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Volume 8, Issue 1

# On Cauchy–Liouville-type theorems

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

## Abstract

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.

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

## 1 Introduction

Any real-valued harmonic function on ${ℝ}^{N}$ ($N\ge 2$) 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 $f:ℝ\to ℝ$ for which the quasilinear PDE

${\mathrm{\Delta }}_{\varphi }u=f\left(u\right)$(1.1)

admits constants as the only possible non-negative solutions in the entire Euclidean space ${ℝ}^{N}$ ($N\ge 2$). In this case we will say that the PDE has a Liouville-type property. Here ${\mathrm{\Delta }}_{\varphi }u:=\mathrm{div}\left(\varphi \left(|\nabla u|\right)\nabla u\right)$ is the so-called ϕ-Laplacian which reduces to the well-known p-Laplacian when $\varphi \left(t\right)=p{t}^{p-2}$ 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 $ℝ$ such that $f\left(0\right)=0$ and $f\left(t\right)>0$ for $t>0$. It is well known (see [9]) that the only non-negative solution of $\mathrm{\Delta }u=f\left(u\right)$ in ${ℝ}^{N}$ is the trivial solution $u\equiv 0$ if and only if

${\int }_{1}^{\mathrm{\infty }}\left({\int }_{0}^{t}f\left(s\right)ds\right){}^{-1/2}dt<\mathrm{\infty }.$(1.2)

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 $f:{ℝ}_{0}^{+}\to {ℝ}_{0}^{+}$ is subcritical in the sense that f satisfies

${f}^{\prime }\left(t\right)\le \left(\alpha -1\right)\frac{f\left(t\right)}{t},t>0$

for some $1<\alpha <{p}^{*}$, where ${p}^{*}$ is the critical Sobolev exponent of $p>1$. If there exists $q>p$ such that $f\left(t\right)\ge {t}^{q-1}$ at infinity, then any non-negative solution of ${\mathrm{\Delta }}_{p}u=-f\left(u\right)$ in ${ℝ}^{N}$ 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 $f:ℝ\to ℝ$ is a differentiable function that has a root and satisfies

${f}^{\prime }\left(t\right)\le \left(p-1\right)\frac{N+1}{N-1}\frac{f\left(t\right)}{t},t>0.$(1.3)

Then any positive solution of ${\mathrm{\Delta }}_{p}u=-f\left(u\right)$ in ${ℝ}^{N}$ 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 $\mathrm{\Delta }u=-f\left(u\right)$. 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 $f:ℝ\to ℝ$, 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 $\varphi :\left(0,\mathrm{\infty }\right)\to \left(0,\mathrm{\infty }\right)$, let us set

$\mathrm{\Psi }\left(t\right):=t\varphi \left(t\right)\mathit{ }\text{and}\mathit{ }\mathrm{\Phi }\left(t\right):={\int }_{0}^{t}\mathrm{\Psi }\left(s\right)𝑑s,t>0.$

We make the following assumptions:

• (ϕ1)

Ψ is a strictly increasing ${C}^{1}$ function in ${ℝ}^{+}:=\left(0,\mathrm{\infty }\right)$.

• (ϕ2)

${lim}_{s\to {0}^{+}}\mathrm{\Psi }\left(s\right)=0$ and ${lim}_{s\to \mathrm{\infty }}\mathrm{\Psi }\left(s\right)=\mathrm{\infty }$.

• (ϕ3)

There exist constants $0<\sigma \le \rho$ such that

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,

(2.1)

for some increasing functions $\lambda \le \mathrm{\Lambda }$. In fact,

Inequalities (2.1), in turn, imply

(2.2)

Another immediate consequence of (2.1) is

$\stackrel{~}{\lambda }\left(s\right)\mathrm{\Phi }\left(t\right)\le \mathrm{\Phi }\left(st\right)\le \stackrel{~}{\mathrm{\Lambda }}\left(s\right)\mathrm{\Phi }\left(t\right),s,t\ge 0,$(2.3)

where

$\stackrel{~}{\lambda }\left(s\right)=\lambda \left(s\right)s\mathit{ }\text{and}\mathit{ }\stackrel{~}{\mathrm{\Lambda }}\left(s\right)=\mathrm{\Lambda }\left(s\right)s.$

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

${\stackrel{~}{\mathrm{\Lambda }}}^{-1}\left(s\right){\mathrm{\Phi }}^{-1}\left(t\right)\le {\mathrm{\Phi }}^{-1}\left(st\right)\le {\stackrel{~}{\lambda }}^{-1}\left(s\right){\mathrm{\Phi }}^{-1}\left(t\right).$(2.4)

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 ${\mathrm{\Delta }}_{2}$-condition, namely, there exists a constant $c>0$ such that

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

Given an open set $\mathrm{\Omega }\subseteq {ℝ}^{N}$, the Orlicz space

is a Banach space under the so-called Luxemburg norm

${\parallel u\parallel }_{\mathrm{\Phi }}:=inf\left\{k>0:{\int }_{\mathrm{\Omega }}\mathrm{\Phi }\left(\frac{|u\left(x\right)|}{k}\right)𝑑x\le 1\right\}.$

The Orlicz–Sobolev space ${W}^{1,\mathrm{\Phi }}\left(\mathrm{\Omega }\right)$ is defined as the set of all weakly differentiable $u\in {L}^{\mathrm{\Phi }}\left(\mathrm{\Omega }\right)$ such that ${D}^{\alpha }u\in {L}^{\mathrm{\Phi }}\left(\mathrm{\Omega }\right)$ for all multi-indices $\alpha =\left({\alpha }_{1},\mathrm{\dots },{\alpha }_{N}\right)$ with $|\alpha |\le 1$. This is a Banach space under the norm

${\parallel u\parallel }_{{W}^{1,\mathrm{\Phi }}\left(\mathrm{\Omega }\right)}={\parallel u\parallel }_{\mathrm{\Phi }}+{\parallel \nabla u\parallel }_{\mathrm{\Phi }}.$

As with the usual Sobolev spaces, ${W}_{0}^{1,\mathrm{\Phi }}\left(\mathrm{\Omega }\right)$ is defined as the closure of ${C}_{c}^{\mathrm{\infty }}\left(\mathrm{\Omega }\right)$ in ${W}^{1,\mathrm{\Phi }}\left(\mathrm{\Omega }\right)$.

We define the local spaces ${L}_{\mathrm{loc}}^{\mathrm{\Phi }}\left(\mathrm{\Omega }\right)$ and ${W}_{\mathrm{loc}}^{1,\mathrm{\Phi }}\left(\mathrm{\Omega }\right)$ by

The dual space ${\left({L}^{\mathrm{\Phi }}\left(\mathrm{\Omega }\right)\right)}^{*}$ is equal to ${L}^{\stackrel{~}{\mathrm{\Phi }}}\left(\mathrm{\Omega }\right)$, where $\stackrel{~}{\mathrm{\Phi }}$ is the N-function given by

$\stackrel{~}{\mathrm{\Phi }}\left(t\right)={\int }_{0}^{t}{\mathrm{\Psi }}^{-1}\left(s\right)𝑑s,$

and is called the complement of Φ. The assumption (ϕ3) shows that $\stackrel{~}{\mathrm{\Phi }}$ satisfies a global ${\mathrm{\Delta }}_{2}$-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 $\mathrm{\Omega }\subseteq {ℝ}^{N}$ be an open set and $g:\mathrm{\Omega }×ℝ\to ℝ$ be a continuous function. We say a weakly differentiable function $v:\mathrm{\Omega }\to ℝ$ is a sub-solution of the PDE

${\mathrm{\Delta }}_{\varphi }u=g\left(x,u\right),x\in \mathrm{\Omega },$(2.5)

if and only if for every open and bounded subset $\mathcal{𝒪}\subseteq \mathrm{\Omega }$, we have $v\in {W}^{1,\mathrm{\Phi }}\left(\mathcal{𝒪}\right)$, with $g\left(x,v\left(x\right)\right)\in {L}^{\stackrel{~}{\mathrm{\Phi }}}\left(\mathcal{𝒪}\right)$, such that

(2.6)

A weakly differentiable function $w:\mathrm{\Omega }\to ℝ$ is said to be a super-solution of (2.5) in Ω if and only if for every open and bounded subset $\mathcal{𝒪}\subseteq \mathrm{\Omega }$, we have $w\in {W}^{1,\mathrm{\Phi }}\left(\mathcal{𝒪}\right)$, with $g\left(x,w\left(x\right)\right)\in {L}^{\stackrel{~}{\mathrm{\Phi }}}\left(\mathcal{𝒪}\right)$, such that the reverse inequality holds in (2.6) for all non-negative $\phi \in {W}_{0}^{1,\mathrm{\Phi }}\left(\mathcal{𝒪}\right)$. A weakly differentiable function $u:\mathrm{\Omega }\to ℝ$ 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

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

${\mathrm{\Delta }}_{\varphi }u=±f\left(u\right),$

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 $\mathrm{\Omega }\mathrm{\subseteq }{\mathrm{R}}^{N}$ be a bounded domain. Suppose that ϕ satisfies (ϕ1) and (ϕ2), and $g\mathrm{\in }{L}_{\mathrm{loc}}^{\mathrm{\infty }}\mathit{}\mathrm{\left(}\mathrm{\Omega }\mathrm{×}\mathrm{R}\mathrm{\right)}$ is such that $t\mathrm{↦}g\mathit{}\mathrm{\left(}x\mathrm{,}t\mathrm{\right)}$ is a non-decreasing function in $\mathrm{R}$ for each $x\mathrm{\in }\mathrm{\Omega }$. Suppose $u\mathrm{,}v\mathrm{\in }{W}^{\mathrm{1}\mathrm{,}\mathrm{\Phi }}\mathit{}\mathrm{\left(}\mathrm{\Omega }\mathrm{\right)}\mathrm{\cap }C\mathit{}\mathrm{\left(}\mathrm{\Omega }\mathrm{\right)}$ satisfy

If $u\mathrm{\le }v$ on $\mathrm{\partial }\mathit{}\mathrm{\Omega }$, then $u\mathrm{\le }v$ 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 $\mathrm{\Omega }\mathrm{\subseteq }{\mathrm{R}}^{N}$ be a bounded open set. Suppose that conditions (ϕ1) and (ϕ3) hold, and $g\mathrm{\in }{L}^{\mathrm{\infty }}\mathit{}\mathrm{\left(}\mathrm{\Omega }\mathrm{×}\mathrm{R}\mathrm{\right)}$. If $u\mathrm{\in }{W}^{\mathrm{1}\mathrm{,}\mathrm{\Phi }}\mathit{}\mathrm{\left(}\mathrm{\Omega }\mathrm{\right)}\mathrm{\cap }{L}^{\mathrm{\infty }}\mathit{}\mathrm{\left(}\mathrm{\Omega }\mathrm{\right)}$ is a solution of (2.5), then $u\mathrm{\in }{C}^{\mathrm{1}\mathrm{,}\alpha }\mathit{}\mathrm{\left(}\mathrm{\Omega }\mathrm{\right)}$ for some constant $\mathrm{0}\mathrm{<}\alpha \mathrm{<}\mathrm{1}$.

