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Advances in Nonlinear Analysis

Editor-in-Chief: Radulescu, Vicentiu / Squassina, Marco

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On Cauchy–Liouville-type theorems

Ataklti Araya / Ahmed Mohammed
Published Online: 2017-08-24 | DOI: https://doi.org/10.1515/anona-2017-0158


In this paper we explore Liouville-type theorems to solutions of PDEs involving the ϕ-Laplace operator in the setting of Orlicz–Sobolev spaces. Our results extend Liouville-type theorems that have been obtained recently.

Keywords: Liouville-type theorems; Orlicz–Sobolev spaces

MSC 2010: 35J60; 35J62; 35J70; 35J75

1 Introduction

Any real-valued harmonic function on N (N2) which is bounded either from above or below is a constant. This is perhaps the earliest formulation of what is commonly known as the Cauchy–Liouville theorem (or simply Liouville’s theorem). Today there is an extensive amount of work that extends this basic result in many different directions. A beautiful account of the history, techniques and some recent results on Liouville-type problems can be found in [9].

Our goal in this paper is to investigate classes of functions f: for which the quasilinear PDE


admits constants as the only possible non-negative solutions in the entire Euclidean space N (N2). In this case we will say that the PDE has a Liouville-type property. Here Δϕu:=div(ϕ(|u|)u) is the so-called ϕ-Laplacian which reduces to the well-known p-Laplacian when ϕ(t)=ptp-2 for p>1.

In our attempt to understand the Liouville-type property of (1.1), we will focus on two classes of functions f. In the first part of the paper we investigate problem (1.1) when f belongs to a class of non-decreasing and non-negative functions whose growth at infinity is dictated by a Keller–Osserman-type condition. To elaborate on this, suppose that f is a non-decreasing continuous real-valued function defined on such that f(0)=0 and f(t)>0 for t>0. It is well known (see [9]) that the only non-negative solution of Δu=f(u) in N is the trivial solution u0 if and only if


This result is essentially due to Keller [14] and Osserman [22], obtained independently. We will use an adaptation of condition (1.2) to the ϕ-Laplacian to obtain a sufficient condition on f in order for (1.1) to have the Liouville-type property.

In the second part of the paper we study another class of functions f that will lead to the PDE (1.1) having a Liouville-type property. To motivate this aspect of our investigation, we mention the paper [29], where the Liouville-type property of the p-Laplacian has been discussed extensively. Among many important results developed therein, we would like to mention the following result which is relevant to the work at hand. Suppose f:0+0+ is subcritical in the sense that f satisfies


for some 1<α<p*, where p* is the critical Sobolev exponent of p>1. If there exists q>p such that f(t)tq-1 at infinity, then any non-negative solution of Δpu=-f(u) in N is the trivial solution, see [29]. In the recent paper [7], an adaptation of a method in [21] was used to obtain a result which complements the aforementioned Liouville-type result. More specifically, the following result was proved. Suppose f: is a differentiable function that has a root and satisfies


Then any positive solution of Δpu=-f(u) in N must be a constant. As pointed out above, the method used in [7] is based on the work of McCoy [21, 20], who investigated the Liouville-type property for Δu=-f(u). This result in [21] is just one among many other Liouville-type results studied for elliptic equations, including some fully nonlinear higher-order equations. We also call attention to the nice paper [3] which discusses, among other things, Liouville-type results for general anisotropic quasilinear equations. The paper [15] contains Liouville comparison principles for quasilinear singular parabolic second-order partial differential inequalities.

The paper is organized as follows. In Section 2 we begin by stating some general assumptions on ϕ and by recalling some results from the literature used in our work. In Section 3 we show that (1.1) has the Liouville-type property when ϕ meets a suitable condition and f is continuous and non-decreasing, and satisfies a generalized Keller–Osserman condition. Some examples will be presented to illustrate the applicability of the main result of the section. Finally, in Section 4, we discuss a Liouville-type property of (1.1) when f belongs to a class of differentiable functions f:, which are not necessarily monotonic, may change sign and satisfy a condition that generalizes (1.3). An appropriate condition on ϕ will also be required. We conclude the section with some illustrative examples.

2 Preliminaries

Given ϕ:(0,)(0,), let us set


We make the following assumptions:

  • (ϕ1)

    Ψ is a strictly increasing C1 function in +:=(0,).

