An integral system of Matukuma type with negative exponents

1.   Introduction and the main results

In this paper, we are concerned with the study of positive solutions to the following integral system:

where $p, q > 0$, $\alpha, \beta\in (0, N),$ and $K, L: \mathbb R^N\to (0, \infty)$ are continuous functions which satisfy:

for some $\gamma > 0$ and $C_2 > C_1 > 0$.

We investigated positive solutions $(u, v)$ of Eq (1.1), that is, functions $u, v\in C(\mathbb R^N)$ such that $u, v > 0$ and satisfying Eq (1.1) pointwise in $\mathbb R^N$. Our study is motivated by the Matukuma equation ^[1,2] introduced in the 1930s to describe the dynamics of globular clusters of stars. Mathematically, the Matukuma equation reads as

where $p > 1$. Solutions of Eq (1.3) which tend to zero at infinity satisfy the integral equation

Also, the quantity

is called the total mass of the solution $u$ of Eq (1.3) (see ^[3,4]). The extended integral equation

was studied in ^[5], and the behavior of positive solutions was discussed in ^[6,7,8] (see also ^{[9,10,11,12,13]}). The case of negative exponents, that is,

was recently investigated in ^[14] under the conditions $p > 0$, $\alpha\in (0, N),$ and $C_1(1+|x|)^{-\beta}\leq K(x)\leq C_2(1+|x|)^{-\beta}$ in $\mathbb R^N$, for some $\beta > 0$ and $C_2 > C_1 > 0$. Integral systems with negative exponents were considered in the case of half-space in ^[15,16] and in the case of whole-space in ^[17,18,19] (see also ^{[20,21,22,23,24]} for further results).

The aim of the present paper is to discuss various qualitative properties of the positive solutions to Eq (1.1). We start our study of Eq (1.1) with the following nonexistence result which extends and improves Theorem 1.1 in ^[14].

Theorem 1.1. (Nonexistence)

Let $1\leq s_1, s_2\leq \infty$ be such that

Then, Eq (1.1) has no positive solutions $(u, v) \in L^{s_1}(\mathbb R^N) \times L^{s_2}(\mathbb R^N)$.

We notice that if $\max\{\alpha, \beta\} > \gamma$, then Eq (1.6) is clearly fulfilled and we directly obtain the following corollary.

Corollary 1.2. Let $1\leq s_1, s_2\leq \infty$. If one of the conditions below holds,

(ⅰ) $\max\{\alpha, \beta\} > \gamma$ and $1\leq s_1, s_2 \leq \infty$;

(ⅱ) $\max\{\alpha, \beta\} \geq \gamma$ and $1\leq s_1, s_2 \leq \infty$, $(s_1, s_2) \neq (\infty, \infty)$;

then Eq (1.1) has no positive solutions $(u, v) \in L^{s_1}(\mathbb R^N) \times L^{s_2}(\mathbb R^N)$.

Next, using some integral identities from harmonic analysis, we can construct an exact solution to Eq (1.1).

Theorem 1.3. (Exact solution)

Assume that $\gamma > 2N$ and let

Then, there exist $C, D > 0,$ and $p, q > 0$ depending on $\alpha, \beta, \gamma,$ and $N$ such that

is an exact solution to Eq (1.1).

The condition $\gamma > 2N$ is needed to ensure that $pq\neq 1,$ which is a restriction in our argument.

Next, we turn to the existence of a solution to Eq (1.1) in a more general framework. Given two positive functions $f, g: \mathbb R^N\to (0, \infty)$ we use the symbol $f\simeq g$ to denote that the quotient $f/g$ is bounded between two positive constants on $\mathbb R^N$. Our result in this case is stated below.

Theorem 1.4. (Existence)

Assume $\max\{\alpha, \beta\} < \gamma$.

(ⅰ) If $\max\{\alpha, \beta\} < \gamma < N$, $0 < p < \frac{\gamma-\alpha}{\gamma-\beta},$ and $0 < q < \frac{\gamma-\beta}{\gamma-\alpha}$, then the integral system (1.1) has a solution $(u, v)$ such that

Moreover, $(u, v)\in C^{0, \mu}(\mathbb R^N)\times C^{0, \nu}(\mathbb R^N)$ for some $\mu, \nu\in (0, 1)$.

