Let  be a function satisfying () and let   be the function given by (12).en for   ( ∞), one has where   .en, using Lemmas 5, 7, and 8, we obtain that     and ∫ ∞   (−)(−) (  ()) (−)   ∞.Hence, it follows from Proposition 9 that where   is the function de�ned in (19) by replacing  by   and  by .is ends the proof.

Proof of Theorem 4
3.1.Existence and Asymptotic Behavior.Let  be a function satisfying () and let   be the function given by (12).By Proposition 10, there exists a constant    such that for each   ( ∞) We look now at the existence of positive solution of problem () satisfying (11).
For the case   , the existence of a positive continuous solution to problem () is due to [5].Now, we look to the existence result of problem () when  ≤    −  and we give precise asymptotic behavior of such solution for    − .For that, we split the proof into two cases.