On nonnegative radial entire solutions of second order quasilinear elliptic systems

Tomomitsu Teramoto

Department of Mathematics, Faculty of Science, Hiroshima University Higashi-Hiroshima 739-8526, Japan e-mail: teramoto@math.sci.hiroshima-u.ac.jp

Abstract In this article, we consider the second order quasilinear elliptic system of

the form

Δ_{p_{i}}u_{i} = H_{i}(|x|)u_{i+1}^{α_{i}}, x ∈ ℝ^{N}, i = 1, 2, ⋯, m

with nonnegative continuous functions H_{i}. Sufficient conditions are given to have nonnegative nontrivial radial entire solutions. When H_{i}, i = 1, 2, ⋯, m, behave like constant multiples of |x|^{λ}, λ ∈ ℝ, we can completely characterize the existence property of nonnegative nontrivial radial entire solutions.

Keywords and phrases: nonnegative entire solution, quasilinear elliptic system AMS Subject Classification: 35J60, 35B05

1 Introduction

This paper is concerned with existence and nonexistence of nonnegative radial entire solutions of second order quasilinear elliptic systems of the form

(1.1)  { Δ_{p_{1}}u_{1} = H_{1}(|x|)u_{2}^{α_{1}}, ; Δ_{p_{2}}u_{2} = H_{2}(|x|)u_{3}^{α_{2}}, ; ⋮ ; Δ_{p_{m}}u_{m} = H_{m}(|x|)u_{m+1}^{α_{m}}, u_{m+1} = u_{1}, x ∈ ℝ^{N},

where Δ_{p}u = div(|Du|^{p−2}Du), |x| denotes the Euclidean length of x ∈ ℝ^{N}, m ≥ 2, N ≥ 1, p_{i} > 1 and α_{i} > 0, i = 1, 2, ⋯, m, are constants satisfying α_{1}α_{2} ⋯ α_{m} > (p_{1} − 1)(p_{2} − 1) ⋯ (p_{m} − 1), and the functions H_{i}, i = 1, 2, ⋯, m, are nonnegative continuous functions on [0, ∞). When p = 2, Δ_{p} reduces to the usual Laplacian.

An entire solution of (1.1) is defined to be a function (u_{1}, u_{2}, ⋯, u_{m}) ∈ (C^{1}(ℝ^{N}))^{m} such that |Du_{i}|^{p_{i}−2}Du_{i} ∈ C^{1}(ℝ^{N}) and satisfy (1.1) at every point of ℝ^{N}. Such a solution is said to be radial if it depends only on |x|.

The problem of existence and nonexistence of nonnegative radial entire solutions for the scalar equation

Δ_{p}u = f(|x|, u), x ∈ ℝ^{N},

has been investigated by several authors, and numerous results have been obtained; see, e.g. [3, 6, 7, 10] and references therein. In particular, when f has the form f(|x|, u) = ±H(|x|)u^{α} with α > 0 and positive function H, critical decay rate of H to admit nonnegative radial entire solutions has been characterized. However, as far as the author knows, very little is known about this problem for the system (1.1) except for the case p_{i} = 2, i = 1, 2, ⋯, m. For p_{i} = 2, we refer to [2, 5, 11, 13, 14]. Recently, in [12], the author has considered the elliptic system (1.1) with m = 2 and has obtained existence and nonexistence criteria of nonnegative nontrivial radial entire solutions. The results in [12] are described roughly as follows :

Theorem 0.1 [12, Theorems 1 and 2] Let m = 2. Suppose that H_{i}, i = 1, 2, satisfy

(1.2)  \frac{C_{1}}{|x|^{λ_{i}}} ≤ H_{i}(|x|) ≤ \frac{C_{2}}{|x|^{λ_{i}}} , |x| ≥ r_{0} > 0, i = 1, 2,

where C_{i} > 0, i = 1, 2, are constants and λ_{i}, i = 1, 2, are parameters.

(i) If λ_{i}, i = 1, 2, satisfy

(1.3) \{ \begin{matrix}λ_{1} − p_{1} + \frac{α_{1}(λ_{2} − p_{2})}{p_{2} − 1} > \frac{α_{1}α_{2} − (p_{1} − 1)(p_{2} − 1)}{(p_{1} − 1)(p_{2} − 1)} max\{0, p_{1} − N\} and \\ λ_{2} − p_{2} + \frac{α_{2}(λ_{1} − p_{1})}{p_{1} − 1} > \frac{α_{1}α_{2} − (p_{1} − 1)(p_{2} − 1)}{(p_{1} − 1)(p_{2} − 1)} max\{0, p_{2} − N\}, \end{matrix}

then the system (1.1) has infinitely many positive radial entire solutions. (ii) If λ_{i}, i = 1, 2, satisfy

\{ \begin{matrix}λ_{1} − p_{1} + \frac{α_{1}(λ_{2} − p_{2})}{p_{2} − 1} ≤ \frac{α_{1}α_{2} − (p_{1} − 1)(p_{2} − 1)}{(p_{1} − 1)(p_{2} − 1)} max\{0, p_{1} − N\} or \\ λ_{2} − p_{2} + \frac{α_{2}(λ_{1} − p_{1})}{p_{1} − 1} ≤ \frac{α_{1}α_{2} − (p_{1} − 1)(p_{2} − 1)}{(p_{1} − 1)(p_{2} − 1)} max\{0, p_{2} − N\}, \end{matrix}

then the system (1.1) does not possess any nonnegative nontrivial radial entire solutions.

Theorem 0.2 [12, Theorems 3 and 4] Let m = 2 and p_{i} = N, i = 1, 2. Suppose that H_{i}, i = 1, 2, satisfy

\frac{C_{1}}{|x|^{N}(log |x|)^{λ_{i}}} ≤ H_{i}(|x|) ≤ \frac{C_{2}}{|x|^{N}(log |x|)^{λ_{i}}} , |x| ≥ r_{0} > 1, i = 1, 2,

where C_{i} > 0, i = 1, 2, are constants and λ_{i}, i = 1, 2, are parameters.

(i) If λ_{i}, i = 1, 2, satisfy

\{ \begin{matrix}λ_{1} − N + \frac{α_{1}(λ_{2} − N)}{N − 1} > \frac{α_{1}α_{2} − (N − 1)^{2}}{N − 1} and \\ λ_{2} − N + \frac{α_{2}(λ_{1} − N)}{N − 1} > \frac{α_{1}α_{2} − (N − 1)^{2}}{N − 1} , \end{matrix}

then the system (1.1) has infinitely many positive radial entire solutions. (ii) If λ_{i}, i = 1, 2, satisfy

\{ \begin{matrix}λ_{1} − N + \frac{α_{1}(λ_{2} − N)}{N − 1} < \frac{α_{1}α_{2} − (N − 1)^{2}}{N − 1} or \\ λ_{2} − N + \frac{α_{2}(λ_{1} − N)}{N − 1} < \frac{α_{1}α_{2} − (N − 1)^{2}}{N − 1} , \end{matrix}

then the system (1.1) has no nonnegative nontrivial radial entire solutions.

Theorem 0.1 characterizes the decay rates of H_{1} and H_{2} for the system (1.1) to admit nonnegative nontrivial radial entire solutions. That is, under the assumption (1.2) the system (1.1) has a nonnegative nontrivial radial entire solution if and only if (1.3) holds.

Considering some results in [11], we conjecture that the conclusion (ii) of Theorem 0.2 is still true even if the condition for (λ_{1}, λ_{2}) is weakened to

\{ \begin{matrix}λ_{1} − N + \frac{α_{1}(λ_{2} − N)}{N − 1} ≤ \frac{α_{1}α_{2} − (N − 1)^{2}}{N − 1} or \\ λ_{2} − N + \frac{α_{2}(λ_{1} − N)}{N − 1} ≤ \frac{α_{1}α_{2} − (N − 1)^{2}}{N − 1} . \end{matrix}

The aim of this paper is to extend Theorems 0.1 and 0.2 to the system (1.1) with m ≥ 3 and to answer the conjecture mentioned above affirmatively.

For nonnegative functions f_{i}, i = 1, 2, there have been a great number of works on qualitative theory for solutions of the elliptic system

\{ \begin{matrix}−Δ_{p_{1}}u_{1} = f_{1}(x, u_{1}, u_{2}), \\ −Δ_{p_{2}}u_{2} = f_{2}(x, u_{1}, u_{2}), \end{matrix} x ∈ ℝ^{N}.

We can find in many works necessary and/or sufficient conditions for this system to have positive entire solutions with (or without) prescribed asymptotic forms near +∞; see, e.g.[1, 8, 9] and references therein.

Let us introduce some notation used throughout this paper. Denote

A = α_{1}α_{2} ⋯ α_{m}

and

P = (p_{1} − 1)(p_{2} − 1) ⋯ (p_{m} − 1).

It follows from these definitions that our assumption is written as A > P. For any sequence {s_{1}, s_{2}, ⋯, s_{m}}, we always make the agreement that s_{m+j} = s_{j}, j = 1, 2, ⋯, m, that is, the suffixes should be taken in the sense ℤ/mℤ. For real constants λ_{1}, λ_{2}, ⋯, λ_{m}, we put

(1.4)  Λ_{i} = λ_{i} − p_{i} + \frac{(λ_{i+1} − p_{i+1})α_{i}}{p_{i+1} − 1} + \frac{(λ_{i+2} − p_{i+2})α_{i}α_{i+1}}{(p_{i+1} − 1)(p_{i+2} − 1)} + ⋯

+\frac{(λ_{i+m−1} − p_{i+m−1})α_{i}α_{i+1} ⋯ α_{i+m−3}α_{i+m−2}}{(p_{i+1} − 1)(p_{i+2} − 1) ⋯ (p_{i+m−2} − 1)(p_{i+m−1} − 1)}

= λ_{i} − p_{i} + ∑_{j=1}^{m−1} \{(λ_{i+j} − p_{i+j}) ∏_{k=0}^{j−1} \frac{α_{i+k}}{p_{i+1+k} − 1}\};

and

(1.5)  β_{i} = \frac{P Λ_{i}}{(A − P)(p_{i} − 1)},

i = 1, 2, ⋯, m. Since our assumptions imposed on H_{i}, i = 1, 2, ⋯, m, take the forms

lim inf_{|x|→∞} |x|^{λ_{i}}H_{i}(|x|) > 0

or

lim sup_{|x|→∞} |x|^{λ_{i}}H_{i}(|x|) < ∞,

all our results are formulated by means of the numbers λ_{i}, Λ_{i}, β_{i}, i = 1, 2, ⋯, m.

This paper is organized as follows. In Section 2, we consider the existence of positive radial entire solutions. In Section 3, we give estimates for nonnegative entire solutions of (1.1). In Section 4, we give nonexistence criteria of nonnegative nontrivial radial entire solutions of (1.1) based on the results in Section 3.

2 Existence results

In this section we consider the existence of positive radial entire solutions of (1.1).

We first observe that (u_{1}, u_{2}, ⋯, u_{m}) is a positive radial entire solution of (1.1) if and only if the function (v_{1}(r), v_{2}(r), ⋯, v_{m}(r)) = (u_{1}(|x|), u_{2}(|x|), ⋯, u_{m}(|x|)), r = |x|, satisfies the system of second order ordinary differential equations

(2.1)  \{ \begin{matrix}r^{1−N}(r^{N−1}|v′_{i}|^{p_{i}−2}v′_{i})′ = H_{i}(r)v_{i+1}^{α_{i}}, r > 0, \\ v′_{i}(0) = 0, \end{matrix} i = 1, 2, ⋯, m,

where ′ = d/dr. Furthermore, integrating (2.1) on [0, r] twice, we obtain the system of integral equations equivalent to (2.1) : (2.2)

v_{i}(r) = a_{i} + ∫_{0}^{r} (s^{1−N} ∫_{0}^{s} t^{N−1}H_{i}(t)v_{i+1}(t)^{α_{i}}dt)^{\frac{1}{p_{i}−1}} ds, r ≥ 0, i = 1, 2, ⋯, m,

where a_{i} = v_{i}(0). Therefore a positive radial entire solution of (1.1) can be obtained, under suitable conditions on H_{i}, by solving the system of integral equations (2.2).