(ⅱ) If $\gamma > N$, $0 < p < \frac{\gamma-N}{N-\beta}$, $0 < q < \frac{\gamma-N}{N-\alpha}$ and $pq < 1$, then the integral system (1.1) has a solution $(u, v)$ such that

Moreover, $(u, v)\in C^{0, \mu}(\mathbb R^N)\times C^{0, \nu}(\mathbb R^N)$ for some $\mu, \nu\in (0, 1)$.

The existence of a positive solution to Eq (1.1) is achieved using the Schauder fixed-point theorem. One major difficulty in our approach is the fact that $C(\mathbb R^N)$ is not a suitable space to work on, due to its lack of compactness. Instead, we shall use an approximation argument and construct solutions to Eq (1.1) over a sequence of expanding balls in $\mathbb R^N$. Further, using some integral estimates which are universal (see Lemma 2.2 below), we are able to obtain the solution to Eq (1.1) through a limit argument. Our argument requires a restriction in the range of exponents $p$ and $q$ as stated in Theorem 1.4 above. Let us point out that the restrictions on $p, q$ in Theorem 1.4(ii) are natural. They appear again in Theorem 1.5 and Theorem 1.6 below.

Next, we show that Eq (1.1) may have a unique solution in some ranges of exponents $p$ and $q$. This is stated in the following result.

Theorem 1.5. (Uniqueness)

Assume $0 < p < \frac{\gamma-N}{N-\beta}$ and $0 < q < \frac{\gamma-N}{N-\alpha}$. If $pq < 1,$ then the integral system (1.1) has a unique positive solution $(u, v)$ in $C(\mathbb R^N)\times C(\mathbb R^N)$.

Finally, and in analogy to the case of a single equation (see Eq (1.4)), we say that a solution $(u, v)$ of system (1.1) has finite total mass if:

Otherwise, we say that $(u, v)$ has infinite total mass.

Our last result in this section is stated below

Theorem 1.6. (Finite and infinite total mass solutions)

Let $(u, v)$ be a positive solution of Eq (1.1).

(ⅰ) If $0 < p < \frac{\gamma-N}{N-\beta}$ and $0 < q < \frac{\gamma-N}{N-\alpha},$ then $(u, v)$ has finite total mass. Moreover

(ⅱ) If either $p = \frac{\gamma-N}{N-\beta}$, $q > \frac{\gamma-N}{N-\alpha}$ or $q = \frac{\gamma-N}{N-\alpha}$, $p > \frac{\gamma-N}{N-\beta}$, then $(u, v)$ has infinite total mass.

The remainder of this paper is organized as follows. In Section 2, we recall some results related to the Riesz potential and its regularity. Section 3 is devoted to the proof of Theorem 1.1. Further, Theorems 1.3–1.5 are proved in Sections 4, 5, and 6. The method we develop in the article can be used to study integral systems of type (1.1) with different signs of exponents. This will be explained in Section 7. Throughout this paper, $C, c, c_1, c_2, ...$ denote positive constants whose values may change from line to line, unless otherwise stated. By $B_r$, we denote the open ball in $\mathbb R^N$ centered at the origin and having radius $r > 0$.

2.   Preliminary results

Let $f\in L^1_{loc}(\mathbb R^N)$. The Riesz potential of order $\gamma\in (0, N)$ of $f$ is given by

For $\alpha, \beta\in (0, N),$ we define

and for $n > 0$ we also set

We recall the following lemma from ^[25] and ^[26].

Lemma 2.1. Assume $s > 1$ is such that $\alpha-1 < \frac{N}{s} < \alpha$ and $\beta-1 < \frac{N}{s} < \beta$. Let $\mu = \alpha-\frac{N}{s}$ and $\nu = \beta-\frac{N}{s}$. Then, the Riesz operators

and

are continuous.

From ^[14], we recall the following integral estimates that extend to those in ^[27].

Lemma 2.2. Let $n\geq 1$ be an integer, $\alpha\in (0, N),$ and $\sigma > \alpha$.

(ⅰ) If $\alpha < \sigma < N,$ then there exist $c_2 > c_1 > 0$ independent of $n$ such that for all $x\in B_n,$ we have

(ⅱ) If $\sigma > N,$ then there exist $c_2 > c_1 > 0$ independent of $n$ such that for all $x\in B_n,$ we have

(ⅲ) There exist $c_2 > c_1 > 0$ independent of $n$ such that for all $x\in B_n,$ we have

3.   Proof of Theorems

3.1.   Proof of Theorem 1.1

Assume that $(u, v) \in L^{s_1}(\mathbb R^N) \times L^{s_2}(\mathbb R^N)$ is a positive solution of Eq (1.1) for some $1\leq s_1, s_2\leq \infty$.