  • (ϕ2)

    lims0+Ψ(s)=0 and limsΨ(s)=.

  • (ϕ3)

    There exist constants 0<σρ such that

    σΦ′′(t)tΦ(t)ρfor all t>0.

Equations involving the ϕ-Laplacian have been used to model different physical phenomena. For instance, they appear in quantum physics [4] and in the modeling of nonlinear elasticity problems or in plasticity [12]. We refer to the papers [26, 2, 6, 10, 11, 24, 25, 28, 27] for more details and other applications.

In the first part of this section we discuss some immediate but useful consequences of conditions (ϕ1)(ϕ3) listed above.

We begin with inequalities which follow directly from (ϕ3), namely,

λ(s)Ψ(t)Ψ(st)Λ(s)Ψ(t)for all s,t0+:=[0,),(2.1)

for some increasing functions λΛ. In fact,

λ(s):=min{sσ,sρ}andΛ(s):=max{sσ,sρ}for s0+.

Inequalities (2.1), in turn, imply

Λ-1(ϱ)Ψ-1(τ)Ψ-1(ϱτ)λ-1(ϱ)Ψ-1(τ)for all ϱ,τ0+.(2.2)

Another immediate consequence of (2.1) is




The following inequality, a direct consequence of (2.3), will be useful in Section 3:


As a result of assumptions (ϕ1)(ϕ2) one can show that Φ is an N-function. From condition (ϕ3) we deduce that Φ satisfies the global Δ2-condition, namely, there exists a constant c>0 such that

Φ(2t)cΦ(t)for all t>0.

In fact, this is a direct consequence of (2.3).

Given an open set ΩN, the Orlicz space

LΦ(Ω):={u:Ω|u is measurable and ΩΦ(|u(x)|)dx<}

is a Banach space under the so-called Luxemburg norm


The Orlicz–Sobolev space W1,Φ(Ω) is defined as the set of all weakly differentiable uLΦ(Ω) such that DαuLΦ(Ω) for all multi-indices α=(α1,,αN) with |α|1. This is a Banach space under the norm


As with the usual Sobolev spaces, W01,Φ(Ω) is defined as the closure of Cc(Ω) in W1,Φ(Ω).

We define the local spaces LlocΦ(Ω) and Wloc1,Φ(Ω) by

LlocΦ(Ω):={u:uLΦ(𝒪) for all 𝒪Ω}andWloc1,Φ(Ω):={u:uW1,Φ(𝒪) for all 𝒪Ω}.

The dual space (LΦ(Ω))* is equal to LΦ~(Ω), where Φ~ is the N-function given by


and is called the complement of Φ. The assumption (ϕ3) shows that Φ~ satisfies a global Δ2-condition. This can be seen by integrating the right inequality in (2.2).

For more discussion of Orlicz–Sobolev spaces we refer the reader to [1, 5, 24] and the references therein.

Let ΩN be an open set and g:Ω× be a continuous function. We say a weakly differentiable function v:Ω is a sub-solution of the PDE


if and only if for every open and bounded subset 𝒪Ω, we have vW1,Φ(𝒪), with g(x,v(x))LΦ~(𝒪), such that

𝒪ϕ(|v|)vφ-𝒪g(x,v)φfor all 0φW01,Φ(𝒪).(2.6)

A weakly differentiable function w:Ω is said to be a super-solution of (2.5) in Ω if and only if for every open and bounded subset 𝒪Ω, we have wW1,Φ(𝒪), with g(x,w(x))LΦ~(𝒪), such that the reverse inequality holds in (2.6) for all non-negative φW01,Φ(𝒪). A weakly differentiable function u:Ω is said to be a solution of (2.5) in Ω if u is both a sub-solution and a super-solution of (2.5) in Ω.

We follow common practice and write

Δϕvg(x,v)in Ω  and  Δϕwg(x,w)in Ω

to indicate that v is a sub-solution and w is a super-solution of (2.5), respectively, in Ω.

As noted in the introduction, in this paper we are interested in the investigation of Liouville-type property for the PDE


where the absorption term will be required to satisfy appropriate conditions.

We recall two results that will be useful in this paper. We begin with the following comparison principle, which is taken from [23, Theorem 2.4.1 and Proposition 2.4.2].

Theorem A (Comparison principle).