Theorem 2.1 Suppose that H_{i}, i = 1, 2, ⋯, m, satisfy

(2.3)  H_{i}(|x|) ≤ \frac{C_{i}}{|x|^{λ_{i}}} , |x| ≥ r_{0} > 0,

where C_{i} > 0 and λ_{i}, i = 1, 2, ⋯, m, are constants. Moreover, for these λ_{i}, Λ_{i} defined by (1.4) satisfy

Λ_{i} > \frac{A − P}{P} max\{0, p_{i} − N\}, i = 1, 2, ⋯, m.

Then (1.1) has infinitely many positive radial entire solutions.

Theorem 2.2 Let p_{i} ≤ N, i = 1, 2, ⋯, m. Suppose that H_{i}, i = 1, 2, ⋯, m, satisfy

(2.4)  H_{i}(|x|) ≤ \frac{C_{i}}{|x|^{p_{i}}(log |x|)^{λ_{i}}} , |x| ≥ r_{0} > 1,

where C_{i} > 0 and λ_{i}, i = 1, 2, ⋯, m, are constants. Moreover

Λ_{i} > \frac{(A − P)(p_{i} − 1)}{P}, i = 1, 2, ⋯, m.

Then (1.1) has infinitely many positive radial entire solutions.

Remark 2.1 (i) When m = 2, Theorem 2.1 reduces to Theorem 1 of [12]. (ii) When p_{i} = 2, i = 1, 2, ⋯, m, and N ≠ 2, Theorem 2.1 reduces to Theorems 3.1 and 3.3 of [13]. (iii) When p_{i} = N = 2, i = 1, 2, ⋯, m, Theorem 2.2 reduces to Theorem 3.2 of [13].

Proof of Theorem 2.1. Without loss of generality, we may assume that r_{0} = 1 in (2.3). Choose constants a_{i} > 0, i = 1, 2, ⋯, m, so that

(2.5) \{ \begin{matrix}((2a_{i+1})^{α_{i}} ∫_{0}^{1} H_{i}(s)ds)^{\frac{1}{p_{i}−1}} ≤ \frac{a_{i}}{2} , \\ M_{i} (2(2a_{i+1})^{α_{i}} max \{∫_{0}^{1} s^{N−1}H_{i}(s)ds, \frac{C_{i}}{N − λ_{i} + α_{i}β_{i+1}} \})^{\frac{1}{p_{i}−1}} ≤ \frac{a_{i}}{2} , \end{matrix}

where

M_{i} = \{ \begin{matrix}\frac{p_{i} − 1}{p_{i} − λ_{i} + α_{i}β_{i+1}} , & p_{i} ≤ N, \\ \frac{p_{i} − 1}{p_{i} − N} , & p_{i} > N, \end{matrix}

and β_{i}, i = 1, 2, ⋯, m, are defined by (1.5). It is possible to choose such constants by the assumption A > P. From the definitions of β_{i} and Λ_{i} we can see that

p_{i} − λ_{i} + α_{i}β_{i+1}
= p_{i} − λ_{i} + \frac{P α_{i}}{(A − P)(p_{i+1} − 1)} \{λ_{i+1} − p_{i+1} + ∑_{j=1}^{m−1} \{(λ_{i+1+j} − p_{i+1+j}) ∏_{k=0}^{j−1} \frac{α_{i+1+k}}{p_{i+2+k} − 1}\}\}

= p_{i} − λ_{i} + \frac{P}{A − P} \{\frac{α_{i}(λ_{i+1} − p_{i+1})}{p_{i+1} − 1} + ∑_{j=1}^{m−1} \{(λ_{i+1+j} − p_{i+1+j}) ∏_{k=−1}^{j−1} \frac{α_{i+1+k}}{p_{i+2+k} − 1}\}\}

= p_{i} − λ_{i} + \frac{P}{A − P} ∑_{j=0}^{m−1} \{(λ_{i+1+j} − p_{i+1+j}) ∏_{k=−1}^{j−1} \frac{α_{i+1+k}}{p_{i+2+k} − 1}\}

= p_{i} − λ_{i} + \frac{P}{A − P} [∑_{j=0}^{m−2} \{(λ_{i+1+j} − p_{i+1+j}) ∏_{k=−1}^{j−1} \frac{α_{i+1+k}}{p_{i+2+k} − 1}\} + \frac{A}{P}(λ_{i} − p_{i})]

= \frac{P (λ_{i} − p_{i})}{A − P} + \frac{P}{A − P} ∑_{j=1}^{m−1} \{(λ_{i+j} − p_{i+j}) ∏_{k=0}^{j−1} \frac{α_{i+k}}{p_{i+1+k} − 1}\}
= \frac{P Λ_{i}}{A − P} > max\{0, p_{i} − N\}.

Define the functions F_{i}, i = 1, 2, ⋯, m, by

F_{i}(r) = \{ \begin{matrix}1, & 0 ≤ r ≤ 1, \\ r^{β_{i}}, & r ≥ 1. \end{matrix}

We regard the space (C[0, ∞))^{m} as Fréchet space equipped with the topology of uniform convergence of functions on each compact subinterval of [0, ∞). Let X ⊂ (C[0, ∞))^{m} denotes the subset defined by

X = {(v_{1}, v_{2}, ⋯, v_{m}) ∈ (C[0, ∞))^{m}; a_{i} ≤ v_{i}(r) ≤ 2a_{i}F_{i}(r), r ≥ 0, 1 ≤ i ≤ m}.

Clearly, X is a non-empty closed convex subset of (C[0, ∞))^{m}. Consider the mapping F : X → (C[0, ∞))^{m} defined by F(v_{1}, v_{2}, ⋯, v_{m}) = (ṽ_{1}, ṽ_{2}, ⋯, ṽ_{m}), where

ṽ_{i}(r) = a_{i} + ∫_{0}^{r} (s^{1−N} ∫_{0}^{s} t^{N−1}H_{i}(t)v_{i+1}(t)^{α_{i}}dt)^{\frac{1}{p_{i}−1}} ds, r ≥ 0, i = 1, 2, ⋯, m.

In order to apply the Schauder-Tychonoff fixed point theorem, we will show that F is a continuous mapping from X into itself such that F(X) is relatively compact.

(I) F maps X into itself. Let (v_{1}, v_{2}, ⋯, v_{m}) ∈ X. Clearly, ṽ_{i}(r) ≥ a_{i}, r ≥ 0. For 0 ≤ r ≤ 1, we have

\begin{matrix}ṽ_{i}(r) ≤ & a_{i} + ∫_{0}^{r} (∫_{0}^{s} H_{i}(t)v_{i+1}(t)^{α_{i}}dt)^{\frac{1}{p_{i}−1}} ds \\ ≤ & a_{i} + ∫_{0}^{1} (∫_{0}^{1} H_{i}(t)(2a_{i+1}F_{i+1}(t))^{α_{i}}dt)^{\frac{1}{p_{i}−1}} ds \\ = & a_{i} + ((2a_{i+1})^{α_{i}} ∫_{0}^{1} H_{i}(t)dt)^{\frac{1}{p_{i}−1}} \\ ≤ & a_{i} + \frac{a_{i}}{2} < 2a_{i}, i = 1, 2, ⋯, m. \end{matrix}

For r ≥ 1, we then write

\begin{matrix}ṽ_{i}(r) = & a_{i} + (∫_{0}^{1} + ∫_{1}^{r}) (s^{1−N} ∫_{0}^{s} t^{N−1}H_{i}(t)v_{i+1}(t)^{α_{i}}dt)^{\frac{1}{p_{i}−1}} ds \\ ≡ & a_{i} + I_{1} + I_{2}. \end{matrix}

A similar computation shows that I_{1} ≤ a_{i}/2, i = 1, 2, ⋯, m. When p_{i} ≤ N, we see that

\begin{matrix}I_{2} ≤ & ∫_{1}^{r} (s^{1−N} ∫_{0}^{s} t^{N−1}H_{i}(t)(2a_{i+1}F_{i+1}(t))^{α_{i}}dt)^{\frac{1}{p_{i}−1}} ds \\ ≤ & ∫_{1}^{r} s^{\frac{1−N}{p_{i}−1}} ((2a_{i+1})^{α_{i}} ∫_{0}^{1} t^{N−1}H_{i}(t)dt + (2a_{i+1})^{α_{i}}C_{i} ∫_{1}^{s} t^{N−1−λ_{i}+a_{i}β_{i+1}}dt)^{\frac{1}{p_{i}−1}} ds \\ ≤ & (2(2a_{i+1})^{α_{i}} max \{∫_{0}^{1} t^{N−1}H_{i}(t)dt, \frac{C_{i}}{N − λ_{i} + α_{i}β_{i+1}} \})^{\frac{1}{p_{i}−1}} ∫_{1}^{r} s^{\frac{1−λ_{i}+α_{i}β_{i+1}}{p_{i}−1}} ds \\ ≤ & M_{i} (2(2a_{i+1})^{α_{i}} max \{∫_{0}^{1} t^{N−1}H_{i}(t)dt, \frac{C_{i}}{N − λ_{i} + α_{i}β_{i+1}} \})^{\frac{1}{p_{i}−1}} r^{\frac{p_{i}−λ_{i}+α_{i}β_{i+1}}{p_{i}−1}} \\ ≤ & \frac{a_{i}}{2} r^{\frac{p_{i}−λ_{i}+α_{i}β_{i+1}}{p_{i}−1}} = \frac{a_{i}}{2} r^{β_{i}}. \end{matrix}

When p_{i} > N, we see that

\begin{matrix}I_{2} ≤ & ∫_{1}^{r} s^{\frac{1−N}{p_{i}−1}} ds (∫_{0}^{r} t^{N−1}H_{i}(t)(2a_{i+1}F_{i+1}(t))^{α_{i}}dt)^{\frac{1}{p_{i}−1}} \\ ≤ & M_{i}r^{\frac{p_{i}−N}{p_{i}−1}} ((2a_{i+1})^{α_{i}} ∫_{0}^{1} t^{N−1}H_{i}(t)dt + (2a_{i+1})^{α_{i}}C_{i} ∫_{1}^{r} t^{N−1−λ_{i}+α_{i}β_{i+1}}dt)^{\frac{1}{p_{i}−1}} \\ ≤ & M_{i}r^{\frac{p_{i}−N}{p_{i}−1}} (2(2a_{i+1})^{α_{i}} max \{∫_{0}^{1} t^{N−1}H_{i}(t)dt, \frac{C_{i}}{N − λ_{i} + α_{i}β_{i+1}} \} r^{N−λ_{i}+α_{i}β_{i+1}})^{\frac{1}{p_{i}−1}} \\ ≤ & \frac{a_{i}}{2} r^{\frac{p_{i}−λ_{i}+α_{i}β_{i+1}}{p_{i}−1}} = \frac{a_{i}}{2} r^{β_{i}}. \end{matrix}

Thus we obtain

ṽ_{i}(r) ≤ \frac{3}{2}a_{i} + \frac{a_{i}}{2} r^{β_{i}} ≤ 2a_{i}r^{β_{i}}, r ≥ 1, i = 1, 2, ⋯, m.

Therefore, F(X) ⊂ X.