## 3 Monotonic non-negative absorption term

Let $f:ℝ\to ℝ$ be a non-decreasing continuous function such that $f\left(0\right)=0$ and $f\left(t\right)>0$ for $t>0$. In this section we extend a classical result of Keller [14] and Osserman [22] who showed independently that $\mathrm{\Delta }u=f\left(u\right)$ can not have a non-trivial solution in ${ℝ}^{N}$ 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 $\sigma \le \rho$, where σ and ρ are the parameters in condition (ϕ3), we note that for $s>1$, we have

${\mathrm{\Lambda }}^{-1}\left(s\right)=\mathrm{min}\left\{{s}^{1/\sigma },{s}^{1/\rho }\right\}={s}^{1/\rho }\mathit{ }\text{and}\mathit{ }{\lambda }^{-1}\left(s\right)=\mathrm{max}\left\{{s}^{1/\sigma },{s}^{1/\rho }\right\}={s}^{1/\sigma }.$

Therefore, for $s>\varrho >1$, we have

$\frac{{\varrho }^{N-1}{\mathrm{\Lambda }}^{-1}\left({\varrho }^{N-1}\right)}{{s}^{N-1}{\lambda }^{-1}\left({s}^{N-1}\right)}=\frac{{\varrho }^{\left(N-1\right)\left(\rho +1\right)/\rho }}{{s}^{\left(N-1\right)\left(\sigma +1\right)/\sigma }}.$(3.1)

Since $\left(\rho +1\right)/\rho \le \left(\sigma +1\right)/\sigma$, we see that the right-hand side of (3.1) is less than one whenever $s>\varrho >1$. Now let us note that

${\stackrel{~}{\mathrm{\Lambda }}}^{-1}\left(s\right)=\mathrm{min}\left\{{s}^{1/\left(\sigma +1\right)},{s}^{1/\left(\rho +1\right)}\right\},$

so that for $0, we have

${\stackrel{~}{\mathrm{\Lambda }}}^{-1}\left(s\right)={s}^{1/\left(\sigma +1\right)}.$

Applying this to (3.1), we find that

${\stackrel{~}{\mathrm{\Lambda }}}^{-1}\left(\frac{{\varrho }^{N-1}{\mathrm{\Lambda }}^{-1}\left({\varrho }^{N-1}\right)}{{s}^{N-1}{\lambda }^{-1}\left({s}^{N-1}\right)}\right)=\frac{{\varrho }^{\left(N-1\right)\left(\rho +1\right)/\rho \left(\sigma +1\right)}}{{s}^{\left(N-1\right)/\sigma }}.$

If $\sigma \ge N-1$, then it is clear that

$\underset{r\to \mathrm{\infty }}{lim}{\int }_{\varrho }^{r}{\stackrel{~}{\mathrm{\Lambda }}}^{-1}\left(\frac{{\varrho }^{N-1}{\mathrm{\Lambda }}^{-1}\left({\varrho }^{N-1}\right)}{{s}^{N-1}{\lambda }^{-1}\left({s}^{N-1}\right)}\right)𝑑s={\varrho }^{\left(N-1\right)\left(\rho +1\right)/\rho \left(\sigma +1\right)}\underset{r\to \mathrm{\infty }}{lim}{\int }_{\varrho }^{r}{s}^{-\left(N-1\right)/\sigma }𝑑s=\mathrm{\infty }.$

On the other hand suppose $\sigma . If, in addition, $\frac{\rho -\sigma }{\rho \sigma \left(\sigma +1\right)}<\frac{1}{N-1}$, then we have

${\int }_{\varrho }^{\mathrm{\infty }}{\stackrel{~}{\mathrm{\Lambda }}}^{-1}\left(\frac{{\varrho }^{N-1}{\mathrm{\Lambda }}^{-1}\left({\varrho }^{N-1}\right)}{{s}^{N-1}{\lambda }^{-1}\left({s}^{N-1}\right)}\right)𝑑s={\varrho }^{\left(N-1\right)\left(\rho +1\right)/\rho \left(\sigma +1\right)}{\int }_{\varrho }^{\mathrm{\infty }}{s}^{-\left(N-1\right)/\sigma }𝑑s=\frac{\sigma }{N-1-\sigma }{\varrho }^{1-\left(N-1\right)\left(\rho -\sigma \right)/\rho \sigma \left(\sigma +1\right)}.$

Consequently, in this case we find

$\underset{\varrho \to \mathrm{\infty }}{lim}{\int }_{\varrho }^{\mathrm{\infty }}{\stackrel{~}{\mathrm{\Lambda }}}^{-1}\left(\frac{{\varrho }^{N-1}{\mathrm{\Lambda }}^{-1}\left({\varrho }^{N-1}\right)}{{s}^{N-1}{\lambda }^{-1}\left({s}^{N-1}\right)}\right)𝑑s=\mathrm{\infty }.$

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

• (ϕ4)

$0\le \frac{\rho -\sigma }{\rho \left(\sigma +1\right)}<\frac{\sigma }{N-1}$.

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 $\frac{\sigma }{N-1}\ge 1$, then for each $\varrho >1$,

$\underset{r\to \mathrm{\infty }}{lim}{\int }_{\varrho }^{r}{\stackrel{~}{\mathrm{\Lambda }}}^{-1}\left(\frac{{\varrho }^{N-1}{\mathrm{\Lambda }}^{-1}\left({\varrho }^{N-1}\right)}{{s}^{N-1}{\lambda }^{-1}\left({s}^{N-1}\right)}\right)𝑑s=\mathrm{\infty }.$

• (ii)

If $\frac{\sigma }{N-1}<1$, then

$\underset{\varrho \to \mathrm{\infty }}{lim}{\int }_{\varrho }^{\mathrm{\infty }}{\stackrel{~}{\mathrm{\Lambda }}}^{-1}\left(\frac{{\varrho }^{N-1}{\mathrm{\Lambda }}^{-1}\left({\varrho }^{N-1}\right)}{{s}^{N-1}{\lambda }^{-1}\left({s}^{N-1}\right)}\right)𝑑s=\mathrm{\infty }.$

Let us now consider a non-decreasing function $f:\left(0,\mathrm{\infty }\right)\to \left(0,\mathrm{\infty }\right)$ such that

(3.2)

Condition (3.2) is easily recognized as a generalization of the well-known Keller–Osserman condition when $\varphi \equiv 1$.

We point out that condition (3.2) is equivalent to

(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

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

${\int }_{t}^{\mathrm{\infty }}\frac{ds}{{\mathrm{\Phi }}^{-1}\left(F\left(s\right)-F\left(t\right)\right)}={\int }_{t}^{\theta +t}\frac{ds}{{\mathrm{\Phi }}^{-1}\left(F\left(s\right)-F\left(t\right)\right)}+{\int }_{\theta +t}^{\mathrm{\infty }}\frac{ds}{{\mathrm{\Phi }}^{-1}\left(F\left(s\right)-F\left(t\right)\right)}$$\le \frac{1}{{\mathrm{\Phi }}^{-1}\left(f\left(t\right)\right)}{\int }_{t}^{\theta +t}\frac{ds}{{\stackrel{~}{\mathrm{\Lambda }}}^{-1}\left(s-t\right)}+{\int }_{\theta +t}^{\mathrm{\infty }}\frac{ds}{{\mathrm{\Phi }}^{-1}\left(F\left(s-t\right)\right)}$$\le \frac{1}{{\mathrm{\Phi }}^{-1}\left(f\left(t\right)\right)}{\int }_{t}^{\theta +t}\frac{ds}{{\left(s-t\right)}^{1/\left(\sigma +1\right)}}+{\int }_{\theta }^{\mathrm{\infty }}\frac{ds}{{\mathrm{\Phi }}^{-1}\left(F\left(s\right)\right)}$$=\frac{{\theta }^{\sigma /\left(\sigma +1\right)}}{{\mathrm{\Phi }}^{-1}\left(f\left(t\right)\right)}+{\int }_{\theta }^{\mathrm{\infty }}\frac{ds}{{\mathrm{\Phi }}^{-1}\left(F\left(s\right)\right)}.$(3.4)

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:

${\mathrm{\Delta }}_{\mathrm{\Phi }}u=f\left(u\right).$(3.5)

We assume that $f:{ℝ}_{0}^{+}\to {ℝ}_{0}^{+}$ is a continuous function such that $f\left(0\right)=0$ and $f\left(t\right)>0$ 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 $u\mathrm{\in }{W}_{\mathrm{loc}}^{\mathrm{1}\mathrm{,}\mathrm{\Phi }}\mathit{}\mathrm{\left(}{\mathrm{R}}^{N}\mathrm{\right)}\mathrm{\cap }C\mathit{}\mathrm{\left(}{\mathrm{R}}^{N}\mathrm{\right)}$ is a non-negative sub-solution of (3.5) in ${\mathrm{R}}^{N}$, then $u\mathrm{\equiv }\mathrm{0}$ in ${\mathrm{R}}^{N}$.

#### Proof.

Let $z\in {ℝ}^{N}$ and let $\epsilon >0$ be arbitrary. Let v be a solution of

$\left\{\begin{array}{cc}& {\left({r}^{N-1}\varphi \left(|{v}^{\prime }|\right){v}^{\prime }\right)}^{\prime }={r}^{N-1}f\left(v\left(r\right)\right),r>0,\hfill \\ & v\left(0\right)=\epsilon ,{v}^{\prime }\left(0\right)=0.\hfill \end{array}$

Let $\left(0,R\right)$ be the maximal interval of existence of v. Note that v cannot be a constant. We shall show that $R<\mathrm{\infty }$. Note that

${r}^{N-1}\varphi \left(|{v}^{\prime }\left(r\right)|\right){v}^{\prime }\left(r\right)={\int }_{0}^{r}{s}^{N-1}f\left(v\left(s\right)\right)𝑑s,0(3.6)

Therefore, ${v}^{\prime }>0$ in $\left(0,R\right)$, and ${r}^{N-1}\mathrm{\Psi }\left({v}^{\prime }\right)$ is a non-decreasing function in $\left(0,R\right)$, and hence ${\mathrm{\Psi }}^{-1}\left({r}^{N-1}\mathrm{\Psi }\left({v}^{\prime }\right)\right)$ is non-decreasing on $\left(0,R\right)$ as well. On multiplying both sides of (3.6) by ${\mathrm{\Psi }}^{-1}\left({r}^{N-1}\mathrm{\Psi }\left({v}^{\prime }\right)\right)$, for any $\varrho , we estimate

${\mathrm{\Psi }}^{-1}\left({r}^{N-1}\mathrm{\Psi }\left({v}^{\prime }\right)\right){r}^{N-1}\mathrm{\Psi }\left({v}^{\prime }\right)={\mathrm{\Psi }}^{-1}\left({r}^{N-1}\mathrm{\Psi }\left({v}^{\prime }\left(r\right)\right)\right){\int }_{0}^{r}{s}^{N-1}f\left(v\left(s\right)\right)𝑑s$$\ge {\int }_{0}^{r}{s}^{N-1}{\mathrm{\Psi }}^{-1}\left({s}^{N-1}\mathrm{\Psi }\left({v}^{\prime }\left(s\right)\right)\right)f\left(v\left(s\right)\right)𝑑s$$\ge {\int }_{\varrho }^{r}{s}^{N-1}{\mathrm{\Lambda }}^{-1}\left({s}^{N-1}\right)f\left(v\left(s\right)\right){v}^{\prime }\left(s\right)𝑑s$(3.7)$\ge {\varrho }^{N-1}{\mathrm{\Lambda }}^{-1}\left({\varrho }^{N-1}\right){\int }_{\rho }^{r}f\left(v\left(s\right)\right){v}^{\prime }\left(s\right)𝑑s$$={\varrho }^{N-1}{\mathrm{\Lambda }}^{-1}\left({\varrho }^{N-1}\right)\left(F\left(v\left(r\right)\right)-F\left(v\left(\varrho \right)\right)\right).$(3.8)