Case 1: $s_1 = s_2 = \infty$. Then, $u$ and $v$ are bounded, and Eq (1.6) yields $\alpha > \gamma$.

For all $x \in \mathbb R^N \setminus B_1$, we deduce

We notice that

Together with $\alpha > \gamma$, they yield

This contradicts the assumption that $u$ is bounded, implying that Eq (1.1) has no positive solution in $L^{s_1}(\mathbb R^N) \times L^{s_2}(\mathbb R^N)$.

Case 2: $1\leq s_1, s_2 < \infty$. Without loss of generality, from Eq (1.6) we may assume

Let $R > 1$. Then,

If $|x|, |y| < R$, then $|x-y|\leq 2R$ and $1+|y|\leq 1+R \leq 2R$. We estimate

By Jensen's inequality, we obtain

This yields

Since $v\in L^{s_2}(\mathbb R^N)$, the above estimate implies

This yields

and then

Since $\alpha-\gamma+N(\frac{p}{s_2}+\frac{1}{s_1}) > 0$ holds, letting $R\to \infty$ in the above estimate, we deduce

This contradicts the assumption that $u\in L^{s_1}(\mathbb R^N)$, implying that Eq (1.1) has no positive solution in $L^{s_1}(\mathbb R^N) \times L^{s_2}(\mathbb R^N)$.

Case 3: $1\leq s_1 < \infty = s_2$. In this case, Eq (1.6) yields

If $\alpha+\frac{N}{s_1} > \gamma$, as in the estimate Eq (3.1), we obtain

This further yields $\alpha < \gamma$ and $u(x)\geq C|x|^{\alpha-\gamma}$ for $|x| > 1$. Then, $u(x)^{s_1}\geq C|x|^{(\alpha-\gamma)s_1}$ for all $x\in \mathbb R^N\setminus B_1,$ which yields $u\not\in L^{s_1}(\mathbb R^N)$, a contradiction.

If $\beta +\frac{Nq}{s_1} > \gamma$, as in the estimate Eq (3.2) which we obtained in Case 2 above, we find

which contradicts the fact that $v\in L^\infty(\mathbb R^N)$.

Case 4: $1\leq s_2 < \infty = s_1$. We use the same argument as in Case 3 above in which we replace $s_1$ by $s_2$ and $u$ by $v$ to raise a contradiction.

The proof of Corollary 1.2 follows directly from Theorem 1.1 since either of the conditions (i) and (ii) imply Eq (1.6).

3.2.   Proof of Theorem 1.3

We will employ the following integral identities, see ^[28].

where $\Gamma$ stands for the Gamma function. Based on the $\alpha, \beta$ in Eq (1.1), we set

Let us take

where $\gamma > 2N,$ and next define

Since $\gamma > \max\{2N, \alpha+\beta\},$ we have $pq\neq 1$. Next, let $C, D > 0$ be such that $CD^p = A(\alpha)$ and $DC^q = A(\beta)$. We then deduce

Next, we claim that

is a solution of Eq (1.1). Indeed, we have

Similarly, we have

This concludes the proof of our result.

3.3.   Proof of Theorem 1.4

The existence of a positive solution to Eq (1.1) will be obtained by combining the Schauder fixed-point theorem with the properties of the Riesz potentials stated in Section 2. We split our discussion into two cases.

(ⅰ) Assume $0 < p < \frac{\gamma-\alpha}{\gamma-\beta}$ and $0 < q < \frac{\gamma-\beta}{\gamma-\alpha}$. Let

For $n\geq1$, we define the closed and convex set $\mathcal{A}_n\subset C(\overline B_n)\times C(\overline B_n)$ by

where $0 < m_1 < M_1$, $0 < m_2 < M_2$ are constants depending on $\alpha, \beta, \gamma, p, q,$ and $N$ that will be chosen in Lemma 3.1 below.

For all $u, v\in \mathcal{A}_n,$ we define

By Lemma 2.2 (ⅰ) for $\sigma = \gamma-p\kappa_2 > \alpha$ and $\sigma = \gamma-q\kappa_1 > \beta,$ respectively, there exists $c_2 > c_1 > 0$ independent of $n\geq 1$ such that

and

Lemma 3.1. Let

where $c_1, c_2$ are defined in Eqs (3.6) and (3.7). If $pq < 1$, then $J_n(\mathcal{A}_n)\subset \mathcal{A}_n$.

