Positive decaying solutions for differential equations with phi-laplacian

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R E S E A R C H

Open Access

Positive decaying solutions for differential

equations with phi-Laplacian

Zuzana Došlá

1*

_{and Mauro Marini}

2

*_{Correspondence:}

[email protected]

1_{Faculty of Science, Masaryk}

University Brno, Kotláˇrská 2, Brno, 611 37, The Czech Republic Full list of author information is available at the end of the article

Abstract

We solve a nonlocal boundary value problem on the half-close interval [1,∞) associated to the diﬀerential equation (a(t)|x|α_{sgn x}₎_{+ b(t)}_|x|β_{sgn x = 0, in the}

superlinear case

α

<

β

. By using a new approach, based on a special energy-type function E, the existence of slowly decaying solutions is examined too.

MSC: 34B40; 34B15; 34B18

Keywords: nonlinear boundary value problem; globally positive solution; decaying

solution; oscillatory solution

1 Introduction

Consider the Emden-Fowler type diﬀerential equation

tαxαsgnx+ b(t)|x|βsgnx= , () where  < α < β and b is a positive continuous function on [,∞) satisfying

Jb=

∞ 

b(s) ds =∞. ()

Jointly with () consider also the more general equation

a(t)xαsgnx+ b(t)|x|βsgnx= , () where a is a positive continuous function on [,∞) such that

Ja=

_∞ 

a–/α(s) ds <∞. ()

Equations () and () arise in the study of radially symmetric solutions of elliptic diﬀer-ential equations with phi-Laplacian operator inR_{; see, e.g., [, ].}

By a solution of () we mean a diﬀerentiable function x on an interval Ix⊆ [, ∞), such

that a(·)|x(·)|α_{is continuously diﬀerentiable and satisﬁes () on I}

x. In addition, x is called

local solutionif Ixis bounded and proper solution if Ixis unbounded and sup{|x(t)| : t ≥

T} >  for any large T ≥ . As usual, a proper solution of () is said to be oscillatory if it has

©2015 Došlá and Marini. This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, pro-vided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

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a sequence of zeros tending to inﬁnity, otherwise it is said to be nonoscillatory. Equation () is said to be oscillatory if any its proper solution is oscillatory.

Deﬁne

A(t) = ∞

t

a–/α(s) ds.

LetP be the class of eventually positive proper solutions x of (). In view of (), the class P can be divided into three subclasses, according to the asymptotic behavior of x as t→ ∞, see, e.g., [], Lemma .. More precisely, any proper solution x∈ P satisﬁes one of the following asymptotic properties:

lim t→∞x(t) = x,  < x<∞, () lim t→∞x(t) = , tlim→∞ x(t) A(t)=∞, () lim t→∞ x(t) A(t)= x,  < x<∞, ()

where xis a positive constant depending on x.

Let x, y∈ P satisfy (), (), respectively. Then x, y tend to zero as t → ∞ and  < y(t) < x(t) for large t. Hence, proper solutions of () satisfying () are called slowly decaying solutions, and proper solutions satisfying () strongly decaying solutions.

Here, we consider the nonlocal BVP on the half-line [,∞) ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ (a(t)|x_|α_sgn_x₎_{+ b(t)|x|}β_sgn_x_{= } _α_{< β,} x(t) > , x(t) < ,

limt→∞x(t) = , limt→∞x(t)/A(t) =∞.

()

Using a suitable change of variable and certain monotonicity properties of a energy-type function E, we prove that () has infinitely many solutions. Consequently, we get also a global multiplicity existence result for slowly decaying solutions of (), which are posi-tive decreasing on the whole interval [,∞). We recall that in the superlinear case α < β, sufficient conditions for existence of slowly decaying solutions are difficult to establish, due to the problem to find sharp upper and lower bounds; see, e.g., [], p., [], p..

Observe that necessary and suﬃcient conditions for existence of solutions of (), which satisfy () or (), can easily be produced; see, e.g., [, ] or [], Section . Moreover, the same is true for slowly decaying solutions in the sublinear case α > β; see, e.g., [, ]. In the opposite situation, that is, in the superlinear case α < β, in spite of many examples of equations of type () having solutions of type (), which can be easily produced, until now no general suﬃcient conditions for their existence are known.

The paper is completed by the solvability of a special BVP, in which also the initial start-ing point is ﬁxed. Moreover, some examples and suggestions for future research complete the paper.

Our results are also motivated by the papers [, ], in which the special case α = β is considered. More precisely in [, ] necessary and suﬃcient conditions for existence of

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slowly decaying solutions for the half-linear equation

a(t)yαsgny+ b(t)|y|αsgny= , () are established, according to α <  or α > , respectively.