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

${\mathrm{\Psi }}^{-1}\left({r}^{N-1}\mathrm{\Psi }\left({v}^{\prime }\right)\right){r}^{N-1}\mathrm{\Psi }\left({v}^{\prime }\right)\le {\lambda }^{-1}\left({r}^{N-1}\right){r}^{N-1}\mathrm{\Psi }\left({v}^{\prime }\left(r\right)\right){v}^{\prime }\left(r\right),\varrho

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

$\mathrm{\Psi }\left({v}^{\prime }\left(r\right)\right){v}^{\prime }\left(r\right)\ge \frac{{\varrho }^{N-1}{\mathrm{\Lambda }}^{-1}\left({\varrho }^{N-1}\right)}{{r}^{N-1}{\lambda }^{-1}\left({r}^{N-1}\right)}\left(F\left(v\left(r\right)\right)-F\left(v\left(\varrho \right)\right)\right),\varrho

On noting that $\mathrm{\Phi }\left(2{v}^{\prime }\left(r\right)\right)\ge \mathrm{\Psi }\left({v}^{\prime }\left(r\right)\right){v}^{\prime }\left(r\right)$, we use (2.4) to get

${v}^{\prime }\left(r\right)\ge \frac{1}{2}{\mathrm{\Phi }}^{-1}\left(\frac{{\varrho }^{N-1}{\mathrm{\Lambda }}^{-1}\left({\varrho }^{N-1}\right)}{{r}^{N-1}{\lambda }^{-1}\left({r}^{N-1}\right)}\left(F\left(v\left(r\right)\right)-F\left(v\left(\varrho \right)\right)\right)\right)$$\ge \frac{1}{2}{\stackrel{~}{\mathrm{\Lambda }}}^{-1}\left(\frac{{\varrho }^{N-1}{\mathrm{\Lambda }}^{-1}\left({\varrho }^{N-1}\right)}{{r}^{N-1}{\lambda }^{-1}\left({r}^{N-1}\right)}\right){\mathrm{\Phi }}^{-1}\left(F\left(v\left(r\right)\right)-F\left(v\left(\varrho \right)\right)\right),0<\varrho

Integrating this last inequality on $\left(\varrho ,r\right)$ yields

${\int }_{v\left(\varrho \right)}^{v\left(r\right)}\frac{ds}{{\mathrm{\Phi }}^{-1}\left(F\left(s\right)-F\left(v\left(\varrho \right)\right)\right)}\ge \frac{1}{2}{\int }_{\varrho }^{r}{\stackrel{~}{\mathrm{\Lambda }}}^{-1}\left(\frac{{\varrho }^{N-1}{\mathrm{\Lambda }}^{-1}\left({\varrho }^{N-1}\right)}{{s}^{N-1}{\lambda }^{-1}\left({s}^{N-1}\right)}\right)𝑑s.$

Consequently, we see that

(3.9)

Note that

$\mathcal{𝒬}\left(t\right)={\int }_{t}^{\mathrm{\infty }}\frac{ds}{{\mathrm{\Phi }}^{-1}\left(F\left(s\right)-F\left(t\right)\right)}={\int }_{0}^{\mathrm{\infty }}\frac{dx}{{\mathrm{\Phi }}^{-1}\left(x\right){F}^{-1}\left(x+F\left(t\right)\right)}$

is a non-increasing function. Therefore, since $\epsilon =v\left(0\right)\le v\left(\varrho \right)$, we have $\mathcal{𝒬}\left(\epsilon \right)\ge \mathcal{𝒬}\left(v\left(\varrho \right)\right)$. Hence, from inequality (3.9), we obtain

(3.10)

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

Now assume that $R=\mathrm{\infty }$. Then we can take $\varrho >1$ in (3.10). If $\sigma \ge N-1$, then taking the limit as $r\to \mathrm{\infty }$ in (3.10) leads to a contradiction, by Remark 3.1.

Now let us suppose that $\sigma . Taking the limit in (3.10) as $r\to \mathrm{\infty }$, we find that

${\int }_{\varrho }^{\mathrm{\infty }}{\stackrel{~}{\mathrm{\Lambda }}}^{-1}\left(\frac{{\varrho }^{N-1}{\mathrm{\Lambda }}^{-1}\left({\varrho }^{N-1}\right)}{{s}^{N-1}{\lambda }^{-1}\left({s}^{N-1}\right)}\right)𝑑s\le 2{\int }_{\epsilon }^{\mathrm{\infty }}\frac{ds}{{\mathrm{\Phi }}^{-1}\left(F\left(s\right)-F\left(\epsilon \right)\right)}.$(3.11)

Then we use condition (ϕ4) and Remark 3.1 to see that inequality (3.11) leads to a contradiction upon taking the limit as $\varrho \to \mathrm{\infty }$.

Therefore, we conclude that indeed $R<\mathrm{\infty }$, and as a consequence $v\left(r\right)\to \mathrm{\infty }$ as $r\to {R}^{-}$. Let us set $w\left(x\right):=v\left(|x-z|\right)$ for $x\in {ℝ}^{N}$, and suppose that u is any non-negative sub-solution of (3.5) in ${ℝ}^{N}$. Then, for $0<\delta <1$ such that $1-\delta$ is sufficiently small, we have $u on $\partial B\left(z,\delta R\right)$. Since w is a solution of (3.5) in the ball $B\left(z,\delta R\right)$ we invoke the comparison principle, Theorem A, to conclude

$u\left(x\right)\le w\left(x\right),x\in B\left(z,\delta R\right).$

In particular, we have $0\le u\left(z\right)\le \epsilon$. Since $\epsilon >0$ is arbitrary, we find that $u\left(z\right)=0$, and since $z\in {ℝ}^{N}$ is arbitrary, we conclude $u\equiv 0$ in ${ℝ}^{N}$. ∎

#### Theorem 3.3.

Suppose that (ϕ1)(ϕ4) hold. Let $f\mathrm{:}\mathrm{R}\mathrm{\to }\mathrm{R}$ be such that $f\mathit{}\mathrm{\left(}t\mathrm{\right)}\mathrm{>}\mathrm{0}$ for $t\mathrm{>}\mathrm{0}$ and $f\mathit{}\mathrm{\left(}\mathrm{0}\mathrm{\right)}\mathrm{=}\mathrm{0}$. Assume also that f satisfies (3.2).

• (a)

If u is a sub-solution of ( 3.5 ), then $u\le 0$ in ${ℝ}^{N}$.

• (b)

If f is an odd function and u satisfies ( 3.5 ), then $u\equiv 0$ in ${ℝ}^{N}$.

#### Proof.

(a)  Let u be any sub-solution of (3.5) in ${ℝ}^{N}$. To show that $u\le 0$ in ${ℝ}^{N}$, it suffices to demonstrate that ${u}^{+}$ is a sub-solution of (3.5) in ${ℝ}^{N}$. Once this is proven, then we can invoke Theorem 3.2 to conclude that ${u}^{+}\equiv 0$, and hence $u=-{u}^{-}\le 0$ in ${ℝ}^{N}$. First, since $u\in {W}_{\mathrm{loc}}^{1,\mathrm{\Phi }}\left({ℝ}^{N}\right)$, let us notice that ${u}^{+}\in {W}_{\mathrm{loc}}^{1,\mathrm{\Phi }}\left({ℝ}^{N}\right)$. To see that ${u}^{+}$ is a sub-solution of (3.5), let $\mathrm{\Omega }:=\left\{x\in {ℝ}^{N}:u\left(x\right)>0\right\}.$ We suppose that Ω is non-empty for otherwise there is nothing to prove. For each positive integer j, set ${\vartheta }_{j}\left(t\right):=\vartheta \left(jt\right)$, $t\in ℝ$, where $\vartheta \in {C}^{1}\left(ℝ\right)$, with

Given $\mathcal{𝒪}\subset \subset {ℝ}^{N}$, let $\phi \in {W}_{0}^{1,\mathrm{\Phi }}\left(\mathcal{𝒪}\right)$ with $\phi \ge 0$ in $\mathcal{𝒪}$. We have

${\int }_{\mathcal{𝒪}}〈\varphi \left(|\nabla {u}^{+}|\right)\nabla {u}^{+},\nabla \phi 〉={\int }_{\mathcal{𝒪}\cap \mathrm{\Omega }}〈\varphi \left(|\nabla u|\right)\nabla u,\nabla \phi 〉$$=\underset{j\to \mathrm{\infty }}{lim}{\int }_{\mathcal{𝒪}}{\vartheta }_{j}\left(u\right)〈\varphi \left(|\nabla u|\right)\nabla u,\nabla \phi 〉.$(3.12)

Let us note that ${\vartheta }_{j}\left(u\right)\phi \in {W}_{0}^{1,\mathrm{\Phi }}\left(\mathcal{𝒪}\right)$ and

${\vartheta }_{j}\left(u\right)〈\varphi \left(|\nabla u|\right)\nabla u,\nabla \phi 〉=〈\varphi \left(|\nabla u|\right)\nabla u,{\vartheta }_{j}\left(u\right)\nabla \phi 〉$$=〈\varphi \left(|\nabla u|\right)\nabla u,\nabla \left({\vartheta }_{j}\left(u\right)\phi \right)〉-{\vartheta }_{j}^{\prime }\left(u\right)\phi 〈\varphi \left(|\nabla u|\right)\nabla u,\nabla u〉$$\le 〈\varphi \left(|\nabla u|\right)\nabla u,\nabla \left({\vartheta }_{j}\left(u\right)\phi \right)〉.$(3.13)

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

${\int }_{\mathcal{𝒪}}〈\varphi \left(|\nabla {u}^{+}|\right)\nabla {u}^{+},\nabla \phi 〉\le -\underset{j\to \mathrm{\infty }}{lim}{\int }_{\mathcal{𝒪}}{\vartheta }_{j}\left(u\right)f\left(u\right)\phi =-{\int }_{\mathcal{𝒪}\cap \mathrm{\Omega }}f\left(u\right)\phi =-{\int }_{\mathcal{𝒪}}f\left({u}^{+}\right)\phi ,$

since $f\left(0\right)=0$. 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 ${ℝ}^{N}$. By what was proved in (a), we observe that $u\le 0$ in ${ℝ}^{N}$. On the other hand, it is easily seen that $-u$ is a solution of (3.5) in ${ℝ}^{N}$. Therefore, $-u\le 0$ in ${ℝ}^{N}$ again, completing the proof that $u\equiv 0$ in ${ℝ}^{N}$. ∎

## 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 $0<\sigma <\frac{1}{2}\left(\sqrt{4N-3}-1\right)$. Then (ϕ4) holds if and only if

$0<\sigma \le \rho <\frac{\sigma \left(N-1\right)}{N-1-\sigma \left(\sigma +1\right)}.$

• (ii)

Suppose $\sigma \ge \frac{1}{2}\left(\sqrt{4N-3}-1\right)$. Then (ϕ4) holds if and only if $\rho \ge \sigma$.