Proof. Since $pq < 1$, it is obvious to see that $m_1 < M_1$ and $m_2 < M_2$. We also have

To prove $J_n(\mathcal{A}_n)\subset \mathcal{A}_n$, let $(u, v)\in \mathcal{A}_n$.

Since $v(x)\leq M_2 (1+|x|)^{-\kappa_2}$ in $B_n$, by Eqs (3.6) and (3.9)$_1$ we have

Also, $v(x)\geq m_2 (1+|x|)^{-\kappa_2}$ in $B_n$ together with Eqs (3.6) and (3.9)$_2$ implies

Similarly, $u(x)\leq M_1 (1+|x|)^{-\kappa_1}$ in $B_n$ combined with Eqs (3.7) and (3.9)$_3$ yields

Finally, since $u(x)\geq m_1 (1+|x|)^{-\kappa_1}$ in $B_n,$ Eqs (3.7) and (3.9)$_4$ produces

Hence, $J_n(\mathcal{A}_n)\subset \mathcal{A}_n$.

From the definition of $\kappa_1$ and $\kappa_2$ in Eq (3.4), we can easily check that

Thus, we can select $s > 1$ such that

Let $\mu = \alpha-\frac{N}{s}\in (0, 1)$ and $\nu = \beta-\frac{N}{s}\in (0, 1)$. By Lemma 2.1, we obtain that

and $J_{n}(\mathcal{A}_n) \subset \mathcal{A}_n$.

Hence, we can use the Schauder fixed-point theorem for $J_{n}$, which implies the existence of $(u_n, v_n) \in \mathcal{A}_n$ such that

Next, we argue as follows.

● Since $\{(u_n, v_n)\}$ is bounded in $C^{0, \mu}(\overline B_1)\times C^{0, \nu}(\overline B_1)$. Since the embedding $C^{0, \mu}(\overline B_1)\times C^{0, \nu}(\overline B_1) \hookrightarrow C(\overline B_1)\times C(\overline B_1)$ is compact, there exists a subsequence $\{(u_n^1, v_n^1)\}_{n\geq 1}$ of $\{(u_n, v_n)\}_{n\geq 1}$ which converges in $C(\overline B_1)\times C(\overline B_1)$.

● Since $\{(u_n^1, v_n^1)\}_{n\geq 2}$ is bounded in $C^{0, \mu}(\overline B_2)\times C^{0, \nu}(\overline B_2)$ and the embedding $C^{0, \mu}(\overline B_2)\times C^{0, \nu}(\overline B_2) \hookrightarrow C(\overline B_2)\times C(\overline B_2)$ is compact, there exists a subsequence $\{(u_n^2, v_n^2)\}_{n\geq 2}$ of $\{(u_n^1, v_n^1)\}_{n\geq 2}$ which converges in $C(\overline B_2)\times C(\overline B_2)$.

● Since $\{(u_n^2, v_n^2)\}_{n\geq 3}$ is bounded in $C^{0, \mu}(\overline B_3)\times C^{0, \nu}(\overline B_3)$ and the embedding $C^{0, \mu}(\overline B_3)\times C^{0, \nu}(\overline B_3) \hookrightarrow C(\overline B_3)\times C(\overline B_3)$ is compact, we can find a subsequence $\{(u_n^3, v_n^3)\}_{n\geq 3}$ of $\{(u_n^2, v_n^2)\}_{n\geq 3}$ which converges in $C(\overline B_3)\times C(\overline B_3)$.

● Inductively, we deduce that, for all $k\geq1$, there exists a subsequence $\{(u_n^k, v_n^k)\}_{n\geq k}$ of $\{(u_n^{k-1}, v_n^{k-1})\}_{n\geq k}$ which converges in $C(\overline B_k)\times C(\overline B_k)$.

Let $\{(U_n, V_n)\} = \{(u_n^n, v_n^n)\}_{n\geq 1},$ which converges to a certain $(U, V) \in C(\mathbb R^N)\times C(\mathbb R^N)$ and fulfills

Since $(U_n, V_n)\in \mathcal{A}_n,$ we can apply the Lebesgue dominated convergence theorem to obtain that $(U, V)$ is a continuous solution of Eq (1.1). Moreover,

Finally, from Lemma 2.1 we deduce $(U, V) \in C^{0, \mu}(\mathbb R^N)\times C^{0, \nu}(\mathbb R^N)$. This concludes the proof in (i).