Let ΩRN be a bounded domain. Suppose that ϕ satisfies (ϕ1) and (ϕ2), and gLloc(Ω×R) is such that tg(x,t) is a non-decreasing function in R for each xΩ. Suppose u,vW1,Φ(Ω)C(Ω) satisfy

Δϕug(x,u)xΩ  𝑎𝑛𝑑  Δϕvg(x,v)in Ω.

If uv on Ω, then uv in Ω.

The following regularity result of Lieberman [17, Theorem 1.7], reminiscent of the regularity results for p-Laplacian-type quasilinear equations of DiBenedetto [8], Lieberman [16] and Tolksdorff [30], will also be useful.

Theorem B (Regularity theorem).

Let ΩRN be a bounded open set. Suppose that conditions (ϕ1) and (ϕ3) hold, and gL(Ω×R). If uW1,Φ(Ω)L(Ω) is a solution of (2.5), then uC1,α(Ω) for some constant 0<α<1.

3 Monotonic non-negative absorption term

Let f: be a non-decreasing continuous function such that f(0)=0 and f(t)>0 for t>0. In this section we extend a classical result of Keller [14] and Osserman [22] who showed independently that Δu=f(u) can not have a non-trivial solution in N if f satisfies a growth condition at infinity that has come to be known as the Keller–Osserman condition. We refer to the paper [9] for an excellent account on this and other related Liouville-type results.

Since σρ, where σ and ρ are the parameters in condition (ϕ3), we note that for s>1, we have


Therefore, for s>ϱ>1, we have


Since (ρ+1)/ρ(σ+1)/σ, we see that the right-hand side of (3.1) is less than one whenever s>ϱ>1. Now let us note that


so that for 0<s<1, we have


Applying this to (3.1), we find that


If σN-1, then it is clear that


On the other hand suppose σ<N-1. If, in addition, ρ-σρσ(σ+1)<1N-1, then we have


Consequently, in this case we find


The above discussion leads us to consider the following condition on the parameters σ and ρ in condition (ϕ3):

  • (ϕ4)


For easy reference, let us summarize the above discussion with regard to condition (ϕ4) in the following remark.

Remark 3.1.

Let us suppose that condition (ϕ4) holds. Then direct computations lead to the following conclusions:

  • (i)

    If σN-11, then for each ϱ>1,


  • (ii)

    If σN-1<1, then


Let us now consider a non-decreasing function f:(0,)(0,) such that

1dsΦ-1(F(s))<,where F(t):=0tf(s)𝑑s.(3.2)

Condition (3.2) is easily recognized as a generalization of the well-known Keller–Osserman condition when ϕ1.

We point out that condition (3.2) is equivalent to

tdsΦ-1(F(s)-F(t))<for all t>0.(3.3)

That (3.3) implies (3.2) is obvious. Therefore, we only need to show that (3.3) is implied by (3.2). For this, we first observe that

F(s)-F(t)f(t)(s-t)andF(s)-F(t)F(s-t)for all st.

Let t>0 and fix 0<θ<1. Using (2.4), together with the above inequalities, we have


Thus, condition (3.2) on f implies that the right-hand side of (3.4) is finite for any t>0. Therefore, (3.3) holds.

In this section we study Liouville-type properties of solutions to the following equation:


We assume that f:0+0+ is a continuous function such that f(0)=0 and f(t)>0 for t>0. Our first Liouville-type result in this section is provided by the following theorem.

Theorem 3.2.

Suppose that (ϕ1)(ϕ4) hold and that f is a non-decreasing function that satisfies condition (3.2). If uWloc1,Φ(RN)C(RN) is a non-negative sub-solution of (3.5) in RN, then u0 in RN.