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

(1)  Let $\varphi \left(t\right)=p{t}^{p-2}$, with $p>1$. Then ${\mathrm{\Delta }}_{\varphi }$ is the standard p-Laplacian. In this case, $\sigma =\rho =p-1$, and therefore condition (ϕ4) holds. Condition (3.2) reduces to the requirement that

(3.14)

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

(2)  Let us now consider $\varphi \left(t\right)=p{t}^{p-2}+q{t}^{q-2}$ for $1. Computation shows that $\sigma =p-1$, $\rho =q-1$ and

$\mathrm{\Phi }\left(t\right)={t}^{p}+{t}^{q}\le 2{t}^{q},t>1.$

As a consequence, we see that

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

$1(3.15)

then (ϕ4) holds if and only if

$1

On the other hand, if

$p\ge \frac{1}{2}\left(\sqrt{4N-1}-1\right),$(3.16)

then (ϕ4) holds if and only if $q\ge p$.

(3)  Let $\varphi \left(t\right)=p{t}^{p-1}{\mathrm{log}}^{q}\left(1+t\right)+q{t}^{p-1}{\left(1+t\right)}^{-1}{\mathrm{log}}^{q-1}\left(1+t\right)$ for $p>1$ and $q>0$. Then $\sigma =p-1$ and $\rho =p+q-1$. Moreover, we see that $\mathrm{\Phi }\left(t\right)={t}^{p}{\mathrm{log}}^{q}\left(1+t\right)$. Note that given $\epsilon >0$, there exists a constant ${t}_{\epsilon }>0$, sufficiently large, such that

Therefore, if there exists $r>p$ such that

${\int }_{t}^{\mathrm{\infty }}\frac{ds}{{\left(F\left(s\right)\right)}^{1/r}}<\mathrm{\infty }$

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

$0

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 $f:ℝ\to ℝ$:

${\mathrm{\Delta }}_{\varphi }u=-f\left(u\right).$(4.1)

We do not make any monotonicity assumption on f, but we require f to be a ${C}^{1}$ 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 ${ℝ}^{N}$, then $v\left(x\right):=u\left(Ax\right)$ is also a solution of (4.1) on ${ℝ}^{N}$ 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 $\varphi \in {C}^{2}\left(0,\mathrm{\infty }\right)$ and satisfies the following condition:

• (ϕ5)

We have

and $\sigma ,\rho$ 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 $\varphi \left(t\right)={t}^{p-2}$, $p>1$, 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, $f\mathrm{\in }{C}_{\mathrm{loc}}^{\mathrm{1}\mathrm{,}\gamma }\mathit{}\mathrm{\left(}\mathrm{R}\mathrm{\right)}$ for some $\mathrm{0}\mathrm{<}\gamma \mathrm{<}\mathrm{1}$, and

(Cf)

If $u\mathrm{\in }{W}_{\mathrm{loc}}^{\mathrm{1}\mathrm{,}\mathrm{\Phi }}\mathit{}\mathrm{\left(}{\mathrm{R}}^{N}\mathrm{\right)}\mathrm{\cap }C\mathit{}\mathrm{\left(}{\mathrm{R}}^{N}\mathrm{\right)}$ is a positive solution of (4.1) in ${\mathrm{R}}^{N}$, then u is a constant in ${\mathrm{R}}^{N}$.

#### Remark 4.2.

Suppose that ϕ satisfies the assumptions of Theorem 4.1. If $f\left(t\right)={t}^{\theta }-{t}^{\vartheta }$ for some $0\le \theta \le \sigma \left(N+1\right)/\left(N-1\right)\le \vartheta$, then f satisfies (Cf), and therefore the only positive solution of (4.1) in ${ℝ}^{N}$ is $u\equiv 1$. We should also note that if f is a non-negative and non-increasing function on $ℝ$, 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 $u\in {C}_{\mathrm{loc}}^{1,\alpha }\left({ℝ}^{N}\right)$. Therefore, it follows that equation (4.1) is uniformly elliptic on open sets that are compactly contained in $\mathcal{𝒪}:=\left\{x\in {ℝ}^{N}:|\nabla u\left(x\right)|>0\right\}$. Hence, we observe that $u\in {C}^{3}$ in any open set that is compactly contained in $\mathcal{𝒪}$ (see [18, Corollary 2.2], with $p=q=2$). See also [19].

Let ${x}_{0}\in {ℝ}^{N}$ be an arbitrary but fixed point. We wish to show $\nabla u\left({x}_{0}\right)=0$. Given $a>0$, let us set

We note that $J\ge 0$ in $B:=B\left({x}_{0},a\right)$ and ${J|}_{|x-{x}_{0}|=a}=0$. Therefore, J attains its maximum value on $\overline{B}$ at some interior point ${x}^{*}\in B$. Suppose $\nabla u\left({x}^{*}\right)=0$. Then Θ (and hence J) will be zero at ${x}^{*}$. But then J (and therefore Θ) will be zero on $\overline{B}$. This would imply $|\nabla u|=0$ in $\overline{B}$, and in particular $\nabla u\left({x}_{0}\right)=0$. Therefore, we assume that $|\nabla u|>0$ at ${x}^{*}$. We recall that u is ${\mathcal{𝒞}}_{\mathrm{loc}}^{3}\left(\mathcal{𝒪}\right)$, where $\mathcal{𝒪}:=\left\{x\in \mathrm{\Omega }:|\nabla u|>0\right\}$.

Now, at ${x}^{*}$, we have

$0={J}_{j}=-2\left({a}^{2}-{r}^{2}\right){\left({r}^{2}\right)}_{j}\mathrm{\Theta }+{\left({a}^{2}-{r}^{2}\right)}^{2}{\mathrm{\Theta }}_{j},j=1,\mathrm{\dots },N,$(4.2)

and ${D}^{2}J\le 0.$

Let us first show that the $N×N$ matrix

$H:={I}_{N}+\frac{{\varphi }^{\prime }\left(|\nabla u|\right)|\nabla u|}{\varphi \left(|\nabla u|\right)}{|\nabla u|}^{-2}\nabla u\nabla {u}^{T}$

is positive definite. Here ${I}_{N}$ is the $N×N$ identity matrix, and ${A}^{T}$ stands for the transpose of the matrix A. To see that H is positive definite, we first note that for any $\xi \in {ℝ}^{N}\setminus \left\{0\right\}$,

${\xi }^{T}H\xi ={|\xi |}^{2}+\frac{{\varphi }^{\prime }\left(|\nabla u|\right)|\nabla u|}{\varphi \left(|\nabla u|\right)}\frac{{\left({\xi }^{T}\nabla u\right)}^{2}}{{|\nabla u|}^{2}}.$

If $\frac{{\varphi }^{\prime }\left(|\nabla u|\right)|\nabla u|}{\varphi \left(|\nabla u|\right)}\ge 0$, then ${\xi }^{T}H\xi \ge {|\xi |}^{2}>0$. If, on the other hand, $-1<\frac{{\varphi }^{\prime }\left(|\nabla u|\right)|\nabla u|}{\varphi \left(|\nabla u|\right)}<0$, then on noting that $|{\xi }^{T}\nabla u|\le |\xi ||\nabla u|$ and therefore

$\frac{{\left({\xi }^{T}\nabla u\right)}^{2}}{{|\nabla u|}^{2}}\le {|\xi |}^{2},$

we have (recalling condition ϕ4)

${\xi }^{T}H\xi \ge {|\xi |}^{2}+\frac{{\varphi }^{\prime }\left(|\nabla u|\right)|\nabla u|}{\varphi \left(|\nabla u|\right)}{|\xi |}^{2}={|\xi |}^{2}\left(1+\frac{{\varphi }^{\prime }\left(|\nabla u|\right)|\nabla u|}{\varphi \left(|\nabla u|\right)}\right)>0.$

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

Now recalling that ${D}^{2}J\le 0$ at ${x}^{*}$, we have $H{D}^{2}J\le 0$ at ${x}^{*}$. Therefore, at ${x}^{*}$, we have

$0\ge \sum _{j=1}^{N}{e}_{j}^{T}\left(H{D}^{2}J\right){e}_{j}=\mathrm{\Delta }J+\frac{{\varphi }^{\prime }\left(|\nabla u|\right)|\nabla u|}{\varphi \left(|\nabla u|\right)}{J}_{ij}{u}_{i}{u}_{j}{|\nabla u|}^{-2},$

where ${e}_{j}$ is the unit vector in ${ℝ}^{N}$ with 1 in the jth position. In the last equation we have used the simple fact that ${e}_{k}^{T}B{e}_{k}={b}_{kk}$, where $B=\left[{b}_{ij}\right]$ is an $N×N$ matrix. Thus, at ${x}^{*}$, we have

$\mathrm{\Delta }J+\frac{{\varphi }^{\prime }\left(|\nabla u|\right)|\nabla u|}{\varphi \left(|\nabla u|\right)}{J}_{ij}{u}_{i}{u}_{j}{|\nabla u|}^{-2}\le 0.$(4.3)

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

$|\nabla u|={u}_{1},{u}_{j}=0,j=2,\mathrm{\dots },N.$(4.4)

Through direct computation, we note that

$\mathrm{\Delta }J+\frac{{\varphi }^{\prime }\left({u}_{1}\right){u}_{1}}{\varphi \left({u}_{1}\right)}{J}_{11}=2{|\nabla \left({r}^{2}\right)|}^{2}\mathrm{\Theta }-2\left({a}^{2}-{r}^{2}\right)\mathrm{\Delta }\left({r}^{2}\right)\mathrm{\Theta }-4\left({a}^{2}-{r}^{2}\right)\nabla \left({r}^{2}\right)\cdot \nabla \mathrm{\Theta }+{\left({a}^{2}-{r}^{2}\right)}^{2}\mathrm{\Delta }\mathrm{\Theta }$$+\frac{{\varphi }^{\prime }\left({u}_{1}\right){u}_{1}}{\varphi \left({u}_{1}\right)}\left[2{\left({\left({r}^{2}\right)}_{1}\right)}^{2}\mathrm{\Theta }-4\left({a}^{2}-{r}^{2}\right)\mathrm{\Theta }-4\left({a}^{2}-{r}^{2}\right){\left({r}^{2}\right)}_{1}{\mathrm{\Theta }}_{1}+{\left({a}^{2}-{r}^{2}\right)}^{2}{\mathrm{\Theta }}_{11}\right].$

This, together with inequality (4.3), implies

$\mathrm{\Delta }\mathrm{\Theta }+\frac{{\varphi }^{\prime }\left({u}_{1}\right){u}_{1}}{\varphi \left({u}_{1}\right)}{\mathrm{\Theta }}_{11}\le -\frac{8{r}^{2}}{{\left({a}^{2}-{r}^{2}\right)}^{2}}\mathrm{\Theta }+\frac{4N}{\left({a}^{2}-{r}^{2}\right)}\mathrm{\Theta }+\frac{4\nabla \left({r}^{2}\right)\cdot \nabla \mathrm{\Theta }}{{a}^{2}-{r}^{2}}$$+\frac{{\varphi }^{\prime }\left({u}_{1}\right){u}_{1}}{\varphi \left({u}_{1}\right)}\left[-\frac{2{\left({\left({r}^{2}\right)}_{1}\right)}^{2}}{{\left({a}^{2}-{r}^{2}\right)}^{2}}\mathrm{\Theta }+\frac{4}{{a}^{2}-{r}^{2}}\mathrm{\Theta }+\frac{4{\left({r}^{2}\right)}_{1}}{{a}^{2}-{r}^{2}}{\mathrm{\Theta }}_{1}\right].$(4.5)

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

${|\nabla \left({r}^{2}\right)|}^{2}=4{r}^{2}\mathit{ }\text{and}\mathit{ }\mathrm{\Delta }\left({r}^{2}\right)=2N.$(4.6)