(II) F is continuous. Let {(v_{1,l}, v_{2,l}, ⋯, v_{m,l})}_{l=1}^{∞} be a sequence in X which converges to (v_{1}, v_{2}, ⋯, v_{m}) ∈ X uniformly on each compact subinterval of [0, ∞). We put

ϕ_{i,l}(r) = r^{1−N} ∫_{0}^{r} s^{N−1}H_{i}(s)v_{i+1,l}(s)^{α_{i}}ds

and

ϕ_{i}(r) = r^{1−N} ∫_{0}^{r} s^{N−1}H_{i}(s)v_{i+1}(s)^{α_{i}}ds.

Then we have

|ϕ_{i,l}(r) − ϕ_{i}(r)| ≤ ∫_{0}^{r} H_{i}(s)|v_{i+1,l}(s)^{α_{i}} − v_{i+1}(s)^{α_{i}}|ds.

Let R > 0 be an arbitrary constant. Since {v_{i,l}}_{l=1}^{∞}, i = 1, 2, ⋯, m, converge to v_{i} uniformly on [0, R], it follows that {ϕ_{i,l}}_{l=1}^{∞}, i = 1, 2, ⋯, m, converge to ϕ_{i} uniformly on [0, R]; and hence \{ϕi,l^{\frac{1}{p_{i}−1}} \}_{l=1}^{∞}, i = 1, 2, ⋯, m, converge to ϕi^{\frac{1}{p_{i}−1}} uniformly on [0, R]. From this fact and

|ṽ_{i,l}(r) − ṽ_{i}(r)| ≤ ∫_{0}^{r} |ϕ_{i,l}(s)^{\frac{1}{p_{i}−1}} − ϕ_{i}(s)^{\frac{1}{p_{i}−1}} | ds,

we can see that {ṽ_{i,l}}_{l=1}^{∞}, i = 1, 2, ⋯, m, converge to ṽ_{i} uniformly on [0, R]. These imply that {ṽ_{i,l}}_{l=1}^{∞}, i = 1, 2, ⋯, m, converge to ṽ_{i} uniformly on each compact subinterval of [0, ∞). Therefore F is continuous.

(III) F(X) is relatively compact. It is sufficient to verify the local equicontinuity of F(X), since F(X) is locally uniformly bounded by the fact that F(X) ⊂ X. Let (v_{1}, v_{2}, ⋯, v_{m}) ∈ X and R > 0. Then we have

\begin{matrix}ṽ′_{i}(r) = & (∫_{0}^{r} (\frac{s}{r})^{N−1} H_{i}(s)v_{i+1}(s)^{α_{i}}ds)^{\frac{1}{p_{i}−1}} \\ ≤ & (∫_{0}^{R} H_{i}(s)(2a_{i+1}F_{i+1}(s))^{α_{i}}ds)^{\frac{1}{p_{i}−1}} < ∞, i = 1, 2, ⋯, m. \end{matrix}

Obviously, these imply the local boundedness of the set {(ṽ′_{1}, ṽ′_{2}, ⋯, ṽ′_{m})|(v_{1}, v_{2}, ⋯, v_{m}) ∈ X}. Hence the relative compactness of F(X) is shown by the Ascoli-Arzelà theorem.

Therefore, there exists an element (v_{1}, v_{2}, ⋯, v_{m}) ∈ X such that (v_{1}, v_{2}, ⋯, v_{m}) = F(v_{1}, v_{2}, ⋯, v_{m}) by the Schauder-Tychonoff fixed point theorem, that is, (v_{1}, v_{2}, ⋯, v_{m}) satisfies the system of integral equations (2.2). The function (u_{1}(x), u_{2}(x), ⋯, u_{m}(x)) = (v_{1}(|x|), v_{2}(|x|), ⋯, v_{m}(|x|)) then gives a solution of (1.1). Since infinitely many (a_{1}, a_{2}, ⋯, a_{m}) satisfy (2.5), we can construct an infinitude of positive radial entire solutions of (1.1). This completes the proof.□

Proof of Theorem 2.2. Without loss of generality, we may assume that r_{0} = e in (2.4). Take constants a_{i} > 0, i = 1, 2, ⋯, m, so that

\{ \begin{matrix}e ((2a_{i+1})^{α_{i}} ∫_{0}^{e} H_{i}(t)dt)^{\frac{1}{p_{i}−1}} ≤ \frac{a_{i}}{2} , \\ (2(2a_{i+1})^{α_{i}} max \{∫_{0}^{e} t^{p_{i}−1}H_{i}(t)dt, \frac{C_{i}}{1 − λ_{i} + α_{i}β_{i+1}} \})^{\frac{1}{p_{i}−1}} ≤ \frac{a_{i}}{2} . \end{matrix}

It is possible to take such constants by the assumption A > P.

Define the functions F_{i}, i = 1, 2, ⋯, m, by

F_{i}(r) = \{ \begin{matrix}1, & 0 ≤ r ≤ e, \\ (log r)^{β_{i}}, & r ≥ e. \end{matrix}

Consider the set

Y = {(v_{1}, v_{2}, ⋯, v_{m}) ∈ (C[0, ∞))^{m}; a_{i} ≤ v_{i}(r) ≤ 2a_{i}F_{i}(r), r ≥ 0, 1 ≤ i ≤ m}

and the mapping F : Y → (C[0, ∞))^{m} defined by F(v_{1}, v_{2}, ⋯, v_{m}) = (ṽ_{1}, ṽ_{2}, ⋯, ṽ_{m}), where

ṽ_{i}(r) = a_{i} + ∫_{0}^{r} (s^{1−N} ∫_{0}^{s} t^{N−1}H_{i}(t)v_{i+1}(t)^{α_{i}}dt)^{\frac{1}{p_{i}−1}} ds.

Obviously, the set Y is closed convex subset of Fréchet space (C[0, ∞))^{m}. We first show that F(Y) ⊂ Y. Let (v_{1}, v_{2}, ⋯, v_{m}) ∈ Y. Clearly, ṽ_{i}(r) ≥ a_{i}, r ≥ 0. For 0 ≤ r ≤ e we have

\begin{matrix}ṽ_{i}(r) ≤ & a_{i} + ∫_{0}^{r} (∫_{0}^{s} H_{i}(t)v_{i+1}(t)^{α_{i}}dt)^{\frac{1}{p_{i}−1}} ds \\ ≤ & a_{i} + ∫_{0}^{e} (∫_{0}^{e} H_{i}(t)(2a_{i+1}F_{i+1}(t))^{α_{i}}dt)^{\frac{1}{p_{i}−1}} ds \\ = & a_{i} + e ((2a_{i+1})^{α_{i}} ∫_{0}^{e} H_{i}(t)dt)^{\frac{1}{p_{i}−1}} \\ ≤ & a_{i} + \frac{a_{i}}{2} < 2a_{i}, i = 1, 2, ⋯, m. \end{matrix}

For r ≥ e, we then write

\begin{matrix}ṽ_{i}(r) = & a_{i} + (∫_{0}^{e} + ∫_{e}^{r}) (s^{1−N} ∫_{0}^{s} t^{N−1}H_{i}(t)v_{i+1}(t)^{α_{i}}dt)^{\frac{1}{p_{i}−l}} ds \\ ≡ & a_{i} + I_{1} + I_{2}. \end{matrix}

A similar computation shows that I_{1} ≤ a_{i}/2, i = 1, 2, ⋯, m. The integral I_{2} is estimated as follows:

I_{2} ≤ ∫_{e}^{r} (s^{1−p_{i}} ∫_{0}^{s} t^{p_{i}−1}H_{i}(t)v_{i+1}(t)^{α_{i}}dt)^{\frac{1}{p_{i}−1}} ds
≤ ∫_{e}^{r} s^{−1}ds (∫_{0}^{r} t^{p_{i}−1}H_{i}(t)v_{i+1}(t)^{α_{i}}dt)^{\frac{1}{p_{i}−1}}
≤ ((2a_{i+1})^{α_{i}} ∫_{0}^{e} t^{p_{i}−1}H_{i}(t)dt + (2a_{i+1})^{α_{i}}C_{i} ∫_{e}^{r} t^{−1}(log t)^{−λ_{i}+α_{i}β_{i+1}}dt)^{\frac{1}{p_{i}−1}} log r
≤ (2(2a_{i+1})^{α_{i}} max \{∫_{0}^{e} t^{p_{i}−1}H_{i}(t)dt, \frac{C_{i}}{1 − λ_{i} + α_{i}β_{i+1}} \} (log r)^{1−λ_{i}+α_{i}β_{i+1}})^{\frac{1}{p_{i}−1}} log r
≤ \frac{a_{i}}{2} (log r)^{\frac{p_{i}−λ_{i}+α_{i}β_{i+1}}{p_{i}−1}} = \frac{a_{i}}{2} (log r)^{β_{i}}.

Thus we obtain

ṽ_{i}(r) ≤ \frac{3}{2}a_{i} + \frac{a_{i}}{2} (log r)^{β_{i}} ≤ 2a_{i}(log r)^{β_{i}}, r ≥ e, i = 1, 2, ⋯, m.

Therefore, F(v_{1}, v_{2}, ⋯, v_{m}) ∈ Y.

The continuity of F and the relative compactness of F(Y) can be verified without difficulty, and so by the Schauder-Tychonoff fixed point theorem there exists (v_{1}, v_{2} ⋯, v_{m}) ∈ Y such that (v_{1}, v_{2}, ⋯, v_{m}) = F(v_{1}, v_{2}, ⋯, v_{m}). It is clear that this fixed point (v_{1}, v_{2}, ⋯, v_{m}) gives rise to a positive radial entire solution of (1.1). The proof is finished.□

3 Growth estimates for nonnegative entire solutions

In this section we consider estimates for nonnegative radial entire solutions of (1.1) which will play an important role to prove nonexistence theorems for nonnegative nontrivial radial entire solutions.

Theorem 3.1 Suppose that H_{i}, i = 1, 2, ⋯, m, satisfy

(3.1)  H_{i}(|x|) ≥ \frac{C_{i}}{|x|^{λ_{i}}} , |x| ≥ r_{0} > 0,

where C_{i} > 0 and λ_{i} are constants. Let (u_{1}, u_{2}, ⋯, u_{m}) be a nonnegative radial entire solution of (1.1). Then u_{i}, i = 1, 2, ⋯, m, satisfy

(3.2)  u_{i}(r) ≤ C̃_{i}r^{β_{i}} at ∞,

where C̃_{i} > 0, i = 1, 2, ⋯, m, are constants and β_{i}, i = 1, 2, ⋯, m, are defined by (1.5).

Theorem 3.2 Let p_{i} = N, i = 1, 2, ⋯, m. Suppose that H_{i}, i = 1, 2, ⋯, m, satisfy

(3.3)  H_{i}(|x|) ≥ \frac{C_{i}}{|x|^{N}(log |x|)^{λ_{i}}} , |x| ≥ r_{0} > 1,

where C_{i} > 0 and λ_{i} are constants. Let (u_{1}, u_{2}, ⋯, u_{m}) be a nonnegative radial entire solution of (1.1). Then u_{i}, i = 1, 2, ⋯, m, satisfy

(3.4)  u_{i}(r) ≤ C̃_{i}(log r)^{β_{i}} at ∞,

where C̃_{i} > 0, i = 1, 2, ⋯, m, are constants, and β_{i}, i = 1, 2, ⋯, m, are defined by (1.5).

Proof of Theorem 3.1. Let (u_{1}, u_{2}, ⋯, u_{m}) be a nonnegative radial entire solution of (1.1). We may assume that (u_{1}, u_{2}, ⋯, u_{m}) ≢ (0, 0, ⋯, 0). Then (u_{1}, u_{2}, ⋯, u_{m}) satisfies the following system of ordinary differential equations (3.5)

\{ \begin{matrix}(r^{N−1}|u′_{i}(r)|^{p_{i}−2}u′_{i}(r))′ = r^{N−1}H_{i}(r)u_{i+1}(r)^{α_{i}}, r > 0, \\ u′_{i}(0) = 0, \end{matrix} i = 1, 2, ⋯, m.