Let zN and let ε>0 be arbitrary. Let v be a solution of


Let (0,R) be the maximal interval of existence of v. Note that v cannot be a constant. We shall show that R<. Note that


Therefore, v>0 in (0,R), and rN-1Ψ(v) is a non-decreasing function in (0,R), and hence Ψ-1(rN-1Ψ(v)) is non-decreasing on (0,R) as well. On multiplying both sides of (3.6) by Ψ-1(rN-1Ψ(v)), for any ϱ<r<R, we estimate


We have used (2.2) in obtaining (3.7). Inequality (3.8), together with


which again is a consequence of (2.2), shows that


On noting that Φ(2v(r))Ψ(v(r))v(r), we use (2.4) to get


Integrating this last inequality on (ϱ,r) yields


Consequently, we see that

12ϱrΛ~-1(ϱN-1Λ-1(ϱN-1)sN-1λ-1(sN-1))𝑑sv(ϱ)dsΦ-1(F(s)-F(v(ϱ)))for all r>ϱ.(3.9)

Note that


is a non-increasing function. Therefore, since ε=v(0)v(ϱ), we have 𝒬(ε)𝒬(v(ϱ)). Hence, from inequality (3.9), we obtain

ϱrΛ~-1(ϱN-1Λ-1(ϱN-1)sN-1λ-1(sN-1))𝑑s2εdsΦ-1(F(s)-F(ε))for all ϱ<r<R.(3.10)

Thus, condition (3.2) on f implies that the left-hand side of (3.10) is finite.

Now assume that R=. Then we can take ϱ>1 in (3.10). If σN-1, then taking the limit as r in (3.10) leads to a contradiction, by Remark 3.1.

Now let us suppose that σ<N-1. Taking the limit in (3.10) as r, we find that


Then we use condition (ϕ4) and Remark 3.1 to see that inequality (3.11) leads to a contradiction upon taking the limit as ϱ.

Therefore, we conclude that indeed R<, and as a consequence v(r) as rR-. Let us set w(x):=v(|x-z|) for xN, and suppose that u is any non-negative sub-solution of (3.5) in N. Then, for 0<δ<1 such that 1-δ is sufficiently small, we have u<w on B(z,δR). Since w is a solution of (3.5) in the ball B(z,δR) we invoke the comparison principle, Theorem A, to conclude


In particular, we have 0u(z)ε. Since ε>0 is arbitrary, we find that u(z)=0, and since zN is arbitrary, we conclude u0 in N. ∎

Theorem 3.3.

Suppose that (ϕ1)(ϕ4) hold. Let f:RR be such that f(t)>0 for t>0 and f(0)=0. Assume also that f satisfies (3.2).

  • (a)

    If u is a sub-solution of ( 3.5 ), then u0 in N.

  • (b)

    If f is an odd function and u satisfies ( 3.5 ), then u0 in N.


(a)  Let u be any sub-solution of (3.5) in N. To show that u0 in N, it suffices to demonstrate that u+ is a sub-solution of (3.5) in N. Once this is proven, then we can invoke Theorem 3.2 to conclude that u+0, and hence u=-u-0 in N. First, since uWloc1,Φ(N), let us notice that u+Wloc1,Φ(N). To see that u+ is a sub-solution of (3.5), let Ω:={xN:u(x)>0}. We suppose that Ω is non-empty for otherwise there is nothing to prove. For each positive integer j, set ϑj(t):=ϑ(jt), t, where ϑC1(), with

ϑ(t)>0on (0,1),ϑ1on [1,)  and  ϑ0on (-,0].

Given 𝒪N, let φW01,Φ(𝒪) with φ0 in 𝒪. We have


Let us note that ϑj(u)φW01,Φ(𝒪) and


Using (3.12), (3.13) and recalling that u is a sub-solution of (3.5) in N, we find


since f(0)=0. Therefore, u+ is a sub-solution of (3.5), as claimed.

(b)  Suppose that f is an odd function, and that u is a solution of (3.5) in N. By what was proved in (a), we observe that u0 in N. On the other hand, it is easily seen that -u is a solution of (3.5) in N. Therefore, -u0 in N again, completing the proof that u0 in N. ∎

3.1 Some examples

Here we consider examples to illustrate Theorems 3.2 and 3.3. It would be convenient to use the following equivalent formulation of condition (ϕ4).

Remark 3.4.

  • (i)

    Suppose 0<σ<12(4N-3-1). Then (ϕ4) holds if and only if


  • (ii)

    Suppose σ12(4N-3-1). Then (ϕ4) holds if and only if ρσ.

In the examples below, conditions (ϕ1)(ϕ3) are easily verifiable.

(1)  Let ϕ(t)=ptp-2, with p>1. Then Δϕ is the standard p-Laplacian. In this case, σ=ρ=p-1, and therefore condition (ϕ4) holds. Condition (3.2) reduces to the requirement that

tds(F(s))1/p<for all t>0.(3.14)

Therefore, if (3.14) holds then Theorems 3.2 and 3.3 hold for any p>1.