Moreover, from (4.2), we obtain

${\mathrm{\Theta }}_{1}=\frac{2{\left({r}^{2}\right)}_{1}}{{a}^{2}-{r}^{2}}\mathrm{\Theta }\mathit{ }\text{and}\mathit{ }\nabla \mathrm{\Theta }=\frac{2\mathrm{\Theta }}{{a}^{2}-{r}^{2}}\nabla \left({r}^{2}\right).$

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

$\mathrm{\Delta }\mathrm{\Theta }+\frac{{\varphi }^{\prime }\left({u}_{1}\right){u}_{1}}{\varphi \left({u}_{1}\right)}{\mathrm{\Theta }}_{11}\le \mathrm{\Theta }\left[\frac{24{r}^{2}}{{\left({a}^{2}-{r}^{2}\right)}^{2}}+\frac{4N}{{a}^{2}-{r}^{2}}+\frac{{\varphi }^{\prime }\left({u}_{1}\right){u}_{1}}{\varphi \left({u}_{1}\right)}\left(\frac{6{\left({\left({r}^{2}\right)}_{1}\right)}^{2}}{{\left({a}^{2}-{r}^{2}\right)}^{2}}+\frac{4}{{a}^{2}-{r}^{2}}\right)\right].$

Therefore, at ${x}^{*}$, we have

$\frac{1}{\mathrm{\Theta }}\left(\mathrm{\Delta }\mathrm{\Theta }+\frac{{\varphi }^{\prime }\left({u}_{1}\right){u}_{1}}{\varphi \left({u}_{1}\right)}{\mathrm{\Theta }}_{11}\right)\le \frac{24{r}^{2}}{{\left({a}^{2}-{r}^{2}\right)}^{2}}+\frac{4N}{{a}^{2}-{r}^{2}}+|\frac{{\varphi }^{\prime }\left({u}_{1}\right){u}_{1}}{\varphi \left({u}_{1}\right)}|\left(\frac{24{r}^{2}}{{\left({a}^{2}-{r}^{2}\right)}^{2}}+\frac{4}{{a}^{2}-{r}^{2}}\right).$

Since ${r}^{2}<{a}^{2}$ in $B\left({x}_{0},a\right)$, we have

$\frac{1}{\mathrm{\Theta }}\left(\mathrm{\Delta }\mathrm{\Theta }+\frac{{\varphi }^{\prime }\left({u}_{1}\right){u}_{1}}{\varphi \left({u}_{1}\right)}{\mathrm{\Theta }}_{11}\right)\le \frac{24{a}^{2}}{{\left({a}^{2}-{r}^{2}\right)}^{2}}+\frac{4N{a}^{2}}{{\left({a}^{2}-{r}^{2}\right)}^{2}}+|\frac{{\varphi }^{\prime }\left({u}_{1}\right){u}_{1}}{\varphi \left({u}_{1}\right)}|\left(\frac{24{a}^{2}}{{\left({a}^{2}-{r}^{2}\right)}^{2}}+\frac{4{a}^{2}}{{\left({a}^{2}-{r}^{2}\right)}^{2}}\right).$(4.7)

Let us now notice that for $i,j=1,\mathrm{\dots },N$, we have

${\mathrm{\Theta }}_{i}=2{u}_{ij}{u}_{j}{u}^{-2}-2{|\nabla u|}^{2}{u}_{i}{u}^{-3},$(4.8)${\mathrm{\Theta }}_{ii}=2{u}_{iij}{u}_{j}{u}^{-2}+2{u}_{ij}{u}_{ij}{u}^{-2}-8{u}_{ij}{u}_{i}{u}_{j}{u}^{-3}-2{|\nabla u|}^{2}{u}^{-3}{u}_{ii}+6{|\nabla u|}^{2}{u}^{-4}{u}_{i}^{2}.$(4.9)

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

${\mathrm{\Theta }}_{11}=2{u}_{111}{u}_{1}{u}^{-2}+2{u}_{1j}{u}_{1j}{u}^{-2}-10{u}_{11}{u}_{1}^{2}{u}^{-3}+6{u}_{1}^{4}{u}^{-4},$$\mathrm{\Delta }\mathrm{\Theta }=2{\left(\mathrm{\Delta }u\right)}_{1}{u}_{1}{u}^{-2}+2{u}_{ij}{u}_{ij}{u}^{-2}-8{u}_{11}{u}_{1}^{2}{u}^{-3}-2{u}_{1}^{2}{u}^{-3}\mathrm{\Delta }u+6{u}_{1}^{4}{u}^{-4}.$

From the above computation we see that at ${x}^{*}$,

$\mathrm{\Delta }\mathrm{\Theta }+\frac{{\varphi }^{\prime }\left({u}_{1}\right){u}_{1}}{\varphi \left({u}_{1}\right)}{\mathrm{\Theta }}_{11}=2{u}_{1}{u}^{-2}\left[{\left(\mathrm{\Delta }u\right)}_{1}+\frac{{\varphi }^{\prime }\left({u}_{1}\right){u}_{1}}{\varphi \left({u}_{1}\right)}{u}_{111}\right]+2{u}^{-2}\left({u}_{ij}{u}_{ij}+\frac{{\varphi }^{\prime }\left({u}_{1}\right){u}_{1}}{\varphi \left({u}_{1}\right)}{u}_{1j}{u}_{1j}\right)$$-2{u}_{1}^{2}{u}^{-3}\left[\mathrm{\Delta }u+\left(4+5\frac{{\varphi }^{\prime }\left({u}_{1}\right){u}_{1}}{\varphi \left({u}_{1}\right)}\right){u}_{11}\right]+6\left(1+\frac{{\varphi }^{\prime }\left({u}_{1}\right){u}_{1}}{\varphi \left({u}_{1}\right)}\right){u}_{1}^{4}{u}^{-4}.$(4.10)

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 ${\mathcal{𝒪}}^{\prime }$ that contains ${x}^{*}$, that is,

$\varphi \left(|\nabla u|\right)\mathrm{\Delta }u+\frac{{\varphi }^{\prime }\left(|\nabla u|\right)}{|\nabla u|}{u}_{ij}{u}_{i}{u}_{j}=-f\left(u\right).$(4.11)

where u is ${C}^{3}$-smooth. On recalling (4.4), at ${x}^{*}$, we obtain the following from (4.11):

$\varphi \left({u}_{1}\right)\mathrm{\Delta }u+{\varphi }^{\prime }\left({u}_{1}\right){u}_{11}{u}_{1}=-f\left(u\right),$

that is,

$\mathrm{\Delta }u+\frac{{\varphi }^{\prime }\left({u}_{1}\right){u}_{1}}{\varphi \left({u}_{1}\right)}{u}_{11}=-\frac{f\left(u\right)}{\varphi \left({u}_{1}\right)}.$(4.12)

We now differentiate (4.11) with respect to the variable ${x}_{1}$. On evaluating the resulting expression at ${x}^{*}$, we find

$\varphi \left({u}_{1}\right){\left(\mathrm{\Delta }u\right)}_{1}+{\varphi }^{\prime }\left({u}_{1}\right){u}_{11}\mathrm{\Delta }u+{\varphi }^{\prime \prime }\left({u}_{1}\right){u}_{1}{u}_{11}^{2}-{\varphi }^{\prime }\left({u}_{1}\right){u}_{11}^{2}+{\varphi }^{\prime }\left({u}_{1}\right){u}_{111}{u}_{1}+2{\varphi }^{\prime }\left({u}_{1}\right)\sum _{j=1}^{N}{u}_{1j}^{2}=-{u}_{1}{f}^{\prime }\left(u\right),$

that is

$\varphi \left({u}_{1}\right){\left(\mathrm{\Delta }u\right)}_{1}+{\varphi }^{\prime }\left({u}_{1}\right){u}_{1}{u}_{111}=-2{\varphi }^{\prime }\left({u}_{1}\right)\sum _{j=2}^{N}{u}_{1j}^{2}-{\varphi }^{\prime }\left({u}_{1}\right){u}_{11}\mathrm{\Delta }u-{\varphi }^{\prime \prime }\left({u}_{1}\right){u}_{1}{u}_{11}^{2}-{\varphi }^{\prime }\left({u}_{1}\right){u}_{11}^{2}-{u}_{1}{f}^{\prime }\left(u\right).$(4.13)

On dividing both sides of (4.13) by $\varphi \left({u}_{1}\right)$, using (4.12) to replace $\mathrm{\Delta }u$ in the resulting expression, and rearranging, we find

${\left(\mathrm{\Delta }u\right)}_{1}+\frac{{\varphi }^{\prime }\left({u}_{1}\right){u}_{1}}{\varphi \left({u}_{1}\right)}{u}_{111}$$=-2\frac{{\varphi }^{\prime }\left({u}_{1}\right)}{\varphi \left({u}_{1}\right)}\sum _{j=2}^{N}{u}_{1j}^{2}+\left[\frac{f\left(u\right)}{\varphi \left({u}_{1}\right)}-\left(1-\frac{{\varphi }^{\prime }\left({u}_{1}\right){u}_{1}}{\varphi \left({u}_{1}\right)}\right){u}_{11}\right]\frac{{\varphi }^{\prime }\left({u}_{1}\right)}{\varphi \left({u}_{1}\right)}{u}_{11}-\frac{{\varphi }^{\prime \prime }\left({u}_{1}\right){u}_{1}}{\varphi \left({u}_{1}\right)}{u}_{11}^{2}-\frac{{u}_{1}{f}^{\prime }\left(u\right)}{\varphi \left({u}_{1}\right)}.$(4.14)

Let us observe the following.

${u}_{ij}{u}_{ij}+\frac{{\varphi }^{\prime }\left({u}_{1}\right){u}_{1}}{\varphi \left({u}_{1}\right)}{u}_{1j}{u}_{1j}=2\sum _{j=2}^{N}{u}_{1j}^{2}+{u}_{11}^{2}+\sum _{j=2}^{N}{u}_{jj}^{2}+\sum _{i,j\ge 2,i\ne j}^{N}{u}_{ij}^{2}+\frac{{\varphi }^{\prime }\left({u}_{1}\right){u}_{1}}{\varphi \left({u}_{1}\right)}{u}_{11}^{2}+\frac{{\varphi }^{\prime }\left({u}_{1}\right){u}_{1}}{\varphi \left({u}_{1}\right)}\sum _{j=2}^{N}{u}_{1j}^{2}$$\ge \left(2+\frac{{\varphi }^{\prime }\left({u}_{1}\right){u}_{1}}{\varphi \left({u}_{1}\right)}\right)\sum _{j=2}^{N}{u}_{1j}^{2}+\left(1+\frac{{\varphi }^{\prime }\left({u}_{1}\right){u}_{1}}{\varphi \left({u}_{1}\right)}\right){u}_{11}^{2}+\sum _{j=2}^{N}{u}_{jj}^{2}.$(4.15)

To proceed further, we need the following inequality.