Recently, BVPs on inﬁnite intervals, associated to equations with phi-Laplacian have been considered in [, ]. The case of nonlocal BVPs for the generalized Laplacian, has been studied, e.g., in [, ]. Finally, we refer the reader to [, ] for other references on this topic.

2 Preliminaries

We start with a change of the independent variable in (), which will be useful.

Lemma  Consider the transformation

s(t) = ∞ t  a/α_{(τ )}dτ – , u(s) = xt(s), ()

where t(s) is the inverse of the function s(t). Then x is a solution of the BVP () if and only

if u is a solution of d ds sαu·(s)αsgnu·+a /α_(s)b(s) s |u| β_sgn u=  ()

on[s,∞), s= s() > , and satisﬁes for s≥ s

u(s) > , u·(s) < , lim

s→∞u(s) = , slim→∞su(s) =∞,

where· denotes the derivative with respect to s. Moreover, condition () is satisﬁed for ().

Proof Using (), we have

x(t) =u·ds dt= s a/α_(t) · u(s) ()

and () is transformed into the equation

d ds · u(s)αsgnu·+α s · u(s)αsgnu· +a /α_(s)b(s) s(α+) |u| β_sgn_u_{= ,} _()

see also [], p., with minor changes. Multiplying () by sα_{, we obtain (). Moreover,} for () assumptions () and () are satisﬁed because

Ja= ∞ s  sds<∞, and ∞ s a/α_(s)b(s) s ds= ∞  b(t) dt =∞.

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Thus, in view of Lemma , in the sequel we will study the existence of solutions x of () which satisfy on [,∞) the boundary conditions

x(t) > , x(t) < ,

lim

t→∞x(t) = , tlim→∞x(t)/A(t) = limt→∞tx(t) =∞.

If x is a solution of (), then denote by x[]_{its quasiderivative, that is,}

x[](t) = tαx(t)αsgnx(t). () Moreover, set Z= ∞  s– s  b(r) dr /α ds, Y= ∞  s–βb(s) ds, and B(t) = t  b(s) ds + c, ()

where c is an arbitrary positive constant. Hence B(t) >  for t≥ . The following result is needed in the following.

Lemma 

(i) Equation () has solutions x∈ P which satisfy (), if and only if Z < ∞.

(ii) Equation () has solutions x∈ P which satisfy limt→∞tx(t) = x,  < x<∞ if and

only if Y <∞. Moreover, for any x,  < x<∞, there exists x ∈ P such that

limt→∞tx(t) = x.

(iii) Equation () is oscillatory if and only if Y =∞.

Proof Claims (i) and (ii) follow from [], Theorems ., ., with minor changes. Claim (iii)

follows from [], Theorem ..

3 The main result

Our main result deals with the solvability of the nonlocal BVP ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ (tα_|x_|α_sgn_x₎_{+ b(t)|x|}β_sgn_x_{= ,} _t_{≥ , α < β,} x(t) > , x(t) < , limt→∞x(t) = , limt→∞tx(t) =∞, (BVP)

under the additional assumption

b∈ C[,∞). ()

Remark  In the superlinear case α < β, in virtue of (), any local solution of () is a solution, i.e. it is continuable to inﬁnity and is proper; see, e.g., [], Theorem ., or [], Appendix A. Notice also that, under the weaker assumption b(t)≥ , sup{b(t) : t ≥ T} > 

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for any T≥ , there may exist equations of type () with uncontinuable solutions; see, e.g., [], p..

The following holds.

Theorem  Assume Z=∞. If () is satisﬁed and the function

G(t) =  t_b_(t)B γ (t) is nonincreasing for t≥ , () where γ = + αβ + α α(β + ) , ()

then(BVP) has inﬁnitely many solutions.

To prove this result, several auxiliary results are needed. Deﬁne for any solution x of () the energy-type function

Ex(t) = B(t)x(t) β+ + x(t)x[](t) + kt α b(t)B(t)x _(t)α+ , ()

where the quasiderivate x[]is deﬁned in () and

k=α(β + )

α+  . ()

Lemma  For any solution x of () we have x(t)αx(t) = α |x_(t)| tα d dtx []_{(t) – t}–_x_(t)α+ sgnx(t). ()

Proof Since x[]_{is continuously diﬀerentiable on [t}

x,∞), tx≥ , and t–αx[](t) =|x(t)|α×

sgnx(t), the function xis continuously diﬀerentiable on [tx,∞) as well. If x(t) = , then

the identity () is valid. Now, assume x(t)= . We have x(t) d

dtx

[]_{(t) =}_x_(t)_αtα–_x_(t)α

sgnx(t) + αtαx(t)α–x(t) = αtα–x(t)α+sgnx(t) + αtαx(t)αx(t),

from which the assertion follows.