Integrating (3.5) over [0, r], we have

r^{N−1}|u′_{i}(r)|^{p_{i}−2}u′_{i}(r) = ∫_{0}^{r} s^{N−1}H_{i}(s)u_{i+1}(s)^{α_{i}}ds, i = 1, 2, ⋯, m.

Hence, we see that u′_{i}(r) ≥ 0 for r ≥ 0. Integrating (3.5) twice over [R, r], R ≥ 0, we have (3.6)

u_{i}(r) ≥ u_{i}(R) + ∫_{R}^{r} (s^{1−N} ∫_{R}^{s} t^{N−1}H_{i}(t)u_{i+1}(t)^{α_{i}}dt)^{\frac{1}{p_{i}−1}} ds, i = 1, 2, ⋯, m.

Since u_{i}, i = 1, 2, ⋯, m, are nonnegative and nontrivial, there exists a point x_{∗} ∈ ℝ^{N} such that u_{i_{0}}(r_{∗}) > 0, r_{∗} = |x_{∗}| for some i_{0} ∈ {1, 2, ⋯, m}. We may assume that r_{∗} ≥ r_{0}. Therefore we see from (3.6) with R = r_{∗} that u_{i}(r) > 0 for r > r_{∗}, i = 1, 2, ⋯, m.

Let us fix R > r_{∗} arbitrarily. Using (3.1) and the inequality

(\frac{t}{s})^{N−1} ≥ (\frac{1}{3})^{N−1} , R ≤ t ≤ s ≤ 3R

in (3.6), we have

\begin{aligned}u_{i}(r) & ≥ & u_{i}(R) + ∫_{R}^{r} (∫_{R}^{s} (\frac{1}{3})^{N−1} C_{i}t^{−λ_{i}}u_{i+1}(t)^{α_{i}}dt)^{\frac{1}{p_{i}−1}} ds \\ & ≥ & C̃_{i}R^{− \frac{λ_{i}}{p_{i}−1}} ∫_{R}^{r} (∫_{R}^{s} u_{i+1}(t)^{α_{i}}dt)^{\frac{1}{p_{i}−1}} ds, R ≤ r ≤ 3R, \end{aligned}

where C̃_{i} > 0, i = 1, 2, ⋯, m, are some constants independent of r and R. From now on, we use C to denote various positive constants independent of r and R as we will have no confusion. Put

(3.7)  f_{i}(r) = C̃_{i}R^{− \frac{λ_{i}}{p_{i}−1}} ∫_{R}^{r} (∫_{R}^{s} u_{i+1}(t)^{α_{i}}dt)^{\frac{1}{p_{i}−1}} ds, R ≤ r ≤ 3R.

Clearly, f_{i}, i = 1, 2, ⋯, m, satisfy

u_{i}(r) ≥ f_{i}(r), R ≤ r ≤ 3R,
f_{i}(R) = f′_{i}(R) = 0,
f′_{i}(r) = C̃_{i}R^{− \frac{λ_{i}}{p_{i}−1}} (∫_{R}^{r} u_{i+1}(s)^{α_{i}}ds)^{\frac{1}{p_{i}−1}} ≥ 0, R ≤ r ≤ 3R,
f′′_{i}(r) > 0, R < r ≤ 3R,

and

(3.8)  (f′_{i}(r)^{p_{i}−1})′ = CR^{−λ_{i}}u_{i+1}(r)^{α_{i}}
≥ CR^{−λ_{i}}f_{i+1}(r)^{α_{i}}, R ≤ r ≤ 3R.

From (3.7) and the monotonicity of u_{i}, we see that

(3.9)  f_{i}(r) ≥ CR^{− \frac{λ_{i}}{p_{i}−1}} u_{i+1}(R)^{\frac{α_{i}}{p_{i}−1}} (r − R)^{\frac{p_{i}}{p_{i}−1}} , R ≤ r ≤ 3R.

Let us fix i ∈ {1, 2, ⋯, m}. Multiplying (3.8) by f′_{i+1}(r) ≥ 0 and integrating by parts the resulting inequality on [R + ε, r], ε > 0, we have

f′_{i+1}(r)f′_{i}(r)^{p_{i}−1} ≥ CR^{−λ_{i}}(f_{i+1}(r)^{α_{i}+1} − f_{i+1}(R + ε)^{α_{i}+1}), R + ε ≤ r ≤ 3R.

Letting ε → 0, we get

f′_{i}(r)f′_{i+1}(r)^{\frac{1}{p_{i}−1}} ≥ CR^{− \frac{λ_{i}}{p_{i}−1}} f_{i+1}(r)^{\frac{α_{i}+1}{p_{i}−1}} , R ≤ r ≤ 3R.

Multiplying this inequality by f′_{i+1} and integrating by parts on [R +ε, r] and letting ε → 0, we obtain

f_{i}(r)f′_{i+1}(r)^{\frac{p_{i}}{p_{i}−1}} ≥ CR^{− \frac{λ_{i}}{p_{i}−1}} f_{i+1}(r)^{\frac{α_{i}+p_{i}}{p_{i}−1}} , R ≤ r ≤ 3R.

From (3.8), we have

(f′_{i−1}(r)^{p_{i−1}−1})′f′_{i+1}(r)^{\frac{p_{i}α_{i−1}}{p_{i}−1}} ≥ CR^{− \frac{λ_{i}α_{i−1}}{p_{i}−1} −λ_{i−1}}f_{i+1}(r)^{\frac{(α_{i}+p_{i})α_{i−1}}{p_{i}−1}} , R ≤ r ≤ 3R.

Again, multiplying this relation by f′_{i+1} and integrating by parts on [R + ε, r] and letting ε → 0 twice, we get

\begin{aligned}& f_{i−1}(r)f′_{i+1}(r)^{\frac{p_{i}α_{i−1}}{(p_{i}−1)(p_{i−1}−1)} + \frac{p_{i−1}}{p_{i−1}−1}} \\ ≥ & CR^{− \frac{λ_{i}α_{i−1}}{(p_{i}−1)(p_{i−1}−1)} − \frac{λ_{i−1}}{p_{i−1}−1}} f_{i+1}(r)^{\frac{(α_{i}+p_{i})α_{i−1}}{(p_{i}−1)(p_{i−1}−1)} + \frac{p_{i−1}}{p_{i−1}−1}} , R ≤ r ≤ 3R. \end{aligned}

From (3.8), we obtain

\begin{aligned}& (f′_{i−2}(r)^{p_{i−2}−1})′f′_{i+1}(r)^{\frac{p_{i}α_{i−1}α_{i−2}}{(p_{i}−1)(p_{i−1}−1)} + \frac{p_{i−1}α_{i−2}}{p_{i−1}−1}} \\ ≥ & CR^{− \frac{λ_{i}α_{i−1}α_{i−2}}{(p_{i}−1)(p_{i−1}−1)} − \frac{λ_{i−1}α_{i−2}}{p_{i−1}−1} −λ_{i−2}}f_{i+1}(r)^{\frac{(α_{i}+p_{i})α_{i−1}α_{i−2}}{(p_{i}−1)(p_{i−1}−1)} + \frac{p_{i−1}α_{i−2}}{p_{i−1}−1}} , R ≤ r ≤ 3R. \end{aligned}

By repeating this procedure we get

(f′_{i−(m−1)}(r)^{p_{i−(m−1)}−1})′f′_{i+1}(r)^{K_{i}} (3.10)

= (f′_{i+1}(r)^{p_{i+1}−1})′f′_{i+1}(r)^{K_{i}} ≥ CR^{−L_{i}}f_{i+1}(r)^{M_{i}}, R ≤ r ≤ 3R,

where

K_{i} = ∑_{j=1}^{m−1} \{p_{i−(j−1)} ∏_{k=j}^{m−1} \frac{α_{i−k}}{p_{i+1−k} − 1}\} ,

L_{i} = ∑_{j=1}^{m−1} \{λ_{i−(j−1)} ∏_{k=j}^{m−1} \frac{α_{i−k}}{p_{i+1−k} − 1}\} + λ_{i+1},

and

\begin{matrix}M_{i} = & \frac{A}{∏_{j=0}^{m−2}(p_{i−j} − 1)} + ∑_{j=1}^{m−1} \{p_{i−(j−1)} ∏_{k=j}^{m−1} \frac{α_{i−k}}{p_{i+1−k} − 1}\} \\ = & \frac{A(p_{i+1} − 1)}{P} + K_{i}. \end{matrix}

Multiplying (3.10) by f′_{i+1}(r) ≥ 0 and integrating by parts on [R + ε, r] and letting ε → 0, we have

(3.11)  f′_{i+1}(r)f_{i+1}(r)^{− \frac{M_{i}+1}{K_{i}+p_{i+1}}} ≥ CR^{− \frac{L_{i}}{K_{i}+p_{i+1}}} , R < r ≤ 3R.

Since (M_{i} + 1)/(K_{i} + p_{i+1}) > 1, we can set

\frac{M_{i} + 1}{K_{i} + p_{i+1}} = δ_{i} + 1, δ_{i} = \frac{(A − P)(p_{i+1} − 1)}{(K_{i} + p_{i+1})P}.

Integrating (3.11) on [2R, 3R], we get

f_{i+1}(2R)^{−δ_{i}} ≥ CR^{− \frac{L_{i}}{K_{i}+p_{i+1}} +1}.

From (3.9) with r = 2R and this inequality, we have

u_{i+2}(R) ≤ CR^{τ_{i}},

where

τ_{i} = \frac{p_{i+1} − 1}{α_{i+1}δ_{i}} \{\frac{L_{i}}{K_{i} + p_{i+1}} − 1 + \frac{(λ_{i+1} − p_{i+1})δ_{i}}{p_{i+1} − 1} \} .

From the definitions of K_{i}, L_{i} and δ_{i}, we see that

\begin{matrix}τ_{i} = & \frac{p_{i+1} − 1}{α_{i+1}δ_{i}(K_{i} + p_{i+1})} [L_{i} − K_{i} − p_{i+1} + \frac{(A − P)(λ_{i+1} − p_{i+1})}{P}] \\ = & \frac{P}{α_{i+1}(A − P)} [∑_{j=1}^{m−1} \{(λ_{i−j+1} − p_{i−j+1}) ∏_{k=j}^{m−1} \frac{α_{i−k}}{p_{i+1−k} − 1}\} + \frac{A(λ_{i+1} − p_{i+1})}{P}] \\ = & \frac{P}{α_{i+1}(A − P)} [∑_{j=0}^{m−2} \{(λ_{i−j+1} − p_{i−j+1}) ∏_{k=j}^{m−1} \frac{α_{i−k}}{p_{i+1−k} − 1}\} + \frac{(λ_{i+2} − p_{i+2})α_{i+1}}{p_{i+2} − 1} ] \end{matrix}

= \frac{P}{(A − P)(p_{i+2} − 1)} [∑_{j=0}^{m−2} \{(λ_{i−j+1} − p_{i−j+1}) ∏_{k=j}^{m−2} \frac{α_{i−k}}{p_{i+1−k} − 1}\} + λ_{i+2} − p_{i+2}]
= \frac{P}{(A − P)(p_{i+2} − 1)} [\frac{(λ_{i+1} − p_{i+1})α_{i}α_{i−1} ⋯ α_{i−m+2}}{(p_{i+1} − 1)(p_{i} − 1) ⋯ (p_{i−m+3} − 1)}

+\frac{(λ_{i} − p_{i})α_{i−1}α_{i−2} ⋯ α_{i−m+2}}{(p_{i} − 1)(p_{i−1} − 1) ⋯ (p_{i−m+3} − 1)} + ⋯ + \frac{(λ_{i−m+4} − p_{i−m+4})α_{i−m+3}α_{i−m+2}}{(p_{i−m+4} − 1)(p_{i−m+3} − 1)}

+\frac{(λ_{i−m+3} − p_{i−m+3})α_{i−m+2}}{p_{i−m+3} − 1} + λ_{i+2} − p_{i+2}]

= \frac{P}{(A − P)(p_{i+2} − 1)} [λ_{i+2} − p_{i+2} + ∑_{j=1}^{m−1} \{(λ_{i+2+j} − p_{i+2+j}) ∏_{k=0}^{j−1} \frac{α_{i+2+k}}{p_{i+3+k} − 1}\}]
= \frac{P Λ_{i+2}}{(A − P)(p_{i+2} − 1)}.