(ⅱ) Assume $0 < p < \frac{\gamma-N}{N-\beta}$, $0 < q < \frac{\gamma-N}{N-\alpha},$ and $pq < 1$.

For $n \geq 1$, we define the closed and convex subset $\mathcal{B}_n \subset C(\overline B_n) \times C(\overline B_n)$ by

where $0 < m_1 < M_1$, $0 < m_2 < M_2$ are constants depending on $\alpha, \beta, \gamma, p, q,$ and $N$.

For all $(u, v)\in \mathcal{B}_n,$ we define

By Lemma 2.2 (ⅱ) applied twice for $\sigma = \gamma-p(N-\beta) > N$ and for $\sigma = \gamma-q(N-\alpha) > N,$ respectively, there exist $c_2 > c_1 > 0$ independent of $n\geq 1$ such that

and

We choose $m_1, m_2, M_1,$ and $M_2$ as given by Eq (3.8) with new constants $c_1, c_2$ from Eqs (3.11) and (3.12). Then, Lemma 3.1 holds, and we deduce $J_n(\mathcal{B}_n)\subset \mathcal{B}_n$.

Next, we select $s > 1$ such that

Let $\mu = \alpha-\frac{N}{s}\in (0, 1)$ and $\nu = \beta-\frac{N}{s}\in (0, 1)$. By Lemma 2.1, we obtain

and $J_{n}(\mathcal{B}_n) \subset \mathcal{B}_n$.

Hence, we can use the Schauder fixed-point theorem for $J_{n}$, which implies the existence of $(u_n, v_n) \in \mathcal{B}_n$ such that

As in part (ⅰ) above, the diagonal sequence $\{(U_n, V_n)\} = \{(u_n^n, v_n^n)\}_{n\geq 1}$ converges to a certain $(U, V) \in C(\mathbb R^N)\times C(\mathbb R^N)$ which fulfills

Since

we can apply Lemma 2.1, and we deduce $(U, V) \in C^{0, \mu}(\mathbb R^N)\times C^{0, \nu}(\mathbb R^N)$. This finishes the proof of our theorem.

3.4.   Proof of Theorem 1.5

Assume $0 < p < \frac{\gamma-N}{N-\beta}$ and $0 < q < \frac{\gamma-N}{N-\alpha}$. The existence of a positive solution to Eq (1.1) in this range of parameters was obtained in Theorem 1.4. We next discuss the uniqueness.

Let $(u, v)$ be a positive solution of Eq (1.1). We note that if $|x| > 1\geq |y|$, then $|x-y|\leq |x|+|y|\leq 2|x|,$ and we combine this fact with $Kv^{-p}\in L^1_{loc}(\mathbb R^N)$ to obtain

This means that $u(x)\geq C(1+|x|)^{\alpha-N}$ for $|x| > 1$.

If $|x|, |y|\leq 1$, then $|x-y|\leq 2$ and also $Kv^{-p}\in L^1(B_1)$ yield

We have shown that

In the same way, we obtain

From Eqs (1.1) and (3.14), we estimate

We can now apply Eq (2.2) with $\sigma = \gamma - p(N-\beta) > N$ to obtain $u(x) \leq C\big(1+|x|\big) ^{\alpha - N}$ in $\mathbb R^N,$ and similarly $v(x) \leq C\big(1+|x|\big) ^{\beta - N}$ in $\mathbb R^N$. Combining these two estimates with Eqs (3.13) and (3.14), we deduce

Now let $(u_1, v_1)$ and $(u_2, v_2)$ be two positive solutions of Eq (1.1). From Eq (3.15), we have

and so, we can find $C > c > 0$ such that

Then, we define

Clearly, $Mu_1\geq u_2$. Assume by contradiction that $M > 1$. Then, $u_2\leq Mu_1$ implies that for all $x\in \mathbb R^N,$ we have

Hence, $v_2\geq M^{-q} v_1$ in $\mathbb R^N,$ which implies

Together with the previous estimates, we obtain

Thus, $M^{pq} u_1\geq u_2$ in $\mathbb R^N$.