Lemma  Assume() and (). Then for any solution x of () we have for t≥ 

d

dtEx(t)≤ .

Proof Let ϕ be a continuously diﬀerentiable function on [,∞). Then for any positive con-stant σ the function|ϕ(t)|σ+_{is continuously diﬀerentiable and}

d dtϕ(t)

σ+

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Using this equality, we have for t≥  d dt tα b(t)B(t)x _(t)α+ = d dt  t_b_(t)B(t)t α+_x_(t)α+ = B(t)tα+x(t)α+d dt  t_b_(t) +  tt α+_x_(t)α+ + (α + ) B(t) t_b_(t) tα+x(t)α++ tα+x(t)αx(t) sgn x(t). Hence we get d dtEx(t) = (β + )x(t) β x(t)B(t) sgn x(t) + tαx(t)αx(t) sgn x(t) + kB(t)tα+x(t)α+d dt  t_b_(t) + k tt α+_x_(t)α+ + k(α + ) B(t) t_b_(t) tα+x(t)α++ tα+x(t)αx(t) sgn x(t) or d dtEx(t) = t α+_g_{(t) + h(t),} where g(t) = ( + k) tx _(t)α+ + kB(t)x(t)α+d dt  t_b_(t) and h(t) = (β + )x(t)βx(t)B(t) sgn x(t) + k(α + ) B(t) t_b_(t) tα+x(t)α++ tα+x(t)αx(t) sgn x(t). From () we obtain  + k = γ k, k(α + ) = α(β + ). Thus, we get g(t) = k γ  tx _(t)α+ + B(t)x(t)α+d dt  t_b_(t) () and h(t) (β + )B(t)=x(t) β x(t) sgn x(t) + α b(t)t α–_x_(t)α+ + α b(t)t α_x_(t)α x(t) sgn x(t).

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In view of (), we have B–γ +(t)x(t)α+ d dt  t_b_(t)B γ (t) = B(t)x(t)α+d dt  t_b_(t) + γ  tx _(t)α+ ≤ , and so, from () we obtain

g(t)≤ .

In order to complete the proof, it is suﬃcient to show that h(t) = . Using Lemma , we have h(t) (β + )B(t)=x(t) β x(t) sgn x(t) + α b(t)t α–_x_(t)α+ – x(t)x(t)βsgnx(t) – α b(t)t α–_x_(t)α+ = ,

thus the assertion follows.

Lemma  Assume(). Then () has proper solutions x for which Ex() < .

Proof Consider on [,∞) the scalar function

φ(u) = cuα+– muα+ cmβ+, () where c= α(β + ) α+  c b() ()

and m is a positive parameter. A standard calculation shows that when m is suﬃciently small, then φ attains negative values in a neighborhood of the point

u= αm (α + )c

. ()

Indeed, consider the local solution x of () with the initial condition

x() = m, x() = –u. ()

In view of Remark , x is continuable to inﬁnity and proper. From () we obtain

Ex() = cmβ+– m αm (α + )c α + k c b() αm (α + )c α+ , or, in view of (), Ex() = φ(u).

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Using (), we get Ex() = φ(u) = cmβ+– α (α + )c α mα++ c α (α + )c m α+ = mα+ cmβ–α+  (α + )c α – + α α+  = mα+ cmβ–α–  α+   (α + )c α .

From this and β > α, choosing m suﬃciently small such that

 < mβ–α_< c  α+   (α + )c α , ()

we get φ(u) < , which is the assertion.

Proof of Theorem From Lemma , the function Exis nonincreasing on [,∞) for any

solution x of ().

Fixed m satisfying (), consider the local solution x of () with the initial condition (), where u is given by (). In view of Remark , this solution is also continuable to in-ﬁnity and proper. Moreover, it is uniquely determined, because in the superlinear case the uniqueness of solutions with respect to the initial conditions holds; see, e.g., []. Moreover, in virtue of the proof of Lemma , we have Ex() < , and so, from Lemma , we obtain

Ex(t)≤ Ex() <  on [,∞). ()

Let us show that x and xcannot have zeros for t≥ . By contradiction, if there exists

t>  such that x(t) = , then, in virtue of the uniqueness with respect to the initial data, we have x(t)= . Hence E(t) > , which contradicts (). Similarly, if there exists t≥  such that x(t) = , we obtain E(t) > , which is again a contradiction.