Using Lemma 4.3 in inequality (4.15), we obtain

${u}_{ij}{u}_{ij}+\frac{{\varphi }^{\prime }\left({u}_{1}\right){u}_{1}}{\varphi \left({u}_{1}\right)}{u}_{1j}{u}_{1j}\ge \left(2+\frac{{\varphi }^{\prime }\left({u}_{1}\right){u}_{1}}{\varphi \left({u}_{1}\right)}\right)\sum _{j=2}^{N}{u}_{1j}^{2}+\left(\frac{N}{N-1}+\frac{{\varphi }^{\prime }\left({u}_{1}\right){u}_{1}}{\varphi \left({u}_{1}\right)}\right){u}_{11}^{2}$$-\frac{2}{N-1}{u}_{11}\mathrm{\Delta }u+\frac{1}{N-1}{\left(\mathrm{\Delta }u\right)}^{2}.$(4.16)

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

$\mathrm{\Delta }\mathrm{\Theta }+\frac{{\varphi }^{\prime }\left({u}_{1}\right){u}_{1}}{\varphi \left({u}_{1}\right)}{\mathrm{\Theta }}_{11}$$\ge 2{u}_{1}{u}^{-2}\left\{-2\frac{{\varphi }^{\prime }\left({u}_{1}\right)}{\varphi \left({u}_{1}\right)}\sum _{j=2}^{N}{u}_{1j}^{2}+\left[\frac{f\left(u\right)}{\varphi \left({u}_{1}\right)}-\left(1-\frac{{\varphi }^{\prime }\left({u}_{1}\right){u}_{1}}{\varphi \left({u}_{1}\right)}\right){u}_{11}\right]\frac{{\varphi }^{\prime }\left({u}_{1}\right)}{\varphi \left({u}_{1}\right)}{u}_{11}-\frac{{\varphi }^{\prime \prime }\left({u}_{1}\right){u}_{1}}{\varphi \left({u}_{1}\right)}{u}_{11}^{2}-\frac{{u}_{1}{f}^{\prime }\left(u\right)}{\varphi \left({u}_{1}\right)}\right\}$$+2{u}^{-2}\left[\left(2+\frac{{\varphi }^{\prime }\left({u}_{1}\right){u}_{1}}{\varphi \left({u}_{1}\right)}\right)\sum _{j=2}^{N}{u}_{1j}^{2}+\left(\frac{N}{N-1}+\frac{{\varphi }^{\prime }\left({u}_{1}\right){u}_{1}}{\varphi \left({u}_{1}\right)}\right){u}_{11}^{2}-\frac{2}{N-1}{u}_{11}\mathrm{\Delta }u+\frac{1}{N-1}{\left(\mathrm{\Delta }u\right)}^{2}\right]$$-2{u}_{1}^{2}{u}^{-3}\left[\mathrm{\Delta }u+\left(4+5\frac{{\varphi }^{\prime }\left({u}_{1}\right){u}_{1}}{\varphi \left({u}_{1}\right)}\right){u}_{11}\right]+6\left(1+\frac{{\varphi }^{\prime }\left({u}_{1}\right){u}_{1}}{\varphi \left({u}_{1}\right)}\right){u}_{1}^{4}{u}^{-4}.$

Rearranging the terms in the last inequality, we find

$\mathrm{\Delta }\mathrm{\Theta }+\frac{{\varphi }^{\prime }\left({u}_{1}\right){u}_{1}}{\varphi \left({u}_{1}\right)}{\mathrm{\Theta }}_{11}\ge 2{u}^{-2}\left\{\left(2-\frac{{\varphi }^{\prime }\left({u}_{1}\right){u}_{1}}{\varphi \left({u}_{1}\right)}\right)\sum _{j=2}^{N}{u}_{1j}^{2}+\frac{{\varphi }^{\prime }\left({u}_{1}\right){u}_{1}}{\varphi \left({u}_{1}\right)}\cdot \frac{f\left(u\right)}{\varphi \left({u}_{1}\right)}{u}_{11}-\frac{{f}^{\prime }{u}_{1}^{2}}{\varphi \left({u}_{1}\right)}$$-\frac{2}{N-1}\left(\mathrm{\Delta }u\right){u}_{11}+\frac{1}{N-1}{\left(\mathrm{\Delta }u\right)}^{2}+\left[{\left(\frac{{\varphi }^{\prime }\left({u}_{1}\right){u}_{1}}{\varphi \left({u}_{1}\right)}\right)}^{2}-\frac{{\varphi }^{\prime \prime }\left({u}_{1}\right){u}_{1}^{2}}{\varphi \left({u}_{1}\right)}+\frac{N}{N-1}\right]{u}_{11}^{2}$$-\left[\mathrm{\Delta }u+\left(4+5\frac{{\varphi }^{\prime }\left({u}_{1}\right){u}_{1}}{\varphi \left({u}_{1}\right)}\right){u}_{11}\right]{u}_{1}^{2}{u}^{-1}+3\left(1+\frac{{\varphi }^{\prime }\left({u}_{1}\right){u}_{1}}{\varphi \left({u}_{1}\right)}\right){u}_{1}^{4}{u}^{-2}\right\}.$(4.17)

We recall from (4.8) that for $j=2,\mathrm{\dots },N$, we have ${\mathrm{\Theta }}_{j}=2{u}_{1j}{u}_{1}{u}^{-2}$, and therefore

$2{u}^{-2}\sum _{j=2}^{N}{u}_{1j}^{2}=\frac{{\sum }_{j=2}^{N}{\mathrm{\Theta }}_{j}^{2}}{2{\left(\frac{{u}_{1}}{u}\right)}^{2}}=\frac{{\sum }_{j=2}^{N}{\mathrm{\Theta }}_{j}^{2}}{2\mathrm{\Theta }}\le \frac{{|\nabla \mathrm{\Theta }|}^{2}}{2\mathrm{\Theta }}.$

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

$\mathrm{\Delta }\mathrm{\Theta }+\frac{{\varphi }^{\prime }\left({u}_{1}\right){u}_{1}}{\varphi \left({u}_{1}\right)}{\mathrm{\Theta }}_{11}\ge -|2-\frac{{\varphi }^{\prime }\left({u}_{1}\right){u}_{1}}{\varphi \left({u}_{1}\right)}|\frac{{|\nabla \mathrm{\Theta }|}^{2}}{2\mathrm{\Theta }}+2{u}^{-2}\left\{\frac{{\varphi }^{\prime }\left({u}_{1}\right){u}_{1}}{\varphi \left({u}_{1}\right)}\cdot \frac{f\left(u\right)}{\varphi \left({u}_{1}\right)}{u}_{11}-\frac{{f}^{\prime }{u}_{1}^{2}}{\varphi \left({u}_{1}\right)}$$+\left[{\left(\frac{{\varphi }^{\prime }\left({u}_{1}\right){u}_{1}}{\varphi \left({u}_{1}\right)}\right)}^{2}-\frac{{\varphi }^{\prime \prime }\left({u}_{1}\right){u}_{1}^{2}}{\varphi \left({u}_{1}\right)}+\frac{N}{N-1}\right]{u}_{11}^{2}$$+\frac{2}{N-1}\left(\frac{{\varphi }^{\prime }\left({u}_{1}\right){u}_{1}}{\varphi \left({u}_{1}\right)}{u}_{11}+\frac{f\left(u\right)}{\varphi \left({u}_{1}\right)}\right){u}_{11}+\frac{1}{N-1}{\left(\frac{{\varphi }^{\prime }\left({u}_{1}\right){u}_{1}}{\varphi \left({u}_{1}\right)}{u}_{11}+\frac{f\left(u\right)}{\varphi \left({u}_{1}\right)}\right)}^{2}$$-\left[-\frac{{\varphi }^{\prime }\left({u}_{1}\right){u}_{1}}{\varphi \left({u}_{1}\right)}{u}_{11}-\frac{f\left(u\right)}{\varphi \left({u}_{1}\right)}+\left(4+5\frac{{\varphi }^{\prime }\left({u}_{1}\right){u}_{1}}{\varphi \left({u}_{1}\right)}\right){u}_{11}\right]{u}_{1}^{2}{u}^{-1}$$+3\left(1+\frac{{\varphi }^{\prime }\left({u}_{1}\right){u}_{1}}{\varphi \left({u}_{1}\right)}\right){u}_{1}^{4}{u}^{-4}\right\}.$

Rearranging the terms in the above inequality yields

$\mathrm{\Delta }\mathrm{\Theta }+\frac{{\varphi }^{\prime }\left({u}_{1}\right){u}_{1}}{\varphi \left({u}_{1}\right)}{\mathrm{\Theta }}_{11}\ge -|2-\frac{{\varphi }^{\prime }\left({u}_{1}\right){u}_{1}}{\varphi \left({u}_{1}\right)}|\frac{{|\nabla \mathrm{\Theta }|}^{2}}{2\mathrm{\Theta }}+2{u}^{-2}\left\{-\frac{{f}^{\prime }{u}_{1}^{2}}{\varphi \left({u}_{1}\right)}$$+\left[\frac{N}{N-1}{\left(\frac{{\varphi }^{\prime }\left({u}_{1}\right){u}_{1}}{\varphi \left({u}_{1}\right)}\right)}^{2}+\frac{2}{N-1}\frac{{\varphi }^{\prime }\left({u}_{1}\right){u}_{1}}{\varphi \left({u}_{1}\right)}-\frac{{\varphi }^{\prime \prime }\left({u}_{1}\right){u}_{1}^{2}}{\varphi \left({u}_{1}\right)}+\frac{N}{N-1}\right]{u}_{11}^{2}$$+\frac{1}{N-1}{\left(\frac{f\left(u\right)}{\varphi \left({u}_{1}\right)}\right)}^{2}+\left[\frac{2}{N-1}\left(1+\frac{{\varphi }^{\prime }\left({u}_{1}\right){u}_{1}}{\varphi \left({u}_{1}\right)}\right)+\frac{{\varphi }^{\prime }\left({u}_{1}\right){u}_{1}}{\varphi \left({u}_{1}\right)}\right]\frac{f\left(u\right)}{\varphi \left({u}_{1}\right)}{u}_{11}$$-\left[4\left(1+\frac{{\varphi }^{\prime }\left({u}_{1}\right){u}_{1}}{\varphi \left({u}_{1}\right)}\right){u}_{11}-\frac{f\left(u\right)}{\varphi \left({u}_{1}\right)}\right]{u}_{1}^{2}{u}^{-1}+3\left(1+\frac{{\varphi }^{\prime }\left({u}_{1}\right){u}_{1}}{\varphi \left({u}_{1}\right)}\right){u}_{1}^{4}{u}^{-2}\right\}.$

We recall from (4.8) that ${\mathrm{\Theta }}_{1}=2{u}_{1}{u}_{11}{u}^{-2}-2{u}_{1}^{3}{u}^{-3}$, that is