Since $M > 1$ and $0 < pq < 1$, we have $M > M^{pq} > 1,$ and this is a contradiction of the fact that $M$ is the infimum value defined in Eq (3.16). Hence, $M = 1$ and $u_1\geq u_2$. Swapping $u_1$ with $u_2$ in the above argument, we deduce $u_2\geq u_1$, and thus $u_1\equiv u_2$. From Eq (1.1), we also have $v_1\equiv v_2$. This concludes the proof of the uniqueness of a positive solution to Eq (1.1).

3.5.   Proof of Theorem 1.6

First, we start with the proof of the finite total mass solution.

(ⅰ) Recall that in the proof of Theorem 1.5, we established that $(u, v)$ satisfies:

in $\mathbb R^N$, for some $c > 0$. Using Eq (3.17) and $0 < p < \frac{\gamma-N}{N-\beta}$, we have

Similarly,

Hence, $(u, v)$ has finite total mass.

Next, in order to establish Eq (1.9), we note that

Using Fatou's lemma, we infer that

For the converse inequality, take $\varepsilon\in(0, 1)$ small and write

where

We note that $|y|\leq\varepsilon|x|$ implies $|x-y|\geq|x|-|y|\geq (1-\varepsilon)|x|$, so $\frac{|x|}{|x-y|}\leq\frac{1}{1-\varepsilon},$ and thus

Next, using Eq (3.17) and the estimate on $K(y)$, we have

Since $p < \frac{\gamma-N}{N-\beta}$, we have $S_2\to 0$ as $|x|\to\infty$.

In order to estimate $S_3$, we note that $|y| > 2|x|$ yields $|x-y|\geq|y|-|x|\geq|x|$, so $\frac{|x|}{|x-y|}\leq1,$ and then

Now, using Eqs (3.19)–(3.22), we deduce

Since $\varepsilon > 0$ was arbitrarily chosen, this yields

From Eqs (3.18) and (3.23), we complete the proof of Eq (1.9). The proof of Eq (1.10) follows similarly.

(ⅱ) Assume, without loss of the generality, that $p = \frac{\gamma-N}{N-\beta}$, $q > \frac{\gamma-N}{N-\alpha},$ and let $\varepsilon > 0$ be such that

Using Eq (3.17) and Lemma 2.2 (ⅲ), we find:

This last inequality and Eq (3.24) yields,

Hence, $(u, v)$ has infinite total mass. This concludes our proof.

4.   Further extensions

In this section we explain how our approach can be used to investigate the existence of solutions to the system

where $p, q > 0$, $\alpha, \beta\in (0, N),$ and $K, L: \mathbb R^N\to (0, \infty)$ are continuous functions which satisfy Eq (1.2). Our main result in this section is stated below.

Theorem 4.1. (Existence)

Assume $\max\{\alpha, \beta\} < \gamma$.

(ⅰ) If $\gamma > N$, $0 < q < \frac{\gamma-N}{N-\alpha},$ and $pq < 1,$ then the integral system (4.1) has a solution $(u, v)$ such that

Moreover, $(u, v)\in C^{0, \mu}(\mathbb R^N)\times C^{0, \nu}(\mathbb R^N)$ for some $\mu, \nu\in (0, 1)$.

(ⅱ) If $\max\{\alpha, \beta\} < \gamma < N$, $0 < p < \frac{N-\gamma}{\gamma-\beta},$ and $0 < q < \frac{\gamma-\beta}{\gamma-\alpha}$, then the integral system (4.1) has a solution $(u, v)$ such that

Moreover, $(u, v)\in C^{0, \mu}(\mathbb R^N)\times C^{0, \nu}(\mathbb R^N)$ for some $\mu, \nu\in (0, 1)$.

Proof. (ⅰ) Assume $\gamma > N$, $0 < q < \frac{\gamma-N}{N-\alpha},$ and $pq < 1$.

For $n \geq 1$, we define the closed and convex subset $\mathcal{A}_n \subset C(\overline B_n) \times C(\overline B_n)$ by

where $0 < m_1 < M_1$, $0 < m_2 < M_2$ are constants depending on $\alpha, \beta, \gamma, p, q,$ and $N$.