(2)  Let us now consider ϕ(t)=ptp-2+qtq-2 for 1<p<q. Computation shows that σ=p-1, ρ=q-1 and


As a consequence, we see that

tdsΦ-1(F(s))21/qtds(F(s))1/qfor all t>0.

Therefore, if the right-hand side in the above inequality is finite for some t>0, then condition (3.2) holds, and therefore Theorems 3.2 and 3.3 hold provided that condition (ϕ4) is satisfied. According to Remark 3.4, if


then (ϕ4) holds if and only if


On the other hand, if


then (ϕ4) holds if and only if qp.

(3)  Let ϕ(t)=ptp-1logq(1+t)+qtp-1(1+t)-1logq-1(1+t) for p>1 and q>0. Then σ=p-1 and ρ=p+q-1. Moreover, we see that Φ(t)=tplogq(1+t). Note that given ε>0, there exists a constant tε>0, sufficiently large, such that

Φ(t)tp+εfor all t>tε.

Therefore, if there exists r>p such that


for some t>0, then condition (3.2) holds. Thus, if (ϕ4) is satisfied, then Theorems 3.2 and 3.3 apply. Here, again, we invoke Remark 3.4. Suppose that (3.15) holds. Then (ϕ4) holds if and only if


If on the other hand (3.16) holds, then (ϕ4) holds if and only if q>0.

4 Sign-changing absorption term

This section is devoted to the study of the Liouville-type property of the following equation for a given f::


We do not make any monotonicity assumption on f, but we require f to be a C1 function that satisfies a sub-critical-type condition (see condition (Cf) below).

We remark that the solutions of (4.1) are invariant under rotations, in the sense that if u is a solution of (4.1) in N, then v(x):=u(Ax) is also a solution of (4.1) on N for any orthogonal matrix A.

To study the Liouville-type property of (4.1), we start by making some suitable assumptions on ϕ. Specifically, we require that ϕC2(0,) and satisfies the following condition:

  • (ϕ5)

    We have

    -<inft>0ϖ(t)supt>0ϖ(t)<(NN-1)σ2-3ρ+2,where ϖ(t):=ϕ′′(t)t2ϕ(t),

    and σ,ρ are the constants in condition (ϕ3).

The main result of this section is the following Liouville-type theorem for the solutions of (4.1). This theorem extends the result in [7], where the special case ϕ(t)=tp-2, p>1, was considered. In the proof of the theorem we will observe the Einstein summation convention over repeated indices.

Theorem 4.1.

Suppose that conditions (ϕ1)(ϕ3) and (ϕ5) hold, fCloc1,γ(R) for some 0<γ<1, and

f(t)σ(N+1N-1)f(t)tfor all t>0.(Cf)

If uWloc1,Φ(RN)C(RN) is a positive solution of (4.1) in RN, then u is a constant in RN.

Remark 4.2.

Suppose that ϕ satisfies the assumptions of Theorem 4.1. If f(t)=tθ-tϑ for some 0θσ(N+1)/(N-1)ϑ, then f satisfies (Cf), and therefore the only positive solution of (4.1) in N is u1. We should also note that if f is a non-negative and non-increasing function on , then f satisfies (Cf), and therefore in this case only constants are the possible positive solutions of (4.1).

Proof of Theorem 4.1.

Let us first note that, as a consequence of the continuity of u and Theorem B (see also [13, Lemma 3.3]), we conclude that uCloc1,α(N). Therefore, it follows that equation (4.1) is uniformly elliptic on open sets that are compactly contained in 𝒪:={xN:|u(x)|>0}. Hence, we observe that uC3 in any open set that is compactly contained in 𝒪 (see [18, Corollary 2.2], with p=q=2). See also [19].

Let x0N be an arbitrary but fixed point. We wish to show u(x0)=0. Given a>0, let us set

J(x):=(a2-r2)2Θ,where Θ=|u|2u2 and r=|x-x0|.

We note that J0 in B:=B(x0,a) and J||x-x0|=a=0. Therefore, J attains its maximum value on B¯ at some interior point x*B. Suppose u(x*)=0. Then Θ (and hence J) will be zero at x*. But then J (and therefore Θ) will be zero on B¯. This would imply |u|=0 in B¯, and in particular u(x0)=0. Therefore, we assume that |u|>0 at x*. We recall that u is 𝒞loc3(𝒪), where 𝒪:={xΩ:|u|>0}.