Therefore we obtain (3.2) by the definition of β_{i}. Thus the proof is completed.□

The next lemma is needed in proving Theorem 3.2.

Lemma 3.3 Let p_{i} = N, i = 1, 2, ⋯, m, and (u_{1}, u_{2}, ⋯, u_{m}) be a nonnegative radial entire solution of (1.1). Then u_{i}, i = 1, 2, ⋯, m, satisfy

u_{i}(r) ≥ u_{i}(0) + (∫_{0}^{r} s^{N−1}H_{i}(s) (log (\frac{r}{s}))^{N−1} u_{i+1}(s)^{α_{i}}ds)^{\frac{1}{N−1}} , r ≥ 0.

Proof. Let (u_{1}, u_{2}, ⋯, u_{m}) be a nonnegative radial entire solution of (1.1). Then u_{i}, i = 1, 2, ⋯, m, satisfy the following system of ordinary differential equations

\{ \begin{matrix}(r^{N−1}|u′_{i}(r)|^{N−2}u′_{i}(r))′ = r^{N−1}H_{i}(r)u_{i+1}(r)^{α_{i}}, r > 0, \\ u′_{i}(0) = 0, \end{matrix} i = 1, 2, ⋯, m.

Integrating these equations on [0, r] twice, we have

\begin{matrix}u_{i}(r) = & u_{i}(0) + ∫_{0}^{r} (∫_{0}^{s} (\frac{t}{s})^{N−1} H_{i}(t)u_{i+1}(t)^{α_{i}}dt)^{\frac{1}{N−1}} ds \\ = & u_{i}(0) + ∫_{0}^{r} (∫_{0}^{r} Φ_{i}(s, t)dt)^{\frac{1}{N−1}} ds, r ≥ 0, \end{matrix}

where

Φ_{i}(s, t) = \{ \begin{matrix}s^{1−N}t^{N−1}H_{i}(t)u_{i+1}(t)^{α_{i}} & for 0 ≤ t ≤ s, \\ 0 & for t > s. \end{matrix}

Using Minkowski’s inequality (cf. [4, p.148]), we see that

∫_{0}^{r} (∫_{0}^{r} Φ_{i}(s, t)dt)^{\frac{1}{N−1}} ds ≥ (∫_{0}^{r} (∫_{0}^{r} Φ_{i}(s, t)^{\frac{1}{N−1}} ds)^{N−1} dt)^{\frac{1}{N−1}} , r ≥ 0.

Then we have

\begin{matrix}u_{i}(r) ≥ & u_{i}(0) + (∫_{0}^{r} (∫_{0}^{r} Φ_{i}(s, t)^{\frac{1}{N−1}} ds)^{N−1} dt)^{\frac{1}{N−1}} \\ = & u_{i}(0) + (∫_{0}^{r} (∫_{t}^{r} s^{−1}tH_{i}(t)^{\frac{1}{N−1}} u_{i+1}(t)^{\frac{α_{i}}{N−1}} ds)^{N−1} dt)^{\frac{1}{N−1}} \\ = & u_{i}(0) + (∫_{0}^{r} t^{N−1}H_{i}(t) (log \frac{r}{t} )^{N−1} u_{i+1}(t)^{α_{i}}dt)^{\frac{1}{N−1}} . \end{matrix}

Thus the proof is finished.□

Proof of Theorem 3.2. Let (u_{1}, u_{2} ⋯, u_{m}) be a nonnegative radial entire solution of (1.1). We may assume that (u_{1}, u_{2} ⋯, u_{m}) ≢ (0, 0, ⋯, 0). As in the proof of Theorem 3.1 we see that u_{i}(r) > 0, r ≥ r_{∗}, i = 1, 2, ⋯, m, for some r_{∗} > r_{0}.

Let us fix R ≥ r_{∗} arbitrarily. From Lemma 3.3, we see that u_{i}, i = 1, 2, ⋯, m, satisfy

(3.12) u_{i}(r) ≥ u_{i}(0) + (∫_{0}^{r} s^{N−1}H_{i}(s) (log \frac{r}{s})^{N−1} u_{i+1}(s)^{α_{i}}ds)^{\frac{1}{N−1}}
≥ (∫_{e^{R}}^{r} s^{N−1}H_{i}(s)(log r − log s)^{N−1}u_{i+1}(s)^{α_{i}}ds)^{\frac{1}{N−1}} , r ≥ e^{R}.

Let log s = t, log r = ρ. Then (3.12) becomes

u_{i}(e^{ρ}) ≥ (∫_{R}^{ρ} e^{Nt}H_{i}(e^{t})(ρ − t)^{N−1}u_{i+1}(e^{t})^{α_{i}}dt)^{\frac{1}{N−1}} , ρ ≥ R, i = 1, 2, ⋯, m.

Now we discuss only on the interval [R, 3R] for a moment. Let R ≤ ρ ≤ 3R. Then, from (3.3), we have

\begin{matrix}u_{i}(e^{ρ}) ≥ & (C_{i} ∫_{R}^{ρ} t^{−λ_{i}}(ρ − t)^{N−1}u_{i+1}(e^{t})^{α_{i}}dt)^{\frac{1}{N−1}} \\ ≥ & (C̃_{i}R^{−λ_{i}} ∫_{R}^{ρ}(ρ − t)^{N−1}u_{i+1}(e^{t})^{α_{i}}dt)^{\frac{1}{N−1}} , R ≤ ρ ≤ 3R, \end{matrix}

where C̃_{i} > 0 are some constants independent of r and R. From now on we use the same letter C to denote various positive constants.

Define the functions f_{i}, i = 1, 2, ⋯, m, by

(3.13)  f_{i}(ρ) = C̃_{i}R^{−λ_{i}} ∫_{R}^{ρ}(ρ − t)^{N−1}u_{i+1}(e^{t})^{α_{i}}dt, R ≤ ρ ≤ 3R.

Then we see that f_{i}, i = 1, 2, ⋯, m, are of class C^{N}[R, 3R] and satisfy

u_{i}(e^{ρ}) ≥ f_{i}(ρ)^{\frac{1}{N−1}} , R ≤ ρ ≤ 3R,
f_{i}^{(k)}(r) ≥ 0, R ≤ ρ ≤ 3R, f_{i}^{(k)}(R) = 0, k = 0, 1, 2, ⋯, N − 1,

and

(3.14)  f_{i}^{(N)}(ρ) = CR^{−λ_{i}}u_{i+1}(e^{ρ})^{α_{i}}
≥ CR^{−λ_{i}}f_{i+1}(ρ)^{\frac{α_{i}}{N−1}} , R ≤ ρ ≤ 3R.

From (3.13) and the monotonicity of u_{i} we have

(3.15)  f_{i}(ρ) ≥ CR^{−λ_{i}}u_{i+1}(e^{R})^{α_{i}} ∫_{R}^{ρ}(ρ − t)^{N−1}dt
≥ CR^{−λ_{i}}(ρ − R)^{N}u_{i+1}(e^{R})^{α_{i}}, R ≤ ρ ≤ 3R.

Let us fix i ∈ {1, 2, ⋯, m}. Multiplying (3.14) by f′_{i+1} and integrating by parts the resulting inequality on [R, ρ], we have

f_{i}^{(N−1)}(ρ)f′_{i+1}(ρ) ≥ CR^{−λ_{i}}f_{i+1}(ρ)^{\frac{α_{i}}{N−1} +1}, R ≤ ρ ≤ 3R.

By repeating this process (N − 1) times, we get

f_{i}(ρ)f′_{i+1}(ρ)^{N} ≥ CR^{−λ_{i}}f_{i+1}(ρ)^{\frac{α_{i}}{N−1} +N}, R ≤ ρ ≤ 3R.

From (3.14) we have

f_{i−1}^{(N)}(ρ)f′_{i+1}(ρ)^{\frac{Nα_{i−1}}{N−1}} ≥ CR^{− \frac{λ_{i}α_{i−1}}{N−1} −λ_{i−1}}f_{i+1}(ρ)^{\frac{α_{i}α_{i−1}}{(N−1)^{2}} + \frac{Nα_{i−1}}{N−1}} , R ≤ ρ ≤ 3R. Multiplying this inequality by f′_{i+1} and integrating by parts N times on [R, ρ], we have f_{i−1}(ρ)f′_{i+1}(ρ)^{\frac{Nα_{i−1}}{N−1} +N} ≥ CR^{− \frac{λ_{i}α_{i−1}}{N−1} −λ_{i−1}}f_{i+1}(ρ)^{\frac{α_{i}α_{i−1}}{(N−1)^{2}} + \frac{Nα_{i−1}}{N−1} +N}, R ≤ ρ ≤ 3R. From (3.14) we have

\begin{aligned}& f_{i−2}^{(N)}(ρ)f′_{i+1}(ρ)^{\frac{Nα_{i−1}α_{i−2}}{(N−1)^{2}} + \frac{Nα_{i−2}}{N−1}} \\ ≥ & CR^{− \frac{λ_{i}α_{i−1}α_{i−2}}{(N−1)^{2}} − \frac{λ_{i−1}α_{i−2}}{N−1} −λ_{i−2}}f_{i+1}(ρ)^{\frac{α_{i}α_{i−1}α_{i−2}}{(N−1)^{3}} + \frac{Nα_{i−1}α_{i−2}}{(N−1)^{2}} + \frac{Nα_{i−2}}{N−1}} , R ≤ ρ ≤ 3R. \end{aligned}

By repeating this procedure we get f_{i−(m−1)}^{(N)}(ρ)f′_{i+1}(ρ)^{K_{i}} (3.16)

= f_{i+1}^{(N)}(ρ)f′_{i+1}(ρ)^{K_{i}} ≥ CR^{−L_{i}}f_{i+1}(ρ)^{M_{i}}, R ≤ ρ ≤ 3R,

where

\begin{matrix}K_{i} = ∑_{j=1}^{m−1} \{\frac{N}{(N − 1)^{m−j}} ∏_{k=j}^{m−1} α_{i−k}\} , \\ L_{i} = ∑_{j=1}^{m−1} \{\frac{λ_{i−(j−1)}}{(N − 1)^{m−j}} ∏_{k=j}^{m−1} α_{i−k}\} + λ_{i+1}, \end{matrix}

and

\begin{matrix}M_{i} = & \frac{∏_{j=0}^{m−1} α_{i−j}}{(N − 1)^{m}} + ∑_{j=1}^{m−1} \{\frac{N}{(N − 1)^{m−j}} ∏_{k=j}^{m−1} α_{i−k}\} \\ = & \frac{A}{(N − 1)^{m}} + K_{i}. \end{matrix}

Multiplying (3.16) by f′_{i+1} and integrating by parts (N − 1) times on [R, ρ], we get

f′_{i+1}(ρ)f_{i+1}(ρ)^{− \frac{A−(N−1)^{m}}{(K_{i}+N)(N−1)^{m}} −1} ≥ CR^{− \frac{L_{i}}{K_{i}+N}} , R < ρ ≤ 3R.

