Positive, unbounded and monotone solutions of the singular second Painlevé equation on the half-line

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Abstract

In this paper we study singular boundary value problems on the half-line and we prove the existence of a global, monotone, positive and unbounded solution. The latter satisfies a Neumann condition at the origin and has prescribed asymptotic behavior at infinity. Our approach is based on a generalization of the Kneser's property (continuum) of the cross-sections of the solutions funnel, i.e. on the properties of the so-called consequent mapping and on properties of the associated vector field on the face space. Two applications, one on the well-known second Painlevé-type equation (which is related to superconductivity theory) and a second in the theory of colloids, clarify our results.

Introduction

In the study of radially symmetric solutions of nonlinear elliptic equations, boundary value problems on the half-line arise quite naturally. Among many authors who have proposed different results in this area (see [3], [5], [14], [19]) D. Guo [8] has studied a two-point boundary value problem (BVP) for second-order nonlinear impulsive integro-differential equations of mixed type. In particular, he has proven the existence of unique solutions on infinite intervals in a Banach space by employing a Lipschitz condition for the nonlinear term. Also, Liu [13], using the Tychonov fixed point theorem, obtained a result on the existence of unbounded solutions for boundary value problems on the half-line.

In a recent paper Yan [20], based on the Leray–Schauder Theorem, proved the existence of unbounded solutions of the boundary value problem1p(t)(p(t)x′(t))′+f(t,x(t))=0,x(0)=a⩾0,limt→∞p(t)x′(t)=b⩾0,under the assumptionsf(t,x)⩽q0(t)Φx(t)1+0t(1/p(s))ds,supc>0csupt⩾0b0t1/p(s)ds+a1+0t1/p(s)ds+sup0⩽y⩽cΦ(y)(0+∞p(t)q0(t)dt)>1,where q0,ΦC((0,+∞),(0,+∞)) and 0+∞p(t)q0(t)dt<∞.

Motivated by the above-mentioned results, we propose in this paper a new approach to the singular BVP on the half-line. More precisely, we consider the Banach spaceK1[0,+∞)={u∈C([0,+∞),R),pu′∈BC([0,+∞),R)},(where BC stands for the space of continuous and bounded functions and C for the space of those continuous maps with finite result under the norm ||u|| defined below) endowed with the norm||u||1=max{||u||,||pu′||},where||pu′||≔supt⩾0{|p(t)u′(t)|}and||x||supt⩾0|x(t)|1+0t(1/p(s))ds.We notice here that, assuming0⩽p(t)x′(t)⩽M,t⩾0for a solution x=x(t) of1p(t)(p(t)x′(t))′+q(t)f(t,x(t),p(t)x′(t))=0,x(0)=a⩾0,limt→0+p(t)x′(t)=x0′⩾b⩾0,by a simple integration we obtain||x||⩽||px′||+a⩽M+a.

On the other hand, we consider our nonlinearity f(t,x,px′) depending on the additional variable p(t)x′(t) and we restrict our assumptions to simpler and more natural ones than those of Yan. Namely, instead of (1.2) we demandf(t,x,px′)⩽q0(t)Φ(||x||1),sup0⩽y⩽cΦ(y)<ε0c0+∞p(t)q(t)q0(t)dt−1forsomeε0<1.The obtained unbounded solution is then positive and strictly increasing and further its “derivative” p(t)x′(t) is also positive but now strictly decreasing.

Concerning the technique employed in this paper, it is worthy to note that our approach is considerably different than that of Yan since we rely on the Kneser's property (continuum) of the cross-sections of the solutions funnel. More precisely, the continuum property of the latter is taken as the cross-sections of the funnel and the boundary of a certain (unbounded) set in the (x,px′) plane. These cross-sections give rise to the so-called (singular) consequent mapping, the properties of which (see [15], [17], [18]) lead to our existence results.

In the last part of the paper we clarify our methods based on two examples. First, we consider the Painlevé-type boundary value problem in a semi-infinite interval (0,+∞)x″=x4−tx2,x′(0)=0andlimt→+∞x(t)t=1,and we prove that there exists a global strictly positive and monotone solution of it. Hasting and McLeod [9], Helffer and Weissler [10], Levi and Winternitz [12] and Guedda [7], employed, via the shooting method, the monoparametric equationx″=x3−(t−c)x,c∈R,as a very useful mathematical model to the superheating field attached to a semi-infinite superconductor (see [4]). In the special case where c=0, we obtain the existence of a strictly positive and decreasing solution of the above equation, satisfying furthermore the new boundary conditionsx(0)=aandlimt→−∞x(t)=limt→−∞x′(t)=0.Secondly, we study the BVPu″(x)=2sinhu(x),u(0)=c>0,limx→+∞u(x)=0=limx→+∞u′(x),which models a system in the theory of colloids (see the recent paper of Agarwal and O'Regan [2]). A positive decreasing solution with a sharper asymptotic behavior-estimation than the analogous one in [2] (see Remark 2) is obtained.

Section snippets

Preliminaries

Consider the following boundary value problem:1p(t)(p(t)x′(t))′+q(t)f(t,x(t),p(t)x′(t))=0,x(0)=a⩾0,limt→∞p(t)x′(t)=b⩾0,where f:Ω≔[0,+∞)2×R→[0,+∞),p∈C([0,+∞),R)∩C1(0+∞),p(t)>0andq(t)⩾0fort∈(0,+∞).

The two conditions below, will be assumed throughout:0t1p(s)ds<∞and0tp(s)q(s)ds<∞foranyt⩾0.

Let also1p(t)(p(t)x′(t))′+q(t)f(t,x(t),p(t)x′(t))=0,x(0)=x0⩾0,p(0)x′(0)≔limt→0+p(t)x′(t)=x0be the corresponding initial value problem. Then, a vector field is defined with crucial properties for our study. More

Main result

Consider the boundary value problem (2.1). For any function x in C[0,+∞] such that limt→∞p(t)x′(t)<∞, we define||x||1max{||x||,||px′||},where we set (and recall) that||x||supt⩾0|x(t)|1+0t(1/p(s))dsand||px||≔supt⩾0{|p(t)x(t)|}.List now the next three conditions:0⩽f(t,x,px′)⩽q0(t)Φ(||x||1),t⩾0,x⩾aandpx′⩾0,where q0∈C((0,+∞),[0,+∞)),Φ∈C([0,+∞),[0,+∞)),0+∞p(t)q(t)q0(t)dt<∞,sup0⩽y⩽cΦ(y)<ε0c0+∞p(t)q(t)q0(t)dt−1,for some c>a+b and 0<ε0<1−[(a+b)/c].

Theorem 5

Assume that conditions (3.1)–(3.3) hold. Then,

A Painlevé-type equation

Consider the Painlevé-type equation in a semi-infinite interval (0,+∞)x″=x4−tx2with a Neumann condition at the origin:x′(0)=0,having a prescribed behavior at +∞:limt→+∞x(t)t=1.

In [10], Helffer and Weissler studied a similar monoparametric equation, namelyx″=x3−(t−c)x,c∈R,and obtained a family of solutions based on the analytical method of shooting. Also, Guedda [7], demonstrated the uniqueness of solution of the above equation under conditions , , without assuming any hypothesis at infinity. On

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