Now, at x*, we have


and D2J0.

Let us first show that the N×N matrix


is positive definite. Here IN is the N×N identity matrix, and AT stands for the transpose of the matrix A. To see that H is positive definite, we first note that for any ξN{0},


If ϕ(|u|)|u|ϕ(|u|)0, then ξTHξ|ξ|2>0. If, on the other hand, -1<ϕ(|u|)|u|ϕ(|u|)<0, then on noting that |ξTu||ξ||u| and therefore


we have (recalling condition ϕ4)


Thus, in any case, we have shown that H>0.

Now recalling that D2J0 at x*, we have HD2J0 at x*. Therefore, at x*, we have


where ej is the unit vector in N with 1 in the jth position. In the last equation we have used the simple fact that ekTBek=bkk, where B=[bij] is an N×N matrix. Thus, at x*, we have


Recalling that (4.1) is invariant under rotation, we choose coordinates so that at the point x*, we have


Through direct computation, we note that


This, together with inequality (4.3), implies


To obtain (4.5), we have used the easily verifiable identities


Moreover, from (4.2), we obtain


Using these and (4.6) in (4.5), we find


Therefore, at x*, we have


Since r2<a2 in B(x0,a), we have


Let us now notice that for i,j=1,,N, we have


Upon using (4.4), from (4.9) we obtain


From the above computation we see that at x*,


Let us now proceed to find alternative forms for the expressions in the parentheses in (4.10). We start by rewriting equation (4.1) in any open set 𝒪 that contains x*, that is,


where u is C3-smooth. On recalling (4.4), at x*, we obtain the following from (4.11):


that is,


We now differentiate (4.11) with respect to the variable x1. On evaluating the resulting expression at x*, we find


that is


On dividing both sides of (4.13) by ϕ(u1), using (4.12) to replace Δu in the resulting expression, and rearranging, we find


Let us observe the following.


To proceed further, we need the following inequality.

Using Lemma 4.3 in inequality (4.15), we obtain


We use (4.14) and (4.16) in (4.10) to get

ΔΘ+ϕ(u1)u1ϕ(u1)Θ112u1u-2{-2ϕ(u1)ϕ(u1)j=2Nu1j2+[f(u)ϕ(u1)-(1-ϕ(u1)u1ϕ(u1))u11]ϕ(u1)ϕ(u1)u11-ϕ′′(u1)u1ϕ(u1)u112-u1f(u)ϕ(u1)}   +2u-2[(2+ϕ(u1)u1ϕ(u1))j=2Nu1j2+(NN-1+ϕ(u1)u1ϕ(u1))u112-2N-1u11Δu+1N-1(Δu)2]   -2u12u-3[Δu+(4+5ϕ(u1)u1ϕ(u1))u11]+6(1+ϕ(u1)u1ϕ(u1))u14u-4.

Rearranging the terms in the last inequality, we find


We recall from (4.8) that for j=2,,N, we have Θj=2u1ju1u-2, and therefore


We use the above inequality and (4.12) to estimate inequality (4.17) as follows:


Rearranging the terms in the above inequality yields


We recall from (4.8) that Θ1=2u1u11u-2-2u13u-3, that is


Inserting this in the last inequality gives


Further rearrangement leads to


Condition (Cf), together with condition (ϕ5), implies that


To continue it will be convenient to introduce the following notations:


Using inequality (4.19) in the last inequality (4.18), dividing both sides of (4.18) by Θ, and rearranging we find