${u}_{11}=\frac{{\mathrm{\Theta }}_{1}{u}^{2}}{2{u}_{1}}+\frac{{u}_{1}^{2}}{u}.$

Inserting this in the last inequality gives

$\mathrm{\Delta }\mathrm{\Theta }+\frac{{\varphi }^{\prime }\left({u}_{1}\right){u}_{1}}{\varphi \left({u}_{1}\right)}{\mathrm{\Theta }}_{11}\ge -|2-\frac{{\varphi }^{\prime }\left({u}_{1}\right){u}_{1}}{\varphi \left({u}_{1}\right)}|\frac{{|\nabla \mathrm{\Theta }|}^{2}}{2\mathrm{\Theta }}+2{u}^{-2}\left\{-\frac{{f}^{\prime }{u}_{1}^{2}}{\varphi \left({u}_{1}\right)}$$+\left[\frac{N}{N-1}{\left(\frac{{\varphi }^{\prime }\left({u}_{1}\right){u}_{1}}{\varphi \left({u}_{1}\right)}\right)}^{2}+\frac{2}{N-1}\frac{{\varphi }^{\prime }\left({u}_{1}\right){u}_{1}}{\varphi \left({u}_{1}\right)}-\frac{{\varphi }^{\prime \prime }\left({u}_{1}\right){u}_{1}^{2}}{\varphi \left({u}_{1}\right)}+\frac{N}{N-1}\right]{\left(\frac{{\mathrm{\Theta }}_{1}{u}^{2}}{2{u}_{1}}+\frac{{u}_{1}^{2}}{u}\right)}^{2}$$+\frac{1}{N-1}{\left(\frac{f\left(u\right)}{\varphi \left({u}_{1}\right)}\right)}^{2}+\left[\frac{2}{N-1}\left(1+\frac{{\varphi }^{\prime }\left({u}_{1}\right){u}_{1}}{\varphi \left({u}_{1}\right)}\right)+\frac{{\varphi }^{\prime }\left({u}_{1}\right){u}_{1}}{\varphi \left({u}_{1}\right)}\right]\left(\frac{{\mathrm{\Theta }}_{1}{u}^{2}}{2{u}_{1}}+\frac{{u}_{1}^{2}}{u}\right)\frac{f\left(u\right)}{\varphi \left({u}_{1}\right)}$$-\left[4\left(1+\frac{{\varphi }^{\prime }\left({u}_{1}\right){u}_{1}}{\varphi \left({u}_{1}\right)}\right)\left(\frac{{\mathrm{\Theta }}_{1}{u}^{2}}{2{u}_{1}}+\frac{{u}_{1}^{2}}{u}\right)-\frac{f\left(u\right)}{\varphi \left({u}_{1}\right)}\right]{u}_{1}^{2}{u}^{-1}+3\left(1+\frac{{\varphi }^{\prime }\left({u}_{1}\right){u}_{1}}{\varphi \left({u}_{1}\right)}\right){u}_{1}^{4}{u}^{-2}\right\}.$

$\mathrm{\Delta }\mathrm{\Theta }+\frac{{\varphi }^{\prime }\left({u}_{1}\right){u}_{1}}{\varphi \left({u}_{1}\right)}{\mathrm{\Theta }}_{11}\ge -|2-\frac{{\varphi }^{\prime }\left({u}_{1}\right){u}_{1}}{\varphi \left({u}_{1}\right)}|\frac{{|\nabla \mathrm{\Theta }|}^{2}}{2\mathrm{\Theta }}$$+\left[\frac{N}{N-1}{\left(\frac{{\varphi }^{\prime }\left({u}_{1}\right){u}_{1}}{\varphi \left({u}_{1}\right)}\right)}^{2}+\frac{2}{N-1}\frac{{\varphi }^{\prime }\left({u}_{1}\right){u}_{1}}{\varphi \left({u}_{1}\right)}-\frac{{\varphi }^{\prime \prime }\left({u}_{1}\right){u}_{1}^{2}}{\varphi \left({u}_{1}\right)}+\frac{N}{N-1}\right]\left(\frac{2{\mathrm{\Theta }}_{1}{u}_{1}}{u}+\frac{2{u}_{1}^{4}}{{u}^{4}}\right)$$+\frac{2}{N-1}{\left(\frac{f\left(u\right)}{u\varphi \left({u}_{1}\right)}\right)}^{2}+\left[\frac{2}{N-1}\left(1+\frac{{\varphi }^{\prime }\left({u}_{1}\right){u}_{1}}{\varphi \left({u}_{1}\right)}\right)+\frac{{\varphi }^{\prime }\left({u}_{1}\right){u}_{1}}{\varphi \left({u}_{1}\right)}\right]\left(\frac{{\mathrm{\Theta }}_{1}f\left(u\right)}{{u}_{1}\varphi \left({u}_{1}\right)}\right)$$-4\left(1+\frac{{\varphi }^{\prime }\left({u}_{1}\right){u}_{1}}{\varphi \left({u}_{1}\right)}\right)\frac{{\mathrm{\Theta }}_{1}{u}_{1}}{u}-2\left(1+\frac{{\varphi }^{\prime }\left({u}_{1}\right){u}_{1}}{\varphi \left({u}_{1}\right)}\right){u}_{1}^{4}{u}^{-4}$$+2\left[\frac{N}{N-1}{\left(\frac{{\varphi }^{\prime }\left({u}_{1}\right){u}_{1}}{\varphi \left({u}_{1}\right)}\right)}^{2}+\frac{2}{N-1}\frac{{\varphi }^{\prime }\left({u}_{1}\right){u}_{1}}{\varphi \left({u}_{1}\right)}-\frac{{\varphi }^{\prime \prime }\left({u}_{1}\right){u}_{1}^{2}}{\varphi \left({u}_{1}\right)}+\frac{N}{N-1}\right]{\left(\frac{{\mathrm{\Theta }}_{1}{u}^{2}}{2u{u}_{1}}\right)}^{2}$$+2{u}^{-2}\left[-{f}^{\prime }\left(u\right)+\frac{N+1}{N-1}\left(1+\frac{{\varphi }^{\prime }\left({u}_{1}\right){u}_{1}}{\varphi \left({u}_{1}\right)}\right)\frac{f\left(u\right)}{u}\right]\frac{{u}_{1}^{2}}{\varphi \left({u}_{1}\right)}.$(4.18)

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

$-{f}^{\prime }\left(u\right)+\frac{N+1}{N-1}\left(1+\frac{{\varphi }^{\prime }\left({u}_{1}\right){u}_{1}}{\varphi \left({u}_{1}\right)}\right)\frac{f\left(u\right)}{u}\ge -{f}^{\prime }\left(u\right)+\sigma \left(\frac{N+1}{N-1}\right)\frac{f\left(u\right)}{u}\ge 0.$(4.19)

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

$A\left(t\right):=-|2-\frac{{\varphi }^{\prime }\left(t\right)t}{\varphi \left(t\right)}|,$$B\left(t\right):=\frac{2N}{N-1}{\left(\frac{{\varphi }^{\prime }\left(t\right)t}{\varphi \left(t\right)}\right)}^{2}-\frac{2\left(N-3\right)}{N-1}\frac{{\varphi }^{\prime }\left(t\right)t}{\varphi \left(t\right)}-\frac{2{\varphi }^{\prime \prime }\left(t\right){t}^{2}}{\varphi \left(t\right)}+\frac{2}{N-1},$$C:=\frac{2}{N-1},$$D\left(t\right):=\frac{2}{N-1}\left(1+\frac{{\varphi }^{\prime }\left(t\right)t}{\varphi \left(t\right)}\right)+\frac{{\varphi }^{\prime }\left(t\right)t}{\varphi \left(t\right)},$$E\left(t\right):=\frac{2N}{N-1}{\left(\frac{{\varphi }^{\prime }\left(t\right)t}{\varphi \left(t\right)}\right)}^{2}-\frac{4\left(N-2\right)}{N-1}\frac{{\varphi }^{\prime }\left(t\right)t}{\varphi \left(t\right)}-\frac{2{\varphi }^{\prime \prime }\left(t\right){t}^{2}}{\varphi \left(t\right)}-\frac{2\left(N-2\right)}{N-1},$$F\left(t\right):=\frac{N}{N-1}{\left(\frac{{\varphi }^{\prime }\left(t\right)t}{\varphi \left(t\right)}\right)}^{2}+\frac{2}{N-1}\frac{{\varphi }^{\prime }\left(t\right)t}{\varphi \left(t\right)}-\frac{{\varphi }^{\prime \prime }\left(t\right){u}_{1}^{2}}{\varphi \left(t\right)}+\frac{N}{N-1}.$

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

$\frac{1}{\mathrm{\Theta }}\left(\mathrm{\Delta }\mathrm{\Theta }+\frac{{\varphi }^{\prime }\left({u}_{1}\right){u}_{1}}{\varphi \left({u}_{1}\right)}{\mathrm{\Theta }}_{11}\right)$$\ge -|2-\frac{{\varphi }^{\prime }\left({u}_{1}\right){u}_{1}}{\varphi \left({u}_{1}\right)}|\frac{{|\nabla \mathrm{\Theta }|}^{2}}{2{\mathrm{\Theta }}^{2}}+\left[\frac{2N}{N-1}{\left(\frac{{\varphi }^{\prime }\left({u}_{1}\right){u}_{1}}{\varphi \left({u}_{1}\right)}\right)}^{2}-\frac{2\left(N-3\right)}{N-1}\frac{{\varphi }^{\prime }\left({u}_{1}\right){u}_{1}}{\varphi \left({u}_{1}\right)}-\frac{2{\varphi }^{\prime \prime }\left({u}_{1}\right){u}_{1}^{2}}{\varphi \left({u}_{1}\right)}+\frac{2}{N-1}\right]\frac{{u}_{1}^{2}}{{u}^{2}}$$+\frac{2}{N-1}{\left(\frac{f\left(u\right)}{{u}_{1}\varphi \left({u}_{1}\right)}\right)}^{2}+\left[\frac{2}{N-1}\left(1+\frac{{\varphi }^{\prime }\left({u}_{1}\right){u}_{1}}{\varphi \left({u}_{1}\right)}\right)+\frac{{\varphi }^{\prime }\left({u}_{1}\right){u}_{1}}{\varphi \left({u}_{1}\right)}\right]\left(\frac{{\mathrm{\Theta }}_{1}f\left(u\right)}{\mathrm{\Theta }{u}_{1}\varphi \left({u}_{1}\right)}\right)$$+\left[\frac{2N}{N-1}{\left(\frac{{\varphi }^{\prime }\left({u}_{1}\right){u}_{1}}{\varphi \left({u}_{1}\right)}\right)}^{2}-\frac{4\left(N-2\right)}{N-1}\frac{{\varphi }^{\prime }\left({u}_{1}\right){u}_{1}}{\varphi \left({u}_{1}\right)}-\frac{2{\varphi }^{\prime \prime }\left({u}_{1}\right){u}_{1}^{2}}{\varphi \left({u}_{1}\right)}-\frac{2\left(N-2\right)}{N-1}\right]\frac{{\mathrm{\Theta }}_{1}{u}_{1}}{\mathrm{\Theta }u}$$+2\left[\frac{N}{N-1}{\left(\frac{{\varphi }^{\prime }\left({u}_{1}\right){u}_{1}}{\varphi \left({u}_{1}\right)}\right)}^{2}+\frac{2}{N-1}\frac{{\varphi }^{\prime }\left({u}_{1}\right){u}_{1}}{\varphi \left({u}_{1}\right)}-\frac{{\varphi }^{\prime \prime }\left({u}_{1}\right){u}_{1}^{2}}{\varphi \left({u}_{1}\right)}+\frac{N}{N-1}\right]\frac{{\mathrm{\Theta }}_{1}^{2}}{4{\mathrm{\Theta }}^{2}}$$=A\left({u}_{1}\right)\frac{{|\nabla \mathrm{\Theta }|}^{2}}{2{\mathrm{\Theta }}^{2}}+B\left({u}_{1}\right)\frac{{u}_{1}^{2}}{{u}^{2}}+C{\left(\frac{f\left(u\right)}{{u}_{1}\varphi \left({u}_{1}\right)}\right)}^{2}+D\left({u}_{1}\right)\left(\frac{{\mathrm{\Theta }}_{1}f\left(u\right)}{\mathrm{\Theta }{u}_{1}\varphi \left({u}_{1}\right)}\right)+E\left({u}_{1}\right)\frac{{\mathrm{\Theta }}_{1}{u}_{1}}{\mathrm{\Theta }u}+F\left({u}_{1}\right)\frac{{\mathrm{\Theta }}_{1}^{2}}{4{\mathrm{\Theta }}^{2}}$$\ge A\left({u}_{1}\right)\frac{{|\nabla \mathrm{\Theta }|}^{2}}{2{\mathrm{\Theta }}^{2}}+B\left({u}_{1}\right)\frac{{u}_{1}^{2}}{{u}^{2}}+C{\left(\frac{f\left(u\right)}{{u}_{1}\varphi \left({u}_{1}\right)}\right)}^{2}+D\left({u}_{1}\right)\left(\frac{{\mathrm{\Theta }}_{1}f\left(u\right)}{\mathrm{\Theta }{u}_{1}\varphi \left({u}_{1}\right)}\right)+E\left({u}_{1}\right)\frac{{\mathrm{\Theta }}_{1}{u}_{1}}{\mathrm{\Theta }u}.$(4.20)