Thus, x is nonoscillatory. Moreover, in view of Lemma , we have

lim

t→∞x(t) = .

Hence, x is positive decreasing in the half-line [,∞), that is,

x(t) > , x(t) <  for t≥ .

From (), the quasiderivative x[]_{is negative decreasing, i.e.}

lim

t→∞–x

[]_{(t) =}

x,  < x≤ ∞.

If x<∞, we get limt→∞x(t)x[](t) = , and from () we obtain lim inft→∞Ex(t)≥ , that

is, a contradiction with (). Hence x=∞, i.e.

lim

t→∞t

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Using the l’Hospital rule we get

lim

t→∞tx(t) =∞.

Hence, x is a solution of (BVP). Since there are inﬁnitely many solutions which satisfy () with the choice of m taken with (), the proof is now complete.

From Theorem  and its proof, we get the following.

Corollary  Under assumptions of Theorem, () has inﬁnitely many slowly decaying

so-lutions, which are positive decreasing on the whole interval [,∞). Moreover, () has also

inﬁnitely many strongly decaying solutions and every nonoscillatory solution of () tends

to zero as t→ ∞.

Proof In virtue of Theorem  and its proof, the boundary value problem (BVP) is solv-able by every solution x which satisﬁes () and (). Clearly, these solutions are slowly decaying solutions.

Consequently, () has nonoscillatory solutions and, in view of Lemma (iii) we get

Y <∞. Then the existence of inﬁnitely many strongly decaying solutions follows from

Lemma (ii).

When the monotonicity condition () is valid only for large t, reasoning as in the proof of Theorem , we obtain the following.

Corollary  Assume Z=∞. If () is satisﬁed and the function G, given in () is

nonin-creasing for any large t, then () has inﬁnitely many slowly decaying solutions, which are

eventually positive decreasing. Moreover, () has also inﬁnitely many strongly decaying

so-lutions and every nonoscillatory solution of () tends to zero as t→ ∞. Finally, when also the initial starting point is ﬁxed, we have the following.

Corollary  Assume Z=∞. If () and () are satisﬁed, then the BVP ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ (tα_|x_|α_sgn_x₎_{+ b(t)|x|}β_sgn_x_{= ,} _t_{≥ , α < β,} x() = x, x(t) > , x(t) < , limt→∞x(t) = , limt→∞tx(t) =∞,

has inﬁnitely many solutions for every initial data xsuch that

 < (x)β–α<  c α+  α+  b() α(β + ) α . ()

Proof The assertion follows by a reasoning as in the proof of Theorem  and choosing

m= xin (). Taking into account that m satisﬁes () and cis given by (), we get ().

The details are left to the reader.

Remark  It is worth to note that the condition () may depend on the choice of the constant c in (), i.e. on the choice of a primitive to b.

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If () is satisﬁed for a ﬁxed B, then () remains to hold for B(t) + c, where c > . Indeed, setting (t) = t–_b–_{(t), from G}_(t)_{≤  we get}

≥ (t) + γ (t)b(t)  B(t)= _{(t) + γ (t)b(t)}  B(t) + c B(t) + c B(t) or ≥ (t) + γ (t)b(t)  B(t) + c

i.e.() is satisﬁed also for B(t) + c with c > . However, if () is valid for B given by (), then it is possible that () is not valid for B(t) = B(t) +˜c, where  < ˜c < c. This fact is illustrated below in Example .

4 Oscillation and nonoscillation

In this section we discuss assumptions of Theorem , jointly with some consequences to the oscillation. Assumption () guarantees the continuability at inﬁnity of any solution of () and its role is discussed in Remark . Concerning the condition Z =∞, a consequence of a result in [] shows that it is a necessary condition for the solvability of (BVP). The following holds.

Theorem  If Z<∞, then () does not have solutions x ∈ P such that

lim

t→∞x(t) = , tlim→∞tx(t) =∞.

Proof Let x be a solution of () and set

y(t) = –tαx(t)αsgnx(t). () A standard calculation shows that y is a solution of equation

 b/β_(t)y /β sgny +  t|y| /α_sgn_y_{= ,} _()

where /β < /α. From [], Theorem , () does not admit eventually positive solutions

ysuch that

lim

t→∞y(t) =∞, tlim→∞

y(t)

b(t) = ,

that is, in view of (), () does not have solutions x∈ P such that

lim

t→∞x(t) = , tlim→∞t

_x_{(t) = –}_∞. Since the l’Hospital rule gives

lim

t→∞tx(t) = – limt→∞t

_x_(t),

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Now, we discuss the monotonicity condition (). We start by recalling the following nonoscillation result, which is an extension of a well-known Kiguradze criterion [], The-orem ..