Integrating this inequality on [2R, 3R], we have

f_{i+1}(2R)^{− \frac{A−(N−1)^{m}}{(K_{i}+N)(N−1)^{m}}} ≥ CR^{− \frac{L_{i}}{K_{i}+N} +1}.

From (3.15), we get

u_{i+2}(e^{R})^{α_{i+1}} ≤ R^{^{\frac{(N−1)^{m}}{A−(N−1)^{m}}} \{L_{i}−K_{i}+ \frac{A(λ_{i+1}−N)}{(N−1)^{m}} −λ_{i+1}\}}.

From the definitions of K_{i} and L_{i}, we see that

L_{i} − K_{i} + \frac{A(λ_{i+1} − N)}{(N − 1)^{m}} − λ_{i+1}

= ∑_{j=1}^{m−1} \{\frac{λ_{i−j+1}}{(N − 1)^{m−j}} ∏_{k=j}^{m−1} α_{i−k}\} − ∑_{j=1}^{m−1} \{\frac{N}{(N − 1)^{m−j}} ∏_{k=j}^{m−1} α_{i−k}\} + \frac{A(λ_{i+1} − N)}{(N − 1)^{m}}
= ∑_{j=1}^{m−2} \{\frac{(λ_{i+1−j} − N)}{(N − 1)^{m−j}} ∏_{k=j}^{m−1} α_{i−k}\} + \frac{α_{i+1}(λ_{i+2} − N)}{N − 1} + \frac{A(λ_{i+1} − N)}{(N − 1)^{m}}
= \frac{α_{i+1}}{N − 1} \{λ_{i+2} − N + ∑_{j=0}^{m−2} \{\frac{(λ_{i+1−j} − N)}{(N − 1)^{m−j−1}} ∏_{k=j}^{m−2} α_{i−k}\}\}
= \frac{α_{i+1}}{N − 1} \{λ_{i+2} − N + \frac{(λ_{i+1} − N)α_{i}α_{i−1} ⋯ α_{i−(m−2)}}{(N − 1)^{m−1}} + \frac{(λ_{i} − N)α_{i−1}α_{i−2} ⋯ α_{i−(m−2)}}{(N − 1)^{m−2}}

+ ⋯ + \frac{(λ_{i−m+4} − N)α_{i−m+3}α_{i−m+2}}{(N − 1)^{2}} + \frac{(λ_{i−m+3} − N)α_{i−m+2}}{N − 1} \}

= \frac{α_{i+1}}{N − 1} \{λ_{i+2} − N + ∑_{j=1}^{m−1} \{\frac{(λ_{i+2+j} − N)}{(N − 1)^{j}} ∏_{k=0}^{j−1} α_{i+2+k}\}\}
= \frac{α_{i+1}Λ_{i+2}}{N − 1} .

Therefore we see that

u_{i+2}(e^{R})^{α_{i}+1} ≤ CR^{\frac{α_{i+1}(N−1)^{m−1}Λ_{i+2}}{A−(N−1)^{m}}} .

Thus we obtain

u_{i+2}(e^{ρ}) ≤ Cρ^{\frac{(N−1)^{m−1}Λ_{i+2}}{A−(N−1)^{m}}} at ∞, i = 1, 2, ⋯, m.

Hence we obtain (3.4) since ρ = log r. The proof is completed.□ EJQTDE, 2002 No. 16, p. 20

4 Nonexistence results

In this section we study the nonexistence of nonnegative nontrivial radial entire solutions of (1.1).

Theorem 4.1 Suppose that H_{i}, i = 1, 2, ⋯, m, satisfy

(4.1)  H_{i}(|x|) ≥ \frac{C_{i}}{|x|^{λ_{i}}} , |x| ≥ r_{0} > 0,

where C_{i} > 0 and λ_{i}, i = 1, 2, ⋯, m, are constants. Moreover

Λ_{i} ≤ \frac{A − P}{P} max\{0, p_{i} − N\} for some i ∈ \{1, 2, ⋯, m\}.

If (u_{1}, u_{2}, ⋯, u_{m}) is a nonnegative radial entire solution of (1.1), then

(u_{1}, u_{2}, ⋯, u_{m}) ≡ (0, 0, ⋯, 0).

Remark 4.1 (i) When m = 2, Theorem 4.1 reduces to Theorem 2 of [12]. However, the proof presented here is simpler than that of Theorem 2 of [12]. (ii) When p_{i} = 2, i = 1, 2, ⋯, m, and N ≠ 2, Theorem 4.1 reduces to Theorems 2.3 and 2.5 of [13].

Theorem 4.2 Let p_{i} = N, i = 1, 2, ⋯, m. Suppose that H_{i}, i = 1, 2, ⋯, m, satisfy

(4.2)  H_{i}(|x|) ≥ \frac{C_{i}}{|x|^{N}(log |x|)^{λ_{i}}} , |x| ≥ r_{0} > 1,

where C_{i} > 0 and λ_{i}, i = 1, 2, ⋯, m, are constants. Moreover

Λ_{i} ≤ \frac{A − (N − 1)^{m}}{(N − 1)^{m−1}} for some i ∈ \{1, 2, ⋯, m\}.

If (u_{1}, u_{2}, ⋯, u_{m}) is a nonnegative radial entire solution of (1.1), then

(u_{1}, u_{2}, ⋯, u_{m}) ≡ (0, 0, ⋯, 0).

Remark 4.2 (i) Theorem 4.2 shows that the conjecture stated in the introduction is true. (ii) When p_{i} = 2, i = 1, 2, ⋯, m, Theorem 4.2 reduces to Theorem 2.4 of [13].

We give an example to show the sharpness of our results.

Example. Let us consider the elliptic system

(4.3)  \{ Δ_{p_{1}}u_{1} = \frac{1}{(1 + |x|)^{λ_{1}}} u_{2}^{α_{1}}, ; Δ_{p_{2}}u_{2} = \frac{1}{(1 + |x|)^{λ_{2}}} u_{3}^{α_{2}}, ; ⋮ ; Δ_{p_{m}}u_{m} = \frac{1}{(1 + |x|)^{λ_{m}}} u_{1}^{α_{m}}, x ∈ ℝ^{N},

where N ≥ 1, p_{i} > 1, α_{i} > 0, i = 1, 2, ⋯, m, are constants satisfying A > P. Since

\frac{C_{i}}{|x|^{λ_{i}}} ≤ \frac{1}{(1 + |x|)^{λ_{i}}} ≤ \frac{C̃_{i}}{|x|^{λ_{i}}} , |x| ≥ 1, i = 1, 2, ⋯, m

hold for some positive constants C_{i} and C̃_{i}, i = 1, 2, ⋯, m, we can see from Theorems 2.1 and 4.1 that a necessary and sufficient condition for (4.3) to have a positive radial entire solution is

Λ_{i} > \frac{A − P}{P} max\{0, p_{i} − N\}, i = 1, 2, ⋯, m.

Proof of Theorem 4.1. Let (u_{1}, u_{2}, ⋯, u_{m}) be a nonnegative nontrivial radial entire solution of (1.1). From Theorem 3.1 and its proof, we see that u_{i}(r) > 0, r ≥ r_{∗}, i = 1, 2, ⋯, m, for some r_{∗} > r_{0} and u_{i}, i = 1, 2, ⋯, m, satisfy

(4.4)  u_{i}(r) ≤ C_{i}r^{β_{i}} at ∞, i = 1, 2, ⋯, m

for some constants C_{i} > 0, i = 1, 2, ⋯, m.

If there exists an i_{0} ∈ {1, 2, ⋯, m} such that

Λ_{i_{0}} < \frac{A − P}{P} max\{0, p_{i_{0}} − N\},

then we can see from the definition of β_{i_{0}} that

\{ \begin{matrix}β_{i_{0}} < 0 & if p_{i_{0}} ≤ N, \\ β_{i_{0}} < \frac{p_{i_{0}} − N}{p_{i_{0}} − 1} & if p_{i_{0}} > N. \end{matrix}

If p_{i_{0}} ≤ N, then it is found that lim_{r→∞} u_{i_{0}}(r) = 0. On the other hand, since u_{i_{0}} is nondecreasing and u_{i_{0}}(r_{∗}) > 0, we have

u_{i_{0}}(r) ≥ u_{i_{0}}(r_{∗}) > 0, r ≥ r_{∗}.

This is a contradiction. If p_{i_{0}} > N, then integrating (3.5) on [0, r] twice we have

\begin{matrix}u_{i_{0}}(r) = & u_{i_{0}}(0) + ∫_{0}^{r} s^{\frac{1−N}{p_{i_{0}} −1}} (∫_{0}^{s} t^{N−1}H_{i_{0}}(t)u_{i_{0}+1}(t)^{α_{i_{0}}} dt)^{\frac{1}{p_{i_{0}} −1}} ds \\ ≥ & ∫_{r_{∗}}^{r} s^{\frac{1−N}{p_{i_{0}} −1}} ds (∫_{0}^{r_{∗}} t^{N−1}H_{i_{0}}(t)u_{i_{0}+1}(t)^{α_{i_{0}}} dt)^{\frac{1}{p_{i_{0}} −1}} \\ = & (∫_{0}^{r_{∗}} t^{N−1}H_{i_{0}}(t)u_{i_{0}+1}(t)^{α_{i_{0}}} dt)^{\frac{1}{p_{i_{0}} −1}} \frac{p_{i_{0}} − 1}{p_{i_{0}} − N} \{r^{\frac{p_{i_{0}} −N}{p_{i_{0}} −1}} − r∗^{\frac{p_{i_{0}} −N}{p_{i_{0}} −1}} \} \\ ≥ & Cr^{\frac{p_{i_{0}} −N}{p_{i_{0}} −1}} , r ≥ r̃_{∗} > r_{∗} \end{matrix}

for some constant C > 0. This contradicts to (4.4) with β_{i_{0}} < (p_{i_{0}} − N)/(p_{i_{0}} − 1). It remains to discuss the case that

Λ_{i} ≥ \frac{A − P}{P} max\{0, p_{i} − N\}, i = 1, 2, ⋯, m.

From the assumption of Λ_{i}, there exists an i_{0} ∈ {1, 2, ⋯, m} such that

Λ_{i_{0}} = \frac{A − P}{P} max\{0, p_{i_{0}} − N\}.

Without loss of generality, we may assume that i_{0} = m, that is,

Λ_{i} ≥ \frac{A − P}{P} max\{0, p_{i} − N\}, i = 1, 2, ⋯, m − 1

and

Λ_{m} = \frac{A − P}{P} max\{0, p_{m} − N\}.

We first observe that

(4.5) λ_{i} ≤ ∑_{j=1}^{m−i−1} \{(p_{i+j} − λ_{i+j}) ∏_{k=0}^{j−1} \frac{α_{i+k}}{p_{i+1+k} − 1}\} + min\{p_{i}, N\}

+ max\{0, p_{m} − N\} ∏_{k=0}^{m−i−1} \frac{α_{i+k}}{p_{i+1+k} − 1}, i = 1, 2, ⋯, m − 2,

and

(4.6)  λ_{m−1} ≤ \frac{α_{m−1} max\{0, p_{m} − N\}}{p_{m} − 1} + min\{p_{m−1}, N\}.