For all $(u, v)\in \mathcal{B}_n$ we define

By Lemma 2.2 (ⅱ) applied twice for $\sigma = \gamma+p(N-\beta) > N$ and for $\sigma = \gamma-q(N-\alpha) > N,$ respectively, there exist $c_2 > c_1 > 0$ independent of $n\geq 1$ such that

and

Lemma 4.2. Let

where $c_1, c_2$ are defined in Eqs (4.3) and (4.4). If $pq < 1$, then $J_n(\mathcal{A}_n)\subset \mathcal{A}_n$.

Proof. Since $pq < 1$, it is obvious to see that $m_1 < M_1$ and $m_2 < M_2$. We also have

To prove $J_n(\mathcal{A}_n)\subset \mathcal{A}_n$, let $(u, v)\in \mathcal{A}_n$.

Since $v(x)\leq M_2 (1+|x|)^{\beta-N}$ in $B_n$, by Eqs (4.3) and (4.6)$_2$ we have

Also, $v(x)\geq m_2 (1+|x|)^{\beta-N}$ in $B_n$, and by Eqs (4.3) and (4.6)$_1$ we have

Similarly, $u(x)\leq M_1 (1+|x|)^{\alpha-N}$ in $B_n$ combined with Eqs (4.4) and (4.6)$_3$ yields

Finally, since $u(x)\geq m_1 (1+|x|)^{\alpha-N}$ in $B_n,$ Eqs (4.4) and (4.6)$_4$ produces

Hence, $J_n(\mathcal{A}_n)\subset \mathcal{A}_n$.

Let us recall that

Next, we select $s > 1$ such that

Let $\mu = \alpha-\frac{N}{s}\in (0, 1)$ and $\nu = \beta-\frac{N}{s}\in (0, 1)$. By Lemma 2.1, we obtain that

and $J_{n}(\mathcal{A}_n) \subset \mathcal{A}_n$.

Hence, we can use the Schauder fixed-point theorem for $J_{n}$, which implies the existence of $(u_n, v_n) \in \mathcal{A}_n$ such that

We next argue as follows.

● Since $\{(u_n, v_n)\}$ is bounded in $C^{0, \mu}(\overline B_1)\times C^{0, \nu}(\overline B_1)$. Since the embedding $C^{0, \mu}(\overline B_1)\times C^{0, \nu}(\overline B_1) \hookrightarrow C(\overline B_1)\times C(\overline B_1)$ is compact, there exists a subsequence $\{(u_n^1, v_n^1)\}_{n\geq 1}$ of $\{(u_n, v_n)\}_{n\geq 1}$ which converges in $C(\overline B_1)\times C(\overline B_1)$.

● Since $\{(u_n^1, v_n^1)\}_{n\geq 2}$ is bounded in $C^{0, \mu}(\overline B_2)\times C^{0, \nu}(\overline B_2)$ and the embedding $C^{0, \mu}(\overline B_2)\times C^{0, \nu}(\overline B_2) \hookrightarrow C(\overline B_2)\times C(\overline B_2)$ is compact, there exists a subsequence $\{(u_n^2, v_n^2)\}_{n\geq 2}$ of $\{(u_n^1, v_n^1)\}_{n\geq 2}$ which converges in $C(\overline B_2)\times C(\overline B_2)$.

● Since $\{(u_n^2, v_n^2)\}_{n\geq 3}$ is bounded in $C^{0, \mu}(\overline B_3)\times C^{0, \nu}(\overline B_3)$ and the embedding $C^{0, \mu}(\overline B_3)\times C^{0, \nu}(\overline B_3) \hookrightarrow C(\overline B_3)\times C(\overline B_3)$ is compact, we can find a subsequence $\{(u_n^3, v_n^3)\}_{n\geq 3}$ of $\{(u_n^2, v_n^2)\}_{n\geq 3}$ which converges in $C(\overline B_3)\times C(\overline B_3)$.

● Inductively, we deduce that, for all $k\geq1$, there exists a subsequence $\{(u_n^k, v_n^k)\}_{n\geq k}$ of $\{(u_n^{k-1}, v_n^{k-1})\}_{n\geq k}$ which converges in $C(\overline B_k)\times C(\overline B_k)$.

Let $\{(U_n, V_n)\} = \{(u_n^n, v_n^n)\}_{n\geq 1}$ which converges to a certain $(U, V) \in C(\mathbb R^N)\times C(\mathbb R^N)$ and fulfills

Since $(U_n, V_n)\in \mathcal{A}_n,$ we can apply the Lebesgue dominated convergence theorem to obtain that $(U, V)$ is a continuous solution of Eq (4.1). Moreover,

Finally, from Lemma 2.1 we deduce $(U, V) \in C^{0, \mu}(\mathbb R^N)\times C^{0, \nu}(\mathbb R^N)$. This concludes the proof in (ⅰ).