1Θ(ΔΘ+ϕ(u1)u1ϕ(u1)Θ11)-|2-ϕ(u1)u1ϕ(u1)||Θ|22Θ2+[2NN-1(ϕ(u1)u1ϕ(u1))2-2(N-3)N-1ϕ(u1)u1ϕ(u1)-2ϕ′′(u1)u12ϕ(u1)+2N-1]u12u2   +2N-1(f(u)u1ϕ(u1))2+[2N-1(1+ϕ(u1)u1ϕ(u1))+ϕ(u1)u1ϕ(u1)](Θ1f(u)Θu1ϕ(u1))   +[2NN-1(ϕ(u1)u1ϕ(u1))2-4(N-2)N-1ϕ(u1)u1ϕ(u1)-2ϕ′′(u1)u12ϕ(u1)-2(N-2)N-1]Θ1u1Θu   +2[NN-1(ϕ(u1)u1ϕ(u1))2+2N-1ϕ(u1)u1ϕ(u1)-ϕ′′(u1)u12ϕ(u1)+NN-1]Θ124Θ2=A(u1)|Θ|22Θ2+B(u1)u12u2+C(f(u)u1ϕ(u1))2+D(u1)(Θ1f(u)Θu1ϕ(u1))+E(u1)Θ1u1Θu+F(u1)Θ124Θ2A(u1)|Θ|22Θ2+B(u1)u12u2+C(f(u)u1ϕ(u1))2+D(u1)(Θ1f(u)Θu1ϕ(u1))+E(u1)Θ1u1Θu.(4.20)

Note that we have used condition (ϕ5) and FB>0 in the penultimate relation.1 Using the Cauchy–Schwarz inequality we estimate (4.20) as follows (we suppress the dependence of A,B,D,E and F on u1):


We now combine inequalities (4.7) and (4.21) to get


In other words, we have


From (4.2), we recall that at x*, we have

ΘΘ=r2(a2-r2)2,so that |Θ|22Θ2=8r2(a2-r2)2.

Therefore, at x*, we have the estimate




Note that by conditions (ϕ3) and (ϕ5), C0(u1) is bounded by a positive . Moreover, we observe that


Since J(x0)J(x*), we conclude that, at x0, we have


Thus, at x0, we have the estimate


Letting a, we find that |u|=0 at x0. Since x0 was arbitrary, we conclude that u0 on N, as desired. The proof is complete. ∎

4.1 An example

Let us illustrate the above theorem with a couple of examples. The simplest case occurs when ϕ(t)=ptp-2 for some p>1. In this case, σ=ρ=p-1, ϖ(t)=(p-2)(p-3) and


Therefore, we immediately see that condition (ϕ5) holds. This has been investigated in [7].

Now let us consider ϕ(t)=ptp-2+qtq-2 for 1<p<q. Recall that in this case σ=p-1 and ρ=q-1. Let us note that2


We now proceed to find conditions under which (ϕ5) holds. We have


Let us observe that


Therefore, we see that

max{(p-2)(p-3),(q-2)(q-3)}={(p-2)(p-3)if p<5/2,(q-2)(q-3)if p5/2.

So let us first suppose that 1<p<5/2. Then inequality (4.22) reduces to


Thus, if


then condition (ϕ5) holds. Now suppose p5/2. Then inequality (4.22) becomes


Therefore, if


then condition (ϕ5) holds.

Now, given p>1, let us set

p*:={13[(p-1)2N-1+3p]if p<52,(p-1)NN-1+1if p52.

We remark that p<p* for p>1. We now summarize the above discussion in the following corollary.

Corollary 4.4.

Given p>1, suppose that f satisfies condition (Cf) with σ=p-1. If 1<p<q<p*, then any non-negative entire solution of


is a constant on RN.


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  • 1

    Let ω:=ϕ(t)tϕ(t) and ϖ:=ϕ′′(t)t2ϕ(t). Then we see that B=2NN-1ω2-2(N-3)N-1ω-2ϖ+2N-1=2[NN-1(1+ω)2-3(1+ω)+2-ϖ] and F=2NN-1ω2+4N-1ω-2ϖ+2NN-1=2[NN-1(1+ω)2-2(1+ω)+2-ϖ]. 

  • 2

    Let a,b. Since min{a,b}a,bmax{a,b}, we have min{a,b}θa+(1-θ)bmax{a,b} for all 0θ1. 

About the article

Received: 2017-07-04

Accepted: 2017-07-12

Published Online: 2017-08-24

This work was supported by ISP (International Science Program) of Uppsala University, Sweden.

Citation Information: Advances in Nonlinear Analysis, Volume 8, Issue 1, Pages 725–742, ISSN (Online) 2191-950X, ISSN (Print) 2191-9496, DOI: https://doi.org/10.1515/anona-2017-0158.

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© 2019 Walter de Gruyter GmbH, Berlin/Boston. This work is licensed under the Creative Commons Attribution 4.0 Public License. BY 4.0

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