In fact, from the definition of Λ_{i}, we obtain

λ_{i} ≥ − ∑_{j=1}^{m−1} \{(λ_{i+j} − p_{i+j}) ∏_{k=0}^{j−1} \frac{α_{i+k}}{p_{i+1+k} − 1}\} + p_{i} + \frac{A − P}{P} max\{0, p_{i} − N\}
= − ( ∑_{j=1}^{m−i−1} + ∑_{j=m−i+1}^{m−1}) \{(λ_{i+j} − p_{i+j}) ∏_{k=0}^{j−1} \frac{α_{i+k}}{p_{i+1+k} − 1}\}

−(λ_{m} − p_{m}) ∏_{k=0}^{m−i−1} \frac{α_{i+k}}{p_{i+1+k} − 1} + p_{i} + \frac{A − P}{P} max\{0, p_{i} − N\}

≡ −S_{1} − S_{2} − S_{3} + p_{i} + \frac{A − P}{P} max\{0, p_{i} − N\}.

From the assumption of Λ_{m} we have

λ_{m} − p_{m} = − ∑_{j=1}^{m−1} \{(λ_{m+j} − p_{m+j}) ∏_{k=0}^{j−1} \frac{α_{m+k}}{p_{m+1+k} − 1}\} + \frac{A − P}{P} max\{0, p_{m} − N\}.

Substituting this relation to S_{3} we have

S_{3} = − ∑_{j=1}^{m−1} \{(λ_{m+j} − p_{m+j}) ∏_{k=0}^{j−1} \frac{α_{m+k}}{p_{m+1+k} − 1}\} ∏_{k=0}^{m−i−1} \frac{α_{i+k}}{p_{i+1+k} − 1}

+\frac{A − P}{P} max\{0, p_{m} − N\} ∏_{k=0}^{m−i−1} \frac{α_{i+k}}{p_{i+1+k} − 1}

= − ∑_{j=1}^{m−1} \{(λ_{m+j} − p_{m+j}) ∏_{k=0}^{m−i+j−1} \frac{α_{i+k}}{p_{i+1+k} − 1}\}

+\frac{A − P}{P} max\{0, p_{m} − N\} ∏_{k=0}^{m−i−1} \frac{α_{i+k}}{p_{i+1+k} − 1}

= − ∑_{j=m−i+1}^{2m−i−1} \{(λ_{i+j} − p_{i+j}) ∏_{k=0}^{j−1} \frac{α_{i+k}}{p_{i+1+k} − 1}\} + \frac{A − P}{P} max\{0, p_{m} − N\} ∏_{k=0}^{m−i−1} \frac{α_{i+k}}{p_{i+1+k} − 1}

EJQTDE, 2002 No. 16, p. 24

= − ∑_{j=m−i+1}^{m−1} \{(λ_{i+j} − p_{i+j}) ∏_{k=0}^{j−1} \frac{α_{i+k}}{p_{i+1+k} − 1}\} − ∑_{j=m}^{2m−i−1} \{(λ_{i+j} − p_{i+j}) ∏_{k=0}^{j−1} \frac{α_{i+k}}{p_{i+1+k} − 1}\}

+\frac{A − P}{P} max\{0, p_{m} − N\} ∏_{k=0}^{m−i−1} \frac{α_{i+k}}{p_{i+1+k} − 1}

= −S_{2} − ∑_{j=0}^{m−i−1} \{(λ_{i+j} − p_{i+j}) ∏_{k=0}^{j+m−1} \frac{α_{i+k}}{p_{i+1+k} − 1}\}

+\frac{A − P}{P} max\{0, p_{m} − N\} ∏_{k=0}^{m−i−1} \frac{α_{i+k}}{p_{i+1+k} − 1}

= −S_{2} − \frac{A}{P}S_{1} − \frac{A}{P}(λ_{i} − p_{i}) + \frac{A − P}{P} max\{0, p_{m} − N\} ∏_{k=0}^{m−i−1} \frac{α_{i+k}}{p_{i+1+k} − 1}.

Thus, we obtain

\begin{aligned}λ_{i} ≥ & (\frac{A}{P} − 1) S_{1} + \frac{A}{P}(λ_{i} − p_{i}) + p_{i} + \frac{A − P}{P} max\{0, p_{i} − N\} \\ & −\frac{A − P}{P} max\{0, p_{m} − N\} ∏_{k=0}^{m−i−1} \frac{α_{i+k}}{p_{i+1+k} − 1}, \end{aligned}

namely,

0 ≥ S_{1} + λ_{i} − p_{i} + max\{0, p_{i} − N\} − max\{0, p_{m} − N\} ∏_{k=0}^{m−i−1} \frac{α_{i+k}}{p_{i+1+k} − 1}
= ∑_{j=1}^{m−i−1} \{(λ_{i+j} − p_{i+j}) ∏_{k=0}^{j−1} \frac{α_{i+k}}{p_{i+1+k} − 1}\} + λ_{i} − min\{p_{i}, N\}

− max\{0, p_{m} − N\} ∏_{k=0}^{m−i−1} \frac{α_{i+k}}{p_{i+1+k} − 1}.

Therefore we obtain (4.5). Similarly we obtain (4.6). From the above computation we see that if

Λ_{i} > \frac{A − P}{P} max\{0, p_{i} − N\},

then ”<” holds in (4.5) and (4.6), and if

Λ_{i} = \frac{A − P}{P} max\{0, p_{i} − N\},

then ”=” holds in (4.5) and (4.6).

From now on, the letter C denotes various positive constants independent of r and R. Integrating (3.5) twice over [r_{∗}, r], from (4.1), we have

(4.7)u_{i}(r) ≥ u_{i}(r_{∗}) + ∫_{r_{∗}}^{r} (s^{1−N} ∫_{r_{∗}}^{s} t^{N−1}H_{i}(t)u_{i+1}(t)^{α_{i}}dt)^{\frac{1}{p_{i}−1}} ds

≥ C ∫_{r_{∗}}^{r} (s^{1−N} ∫_{r_{∗}}^{s} t^{N−1−λ_{i}}u_{i+1}(t)^{α_{i}}dt)^{\frac{1}{p_{i}−1}} ds, i = 1, 2, ⋯, m.

In what follows of the proof the argument is divided into two cases according to p_{m}. (i) Let p_{m} ≤ N. We first consider the case that

Λ_{m−1} = \frac{A − P}{P} max\{0, p_{m−1} − N\}.

Then from (4.6) we see that λ_{m−1} = min{p_{m−1}, N}. From (4.7) with i = m − 1 we have

u_{m−1}(r) ≥ Cu_{m}(r_{∗})^{\frac{α_{m−1}}{p_{m−1}−1}} ∫_{r_{∗}+1}^{r} (s^{1−N} ∫_{r_{∗}}^{s} t^{N−1−min\{p_{m−1},N\}}dt)^{\frac{1}{p_{m−1}−1}} ds.

Therefore we see that, for p_{m−1} < N

\begin{matrix}u_{m−1}(r) ≥ & C ∫_{r_{∗}+1}^{r} (s^{1−N} ∫_{r_{∗}}^{s} t^{N−1−p_{m−1}}dt)^{\frac{1}{p_{m−1}−1}} ds \\ ≥ & C ∫_{r_{∗}+1}^{r} s^{−1}ds \\ ≥ & C log r, r ≥ r_{1} > r_{∗} + 1, \end{matrix}

for p_{m−1} = N

\begin{matrix}u_{m−1}(r) ≥ & C ∫_{r_{∗}+1}^{r} (s^{1−N} ∫_{r_{∗}}^{s} t^{−1}dt)^{\frac{1}{p_{m−1}−1}} ds \\ ≥ & C ∫_{r_{∗}+1}^{r} s^{−1}(log s)^{\frac{1}{p_{m−1}−1}} ds \\ ≥ & C(log r)^{\frac{p_{m−1}}{p_{m−1}−1}} , r ≥ r_{1} > r_{∗} + 1, \end{matrix}

and for p_{m−1} > N

\begin{matrix}u_{m−1}(r) ≥ & C ∫_{r_{∗}+1}^{r} (s^{1−N} ∫_{r_{∗}}^{s} t^{−1}dt)^{\frac{1}{p_{m−1}−1}} ds \\ ≥ & C ∫_{r_{∗}+1}^{r} s^{\frac{1−N}{p_{m−1}−1}} (log s)^{\frac{1}{p_{m−1}−1}} ds \\ ≥ & Cr^{\frac{p_{m−1}−N}{p_{m−1}−1}} (log r)^{\frac{1}{p_{m−1}−1}} , r ≥ r_{1} > r_{∗} + 1. \end{matrix}

Here, the last inequality is given by integration by parts. On the other hand, from (4.4) with i = m − 1 and the definition of β_{m−1} we see that

u_{m−1}(r) ≤ \{ \begin{matrix}C & if p_{m−1} ≤ N, \\ Cr^{\frac{p_{m−1}−N}{p_{m−1}−1}} & if p_{m−1} > N \end{matrix}

for large r > r_{∗}. This is a contradiction.

Next we consider the case that

Λ_{m−2} = \frac{A − P}{P} max\{0, p_{m−2} − N\}.

Then we see from (4.5) with i = m − 2 and (4.6) that

λ_{m−1} < min\{p_{m−1}, N\} and λ_{m−2} = \frac{(p_{m−1} − λ_{m−1})α_{m−2}}{p_{m−1} − 1} + min\{p_{m−2}, N\}.

From (4.7) with i = m − 1 we have

\begin{matrix}u_{m−1}(r) ≥ & Cu_{m}(r_{∗})^{\frac{α_{m−1}}{p_{m−1}−1}} ∫_{r_{∗}+1}^{r} (s^{1−N} ∫_{r_{∗}}^{s} t^{N−1−λ_{m−1}}dt)^{\frac{1}{p_{m−1}−1}} ds \\ ≥ & C ∫_{r_{∗}+1}^{r} s^{\frac{1−λ_{m−1}}{p_{m−1}−1}} ds \\ ≥ & Cr^{\frac{p_{m−1}−λ_{m−1}}{p_{m−1}−1}} , r ≥ r_{1} > r_{∗} + 1. \end{matrix}

From this estimate and (4.7) with i = m − 2 we obtain

\begin{matrix}u_{m−2}(r) ≥ & C ∫_{r_{1}+1}^{r} (s^{1−N} ∫_{r_{1}}^{s} t^{N−1−λ_{m−2}+ \frac{α_{m−2}(p_{m−1}−λ_{m−1})}{p_{m−1}−1}} dt)^{\frac{1}{p_{m−2}−1}} ds \\ = & C ∫_{r_{1}+1}^{r} (s^{1−N} ∫_{r_{1}}^{s} t^{N−1−min\{p_{m−2},N\}}dt)^{\frac{1}{p_{m−2}−1}} ds. \end{matrix}

Therefore we see that for r ≥ r_{2} > r_{1} + 1

u_{m−2}(r) ≥ \{ \begin{matrix}C log r & if p_{m−2} < N, \\ C(log r)^{\frac{p_{m−2}}{p_{m−2}−1}} & if p_{m−2} = N, \\ Cr^{\frac{p_{m−2}−N}{p_{m−2}−1}} (log r)^{\frac{1}{p_{m−2}−1}} & if p_{m−2} > N. \end{matrix}

On the other hand, from (4.4) with i = m − 2 and the definition of β_{m−2} we see that

u_{m−2}(r) ≤ \{ \begin{matrix}C & if p_{m−2} ≤ N, \\ Cr^{\frac{p_{m−2}−N}{p_{m−2}−1}} & if p_{m−2} > N, \end{matrix}

for large r ≥ r_{∗}. This is a contradiction.

Similarly, suppose that there exists an i_{0} ∈ {1, 2, ⋯, m} such that

Λ_{i_{0}} = \frac{A − P}{P} max\{0, p_{i_{0}} − N\}

and

Λ_{i} > \frac{A − P}{P} max\{0, p_{i} − N\}, i = i_{0} + 1, ⋯, m − 1.