(ⅱ) Assume $\max\{\alpha, \beta\} < \gamma < N$, $0 < p < \frac{N-\gamma}{\gamma-\beta},$ and $0 < q < \frac{\gamma-\beta}{\gamma-\alpha}$. Let

For $n\geq1$, we define the closed and convex set $\mathcal{B}_n\subset C(\overline B_n)\times C(\overline B_n)$ by

where $0 < m_1 < M_1$, $0 < m_2 < M_2$ are constants depending on $\alpha, \beta, \gamma, p, q,$ and $N$ that will be chosen in Lemma 4.2 below.

For all $u, v\in \mathcal{B}_n,$ we define

Since $0 < q < \frac{\gamma-\beta}{\gamma-\alpha}$, we can check

Also, by $N > \gamma$, we deduce $\gamma-\kappa_1 q < N$.

Next, since $0 < p < \frac{N-\gamma}{\gamma-\beta}$, we obtain

Also, by $\max\{\alpha, \beta\} < \gamma < N$, it follows that $\gamma+\kappa_2 p > \alpha$.

Now we have $\beta < \gamma-\kappa_1 q < N$ and $\alpha < \gamma+\kappa_2 p < N$. Hence, by Eq (3.7), there exists $c_2 > c_1 > 0$ independent of $n\geq 1$ such that

and

We choose $m_1, m_2, M_1,$ and $M_2$ as given by Eq (4.5) with new constants $c_1, c_2$ from Eqs (4.9) and (4.10). Then, Lemma 4.2 holds, and we deduce $J_n(\mathcal{B}_n)\subset \mathcal{B}_n$.

Thus, we can select $s > 1$ such that

Let $\mu = \alpha-\frac{N}{s}\in (0, 1)$ and $\nu = \beta-\frac{N}{s}\in (0, 1)$. By Lemma 2.1, we obtain that

and $J_{n}(\mathcal{B}_n) \subset \mathcal{B}_n$.

Hence, we can use the Schauder fixed-point theorem for $J_{n}$, which implies the existence of $(u_n, v_n) \in \mathcal{B}_n$ such that

As in part (ⅰ) above, the diagonal sequence $\{(U_n, V_n)\} = \{(u_n^n, v_n^n)\}_{n\geq 1}$ converges to a certain $(U, V) \in C(\mathbb R^N)\times C(\mathbb R^N),$ which fulfills

Since

we can apply Lemma 2.1 to deduce $(U, V) \in C^{0, \mu}(\mathbb R^N)\times C^{0, \nu}(\mathbb R^N)$. This finishes the proof of our theorem.

With a similar approach, we can also obtain the existence of solutions to the system

where $p, q > 0$, $\alpha, \beta\in (0, N),$ and $K, L: \mathbb R^N\to (0, \infty)$ are continuous functions which satisfy Eq (1.2).

Our main result regarding the system (4.11) reads as follows.

Theorem 4.3. (Existence)

Assume $\max\{\alpha, \beta\} < \gamma$.

(ⅰ) If $\max\{\alpha, \beta\} < \gamma < N$, $p > \frac{N-\gamma}{N-\beta}$, $q > \frac{N-\gamma}{N-\alpha},$ and $pq < 1$, then the integral system (4.11) has a solution $(u, v)$ such that

Moreover, $(u, v)\in C^{0, \mu}(\mathbb R^N)\times C^{0, \nu}(\mathbb R^N)$ for some $\mu, \nu\in (0, 1)$.

(ⅱ) If $\gamma > N$ and $pq < 1,$ then the integral system (4.11) has a solution $(u, v)$ such that

Moreover, $(u, v)\in C^{0, \mu}(\mathbb R^N)\times C^{0, \nu}(\mathbb R^N)$ for some $\mu, \nu\in (0, 1)$.

Use of AI tools declaration

The author declares he has not used Artificial Intelligence (AI) tools in the creation of this article.s

Acknowledgements

The author would like to thank the anonymous a for the careful reading of the manuscript which led to the current form.

This publication has emanated from research conducted with the financial support of Taighde Éireann-Research Ireland under Grant number 18/CRT/6049.

Conflict of interest

The author declares there is no conflict of interest.