Theorem ([], Theorem .) If there exists a positive number ε such that the function ¯G(t) =  t_b_(t) t  b(s) ds γ+ε

is nonincreasing for large t, ()

where γ is given in(), then all solutions of () are nonoscillatory.

A standard calculation shows that if () holds for t≥ t≥ , then () is valid on the same interval [t,∞) as well. Thus, in view of Corollary , we can obtain an existence result for slowly decaying solutions x of (). Nevertheless, condition () can be valid in a larger interval than [t,∞). The next example illustrates this fact.

Example  Consider the equation

tx(t)/sgnx+ x= . () We have Z =∞ and γ = /. Moreover, choosing c =  in (), for the function G given in () we have G(t) = t–/_{. Then () is satisﬁed for t}_{≥  and Theorem  is applicable.} Analogously, () is veriﬁed for any large t and  < ε < /, because

¯G(t) = t–_{(t – )}/_{(t – )}ε

. On the other hand, we have

¯G_{(t) = t}–_{(t – )}/_{(t – )}ε_{/ + ε – t(t – )}

and so () is not valid in a right neighborhood of t = . By Theorem , all solutions of () are nonoscillatory and by Lemma  tend to zero as t→ ∞. Moreover, in view of The-orem , () has both slowly decaying solutions and strongly decaying solutions. Finally, slowly decaying solutions are globally positive on the whole interval [,∞).

Example  Consider the equation

t/x(t)/sgnx+ x(t) = . () We have Z =∞ and γ = . Choosing c =  in (), we obtain G(t) =  and Theorem  can be applied. Indeed, as it is easy to verify, the function x(t) = –_t–/_{is a slowly decaying} solution of (). Moreover, () has both slowly decaying solutions and strongly decaying solutions and slowly decaying solutions are globally positive on the whole interval [,∞). Observe that if we choose c = / in (), then the corresponding function G is increasing and () is not satisﬁed.

Furthermore, the function ¯Ggiven in () is increasing for large t and any ε > . Hence, Theorem  cannot be used. Then it is a question whether () admits or does not admit oscillatory solutions.

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When α = , the coexistence between oscillatory solutions and nonoscillatory solutions can be obtained by using Lemma  and a result from [], Theorem . The following holds.

Theorem  Consider the equation

tx+ b(t)|x|βsgnx= , ()

where β>  and b(t) >  for t≥ . Assume Y < ∞. If the function

H(t) = t(–β)/b(t)

is nondecreasing for large t and limt→∞H(t) =∞, then () has inﬁnitely many oscillatory

solutions and inﬁnitely many strongly decaying solutions. Moreover, every nonoscillatory

solution of () tends to zero as t→ ∞.

Proof In view of [], Theorem , any solution x of () which satisﬁes x(t)x(t) >  at some t≥ , is oscillatory. The remaining part of the statement follows from Lemma . The following example shows that both types of nonoscillatory decaying solutions can coexist with oscillatory solutions.

Example  Consider the equation

tx(t)+ t/x= . () The assumptions in Theorem  are satisﬁed. Hence, () has oscillatory solutions and inﬁnitely many strongly decaying solutions. Moreover, every nonoscillatory solution tends to zero as t→ ∞, according to Lemma , because Z = ∞. Furthermore, it is easy to verify that the function

x(t) = √

  t

–/

is a slowly decaying solution of (). Thus, for () slowly decaying solutions and strongly decaying solutions coexist with oscillatory solutions. Observe that for the function G given in () we have G(t) = t–/  t /_{+ k} / ,

where k is a suitable constant such that k > –/. A standard calculation shows that () is not satisﬁed and Theorem  cannot be applied. A similar argument shows that also () fails.

Open problems Example  suggests that for the existence of at least one slowly decaying solution, the assumption on monotonicity in () could be relaxed. Moreover, Example  (and Theorem ) deal with the case α = , that is, when the diﬀerential operator is the Sturm-Liouville disconjugate operator. Does the coexistence between oscillatory solutions and decaying solutions, illustrated in Example , occur also when α=  (and β > α)?

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Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

All authors contributed equally to the manuscript and read and approved the ﬁnal draft.

Author details

1_{Faculty of Science, Masaryk University Brno, Kotláˇrská 2, Brno, 611 37, The Czech Republic.}2_{Department of Mathematics}

and Informatics, ‘Ulisse Dini’, University of Florence, Via S. Marta 3, Florence, I-50139, Italy.

Acknowledgements

The ﬁrst author is supported by Grant P201/11/0768 of the Czech Science Foundation. Received: 4 February 2015 Accepted: 19 May 2015

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