Then we see from (4.6) and (4.7) with i = m − 1 that

u_{m−1}(r) ≥ Cr^{\frac{p_{m−1}−λ_{m−1}}{p_{m−1}−1}} , r ≥ r_{1} > r_{∗} + 1.

From this estimate, (4.5) with i = m − 2, (4.7) with i = m − 2 we have

\begin{matrix}u_{m−2}(r) ≥ & C ∫_{r_{1}+1}^{r} (s^{1−N} ∫_{r_{1}}^{s} t^{N−1−λ_{m−2}+ \frac{α_{m−2}(p_{m−1}−λ_{m−1})}{p_{m−1}−1}} dt)^{\frac{1}{p_{m−2}−1}} ds \\ ≥ & C ∫_{r_{1}+1}^{r} s^{\frac{1−λ_{m−2}}{p_{m−2}−1} + \frac{α_{m−2}(p_{m−1}−λ_{m−1})}{(p_{m−1}−1)(p_{m−2}−1)}} ds \\ ≥ & Cr^{\frac{p_{m−2}−λ_{m−2}}{p_{m−2}−1} + \frac{α_{m−2}(p_{m−1}−λ_{m−1})}{(p_{m−1}−1)(p_{m−2}−1)}} , r ≥ r_{2} > r_{1} + 1. \end{matrix}

By repeating this procedure, we get a sequence {r_{j}}_{j=2}^{m−i_{0}−1} such that

u_{i}(r) ≥ Cr^{τ_{i}}, r ≥ r_{j} > r_{j−1} + 1, i = m − 2, m − 3, ⋯, i_{0} + 1,

where

\begin{matrix}τ_{i} = & \frac{1}{p_{i} − 1} \{p_{i} − λ_{i} + ∑_{j=1}^{m−i−1} \{(p_{i+j} − λ_{i+j}) ∏_{k=0}^{j−1} \frac{α_{i+k}}{p_{i+1+k} − 1}\}\} \\ = & \frac{p_{i} − λ_{i} + α_{i}τ_{i+1}}{p_{i} − 1} . \end{matrix}

From this estimate, (4.5) with i = i_{0}, (4.7) with i = i_{0} we have

\begin{matrix}u_{i_{0}}(r) ≥ & C ∫_{r_{m−i_{0}−1}+1}^{r} (s^{1−N} ∫_{r_{m−i_{0}−1}}^{s} t^{N−1−λ_{i_{0}}+α_{i_{0}}τ_{i_{0}+1}}dt)^{\frac{1}{p_{i_{0}} −1}} ds \\ = & C ∫_{r_{m−i_{0}−1}+1}^{r} (s^{1−N} ∫_{r_{m−i_{0}−1}}^{s} t^{N−1−min\{p_{i_{0}},N\}}dt)^{\frac{1}{p_{i_{0}} −1}} ds. \end{matrix}

Therefore we see that for r ≥ r_{m−i_{0}} > r_{m−i_{0}−1} + 1

u_{i_{0}}(r) ≥ \{ \begin{matrix}C log r & if p_{i_{0}} < N, \\ C(log r)^{\frac{p_{i_{0}}}{p_{i_{0}} −1}} & if p_{i_{0}} = N, \\ Cr^{\frac{p_{i_{0}} −N}{p_{i_{0}} −1}} (log r)^{\frac{1}{p_{i_{0}} −1}} & if p_{i_{0}} > N. \end{matrix}

On the other hand, from (4.4) with i = i_{0} and the definition of β_{i_{0}} we see that

u_{i_{0}}(r) ≤ \{ \begin{matrix}C & if p_{i_{0}} ≤ N, \\ Cr^{\frac{p_{i_{0}} −N}{p_{i_{0}} −1}} & if p_{i_{0}} > N, \end{matrix}

for large r ≥ r_{∗}. This is a contradiction. Thus the proof is completed for the case p_{m} ≤ N.

(ii) Let p_{m} > N. Then, integrating (3.5) on [0, r] twice, we have

(4.8) u_{m}(r) = u_{m}(0) + ∫_{0}^{r} s^{\frac{1−N}{pm−1}} (∫_{0}^{s} t^{N−1}H_{m}(t)u_{1}(t)^{α_{m}}dt)^{\frac{1}{pm−1}} ds

≥ ∫_{r_{∗}}^{r} s^{\frac{1−N}{pm−1}} ds (∫_{0}^{r_{∗}} t^{N−1}H_{m}(t)u_{1}(t)^{α_{m}}dt)^{\frac{1}{pm−1}}
≥ Cr^{\frac{pm−N}{pm−1}} , r ≥ r_{1} > r_{∗}.

Let us consider the case that

Λ_{m−1} = \frac{A − P}{P} max\{0, p_{m−1} − N\}.

Then from (4.6) we see that

λ_{m−1} = \frac{α_{m−1}(p_{m} − N)}{p_{m} − 1} + min\{p_{m−1}, N\}.

From (4.7) with i = m − 1 and (4.8) we have

\begin{matrix}u_{m−1}(r) ≥ & ∫_{r_{1}+1}^{r} (s^{1−N} ∫_{r_{1}}^{s} t^{N−1−λ_{m−1}+ \frac{α_{m−1}(pm−N)}{pm−1}} dt)^{\frac{1}{p_{m−1}−1}} ds \\ = & ∫_{r_{1}+1}^{r} (s^{1−N} ∫_{r_{1}}^{s} t^{N−1−min\{p_{m−1},N\}}dt)^{\frac{1}{p_{m−1}−1}} ds. \end{matrix}

Therefore we see that for r ≥ r_{2} > r_{1} + 1

u_{m−1}(r) ≥ \{ \begin{matrix}C log r & if p_{m−1} < N, \\ C(log r)^{\frac{p_{m−1}}{p_{m−1}−1}} & if p_{m−1} = N, \\ Cr^{\frac{p_{m−1}−N}{p_{m−1}−1}} (log r)^{\frac{1}{p_{m−1}−1}} & if p_{m−1} > N. \end{matrix}

On the other hand, from (4.4) with i = m − 1 and the definition of β_{m−1} we see that

u_{m−1}(r) ≤ \{ \begin{matrix}C & if p_{m−1} ≤ N, \\ Cr^{\frac{p_{m−1}−N}{p_{m−1}−1}} & if p_{m−1} > N, \end{matrix}

for large r ≥ r_{∗}. This is a contradiction.

Using similar arguments as in (i), we can get a contradiction for the case that

Λ_{i_{0}} = \frac{A − P}{P} max\{0, p_{i_{0}−1} − N\} for some i_{0} ∈ \{1, 2, ⋯, m\},

and

Λ_{i} > \frac{A − P}{P} max\{0, p_{i−1} − N\}, i = i_{0} + 1, i_{0} + 2, ⋯, m − 1.

The proof is finished.□

Proof of Theorem 4.2. Let (u_{1}, u_{2}, ⋯, u_{m}) be a nonnegative nontrivial radial entire solution of (1.1). From Theorem 3.2 and its proof we see that u_{i}(r) > 0, r ≥ r_{∗}, i = 1, 2, ⋯, m, for some r_{∗} > r_{0} and u_{i}, i = 1, 2, ⋯, m, satisfy

(4.9)  u_{i}(r) ≤ C_{i}(log r)^{β_{i}} at ∞,

where C_{i} > 0, i = 1, 2, ⋯, m, are constants. If there exists an i_{0} ∈ {1, 2, ⋯, m} such that

Λ_{i_{0}} < \frac{A − (N − 1)^{m}}{(N − 1)^{m−1}} ,

then we see that β_{i_{0}} < 1 by the definition of β_{i_{0}}. On the other hand, integrating (3.5) on [0, r] twice, we have

(4.10) u_{i_{0}}(r) = u_{i_{0}}(0) + ∫_{0}^{r} s^{−1} (∫_{0}^{s} t^{N−1}H_{i_{0}}(t)u_{i_{0}+1}(t)^{α_{i_{0}}} dt)^{\frac{1}{N−1}} ds

≥ ∫_{r_{∗}}^{r} s^{−1}ds (∫_{0}^{r_{∗}} t^{N−1}H_{i_{0}}u_{i_{0}+1}(t)^{α_{i_{0}}} dt)^{\frac{1}{N−1}}
≥ C log r, r ≥ r_{1} > r_{∗}

for some constant C > 0. This contradicts to (4.9) with β_{i_{0}} < 1. It remains to discuss the case

Λ_{i} ≥ \frac{A − (N − 1)^{m}}{(N − 1)^{m−1}} , i = 1, 2, ⋯, m.

From the assumption of Λ_{i} there exists an i_{0} ∈ {1, 2, ⋯, m} such that

Λ_{i_{0}} = \frac{A − (N − 1)^{m}}{(N − 1)^{m−1}} .

Without loss of generality we may assume that i_{0} = m, that is,

Λ_{i} ≥ \frac{A − (N − 1)^{m}}{(N − 1)^{m−1}} , i = 1, 2, ⋯, m − 1

and

Λ_{m} = \frac{A − (N − 1)^{m}}{(N − 1)^{m−1}} .

A similar computation as was used in the proof of Theorem 4.1 shows that (4.11)

λ_{i} ≤ − ∑_{j=1}^{m−i−1} \{\frac{(λ_{i+j} − N)}{(N − 1)^{j}} ∏_{k=0}^{j−1} α_{i+k}\} + \frac{∏_{k=0}^{m−i−1} α_{i+k}}{(N − 1)^{m−i−1}} + 1, i = 1, 2, ⋯, m − 2,

and

(4.12)  λ_{m−1} ≤ α_{m−1} + 1.

We notice that ”<” holds in (4.11) and (4.12) if

Λ_{i} > \frac{A − (N − 1)^{m}}{(N − 1)^{m−1}} ,

and ”=” holds in (4.11) and (4.12) if

Λ_{i} = \frac{A − (N − 1)^{m}}{(N − 1)^{m−1}} .

First we consider the case that

Λ_{m−1} = \frac{A − (N − 1)^{m}}{(N − 1)^{m−1}} .

Using the same computation as (4.10), we have

u_{m}(r) ≥ C log r, r ≥ r_{1} > r_{∗}

for some constant C > 0. From this estimate, (4.7) with i = m − 1, (4.12) we have

u_{m−1}(r) ≥ C ∫_{r_{1}+1}^{r} s^{−1} (∫_{r_{1}}^{s} t^{−1}(log t)^{−λ_{m−1}+α_{m−1}}dt)^{\frac{1}{N−1}} ds
= C ∫_{r_{1}+1}^{r} s^{−1} (∫_{r_{1}}^{s} t^{−1}(log t)^{−1}dt)^{\frac{1}{N−1}} ds
≥ C ∫_{r_{1}+1}^{r} s^{−1}(log(log s))^{\frac{1}{N−1}} ds
≥ C log r(log(log r))^{\frac{1}{N−1}} , r ≥ r_{2} > r_{1} + 1

for some constant C > 0. On the other hand, from (4.9) with i = m − 1 and the definition of β_{m−1} we see that

u_{m−1}(r) ≤ C_{m−1} log r at ∞.

This is a contradiction.

Using similar arguments as in the proof of Theorem 4.1, we get a contradiction for the case that

Λ_{i_{0}} = \frac{A − (N − 1)^{m}}{(N − 1)^{m−1}} for some i_{0} ∈ \{1, 2, ⋯, m\}

and

Λ_{i} > \frac{A − (N − 1)^{m}}{(N − 1)^{m−1}} , i = i_{0} + 1, i_{0} + 2, ⋯, m − 1.

The proof is completed.□

Acknowledgment. The author would like to express his sincere gratitude to Professor Hiroyuki Usami for many helpful suggestions and stimulating discussions. The author also wishes to thank the referee for some helpful comments.

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