Note that we have used condition (ϕ5) and $F\ge B>0$ 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 ${u}_{1}$):

$\frac{1}{\mathrm{\Theta }}\left(\mathrm{\Delta }\mathrm{\Theta }+\frac{{\varphi }^{\prime }\left({u}_{1}\right){u}_{1}}{\varphi \left({u}_{1}\right)}{\mathrm{\Theta }}_{11}\right)\ge A\frac{{|\nabla \mathrm{\Theta }|}^{2}}{2{\mathrm{\Theta }}^{2}}+B\frac{{u}_{1}^{2}}{{u}^{2}}+C{\left(\frac{f\left(u\right)}{{u}_{1}\varphi \left({u}_{1}\right)}\right)}^{2}-\frac{{D}^{2}}{2C}\left(\frac{{\mathrm{\Theta }}_{1}^{2}}{2{\mathrm{\Theta }}^{2}}\right)-C{\left(\frac{f\left(u\right)}{{u}_{1}\varphi \left({u}_{1}\right)}\right)}^{2}-\frac{{E}^{2}}{B}\left(\frac{{\mathrm{\Theta }}_{1}^{2}}{2{\mathrm{\Theta }}^{2}}\right)-\frac{B}{2}\frac{{u}_{1}^{2}}{{u}^{2}}$$=A\frac{{|\nabla \mathrm{\Theta }|}^{2}}{2{\mathrm{\Theta }}^{2}}+\frac{B}{2}\frac{{u}_{1}^{2}}{{u}^{2}}-\frac{1}{2}\left(\frac{{D}^{2}}{C}+\frac{2{E}^{2}}{B}\right)\frac{{\mathrm{\Theta }}_{1}^{2}}{2{\mathrm{\Theta }}^{2}}.$(4.21)

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

$A\frac{{|\nabla \mathrm{\Theta }|}^{2}}{2{\mathrm{\Theta }}^{2}}+\frac{B}{2}\frac{{u}_{1}^{2}}{{u}^{2}}-\frac{1}{2}\left(\frac{{D}^{2}}{C}+\frac{2{E}^{2}}{B}\right)\frac{{\mathrm{\Theta }}_{1}^{2}}{2{\mathrm{\Theta }}^{2}}\le \frac{24{a}^{2}}{{\left({a}^{2}-{r}^{2}\right)}^{2}}+\frac{4N{a}^{2}}{{\left({a}^{2}-{r}^{2}\right)}^{2}}+|\frac{{\varphi }^{\prime }\left({u}_{1}\right){u}_{1}}{\varphi \left({u}_{1}\right)}|\left(\frac{24{a}^{2}}{{\left({a}^{2}-{r}^{2}\right)}^{2}}+\frac{4{a}^{2}}{{\left({a}^{2}-{r}^{2}\right)}^{2}}\right).$

In other words, we have

$0\le \frac{{u}_{1}^{2}}{{u}^{2}}\le \frac{2}{B}\left[\frac{4{a}^{2}\left(6+N\right)}{{\left({a}^{2}-{r}^{2}\right)}^{2}}+|\frac{{\varphi }^{\prime }\left({u}_{1}\right){u}_{1}}{\varphi \left({u}_{1}\right)}|\frac{28{a}^{2}}{{\left({a}^{2}-{r}^{2}\right)}^{2}}+|\frac{1}{2}\left(\frac{{D}^{2}}{C}+\frac{2{E}^{2}}{B}\right)-A|\frac{{|\nabla \mathrm{\Theta }|}^{2}}{2{\mathrm{\Theta }}^{2}}\right].$

From (4.2), we recall that at ${x}^{*}$, we have

Therefore, at ${x}^{*}$, we have the estimate

$0\le \frac{{|\nabla u|}^{2}}{{u}^{2}}\le \frac{{C}_{0}\left({u}_{1}\right){a}^{2}}{{\left({a}^{2}-{r}^{2}\right)}^{2}},$

where

${C}_{0}\left({u}_{1}\right):=\frac{8}{B}\left[6+N+7\left(\rho +1\right)+2|\frac{1}{2}\left(\frac{{D}^{2}\left({u}_{1}\right)}{C}+\frac{2{E}^{2}\left({u}_{1}\right)}{B\left({u}_{1}\right)}\right)-A\left({u}_{1}\right)|\right].$

Note that by conditions (ϕ3) and (ϕ5), ${C}_{0}\left({u}_{1}\right)$ is bounded by a positive $\mathcal{ℳ}$. Moreover, we observe that

$J\left({x}^{*}\right)=\frac{{|\nabla u|}^{2}}{{u}^{2}}{\left({a}^{2}-{r}^{2}\right)}^{2}\le \mathcal{ℳ}{a}^{2}.$

Since $J\left({x}_{0}\right)\le J\left({x}^{*}\right)$, we conclude that, at ${x}_{0}$, we have

$\frac{{|\nabla u|}^{2}}{{u}^{2}}{a}^{4}=J\left({x}_{0}\right)\le J\left({x}^{*}\right)\le \mathcal{ℳ}{a}^{2}.$

Thus, at ${x}_{0}$, we have the estimate

$\frac{{|\nabla u|}^{2}}{{u}^{2}}\le \frac{\mathcal{ℳ}}{{a}^{2}}.$

Letting $a\to \mathrm{\infty }$, we find that $|\nabla u|=0$ at ${x}_{0}$. Since ${x}_{0}$ was arbitrary, we conclude that $\nabla u\equiv 0$ on ${ℝ}^{N}$, 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 $\varphi \left(t\right)=p{t}^{p-2}$ for some $p>1$. In this case, $\sigma =\rho =p-1$, $\varpi \left(t\right)=\left(p-2\right)\left(p-3\right)$ and

$\frac{N}{N-1}{\sigma }^{3}-3\rho +2=\frac{{\left(p-1\right)}^{2}}{N-1}+\left(p-2\right)\left(p-3\right).$

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

Now let us consider $\varphi \left(t\right)=p{t}^{p-2}+q{t}^{q-2}$ for $1. Recall that in this case $\sigma =p-1$ and $\rho =q-1$. Let us note that2

$\varpi \left(t\right):=\frac{{\varphi }^{\prime \prime }\left(t\right){t}^{2}}{\varphi \left(t\right)}=\frac{p\left(p-2\right)\left(p-3\right){t}^{p-1}+q\left(q-2\right)\left(q-3\right){t}^{q-1}}{p{t}^{p-1}+q{t}^{q-1}}$$\le \mathrm{max}\left\{\left(p-2\right)\left(p-3\right),\left(q-2\right)\left(q-3\right)\right\}.$

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

$\frac{N}{N-1}{\sigma }^{2}-3\rho +2-\varpi \ge \frac{N}{N-1}{\left(p-1\right)}^{2}-3\left(q-1\right)+2-\mathrm{max}\left\{\left(p-2\right)\left(p-3\right),\left(q-2\right)\left(q-3\right)\right\}.$(4.22)

Let us observe that

$\left(q-2\right)\left(q-3\right)=\left(q-p+p-2\right)\left(q-p+p-3\right)$$={\left(q-p\right)}^{2}+\left(q-p\right)\left(2p-5\right)+\left(p-2\right)\left(p-3\right)$$=\left(q-p\right)\left(q+p-5\right)+\left(p-2\right)\left(p-3\right)$$\ge \left(q-p\right)\left(2p-5\right)+\left(p-2\right)\left(p-3\right).$

Therefore, we see that

So let us first suppose that $1. Then inequality (4.22) reduces to

$\frac{N}{N-1}{\sigma }^{2}-3\rho +2-\varpi \ge \frac{N}{N-1}{\left(p-1\right)}^{2}-3\left(q-1\right)+2-\left(p-2\right)\left(p-3\right).$

Thus, if

$p$=\frac{1}{3}\left[\frac{N}{N-1}{\left(p-1\right)}^{2}-{\left(p-1\right)}^{2}+3p\right]$$=\frac{1}{3}\left[\frac{{\left(p-1\right)}^{2}}{N-1}+3p\right],$

then condition (ϕ5) holds. Now suppose $p\ge 5/2$. Then inequality (4.22) becomes

$\frac{N}{N-1}{\sigma }^{2}-3\rho +2-\varpi \ge \frac{N}{N-1}{\left(p-1\right)}^{2}-3\left(q-1\right)+2-\left(q-2\right)\left(q-3\right)$$=\frac{N}{N-1}{\left(p-1\right)}^{2}-{q}^{2}+2q-1=\frac{N}{N-1}{\left(p-1\right)}^{2}-{\left(q-1\right)}^{2}.$

Therefore, if

$p

then condition (ϕ5) holds.

Now, given $p>1$, let us set

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

#### Corollary 4.4.

Given $p\mathrm{>}\mathrm{1}$, suppose that f satisfies condition (Cf) with $\sigma \mathrm{=}p\mathrm{-}\mathrm{1}$. If $\mathrm{1}\mathrm{<}p\mathrm{<}q\mathrm{<}{p}^{\mathrm{*}}$, then any non-negative entire solution of

${\mathrm{\Delta }}_{p}u+{\mathrm{\Delta }}_{q}u=-f\left(u\right)$

is a constant on ${\mathrm{R}}^{N}$.

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## Footnotes

• 1

Let $\omega :=\frac{{\varphi }^{\prime }\left(t\right)t}{\varphi \left(t\right)}$ and $\varpi :=\frac{{\varphi }^{\prime \prime }\left(t\right){t}^{2}}{\varphi \left(t\right)}$. Then we see that $B=\frac{2N}{N-1}{\omega }^{2}-\frac{2\left(N-3\right)}{N-1}\omega -2\varpi +\frac{2}{N-1}=2\left[\frac{N}{N-1}{\left(1+\omega \right)}^{2}-3\left(1+\omega \right)+2-\varpi \right]$ and $F=\frac{2N}{N-1}{\omega }^{2}+\frac{4}{N-1}\omega -2\varpi +\frac{2N}{N-1}=2\left[\frac{N}{N-1}{\left(1+\omega \right)}^{2}-2\left(1+\omega \right)+2-\varpi \right]$.

• 2

Let $a,b\in ℝ$. Since $\mathrm{min}\left\{a,b\right\}\le a,b\le \mathrm{max}\left\{a,b\right\}$, we have $\mathrm{min}\left\{a,b\right\}\le \theta a+\left(1-\theta \right)b\le \mathrm{max}\left\{a,b\right\}$ for all $0\le \theta \le 1$.

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,

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