Existence and uniqueness of radial solution for the elliptic equation system in an annulus

1.   Introduction

In this article we discuss the existence and uniqueness of radial solution for the elliptic equation system

in an annular domain $\Omega = \{x\in \mathbb{R}^{N}:\; r_1 < |x| < r_2\}$, where $N\ge 3, \; 0 < r_1 < r_2 < \infty, \; f, \; g:[r_1, \; r_2]\times \mathbb{R}\times \mathbb{R}\times \mathbb{R}^+\to \mathbb{R}$ are continuous.

This problem arises in many different areas of applied mathematics and physics, for instance, incineration theory of gases, solid state physics, variational methods and optimal control. Therefore, there have been many research results, see ^{[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25]} and references therein.

The authors of ^[1] considered the Dirichlet elliptic system

where $\Omega = \{x\in \mathbb{R}^{N}:\; R_1 < |x| < R_2\}, \; R_1, \; R_2 > 0, \; f, \; g:[0, \infty)\times [0, \infty)\to (0, \infty), \; \lambda$ is a positive real parameter. By establishing the strong maximum principle, applying upper and lower solutions method and fixed point index results proved the existence of positive radial solutions in the condition (A).

$(A)\quad f_\infty\equiv {\lim\limits_{(u, v)\to \infty}}\frac{f(u, v)}{u+v} = \infty, \quad g_\infty\equiv {\lim\limits_{(u, v)\to \infty}}\frac{g(u, v)}{u+v} = \infty.$

In ^[2], Lee replaced the annular domain with an exterior domain.

In ^[4], the authors used topological methods to prove the existence of positive solutions for semilinear elliptic systems of the form

where $\Omega$ is a bounded domain in $\mathbb{R}^2$, $f, \; g:\Omega \times \mathbb{R}^2\to \mathbb{R}$ are continuous. Similarly, in ^[8], the authors also obtain a priori estimates, and then use Leray-Schauder topological degree theory to establish the existence of positive radial solutions vanishing at infinity.

In addition to the above domain, there are ball domain, see ^{[3,12,13,17,18,20,21]}. In ^[3], Hai considered the boundary value problem

where $B$ is the open unit ball in $\mathbb{R}^N$, $f, \; g:\mathbb{R}^+\to \mathbb{R}^+$. They establish upper and lower estimates, and the existence and uniqueness of positive solutions are obtained in the case of $f$ and $g$ superlinear. In ^[17], the above authors proved the existence and multiplicity of positive radial solutions for the infinite semipositone/positone superlinear systems.

Recently, in ^[23], the authors used the fixed point index theory to study the existence of positive radial solutions for a system of boundary value problems with semipositone second order elliptic equations

where $\alpha, \; \beta, \; \gamma, \; \delta\ge 0, \; f, \; g:C(\mathbb{R}^+\times \mathbb{R}^+, \; \mathbb{R})$ and satisfy

In ^[24], Li discussed the existence of positive radial solutions of single elliptic equation. Inspired by the aforementioned article, we extend the results of ^[24] to the equation system.

The purpose of this article is to obtain existence and uniqueness results of radial solution for the elliptic equation system. However, we note that in most of the article on nonlinear differential equations the nonlinear terms are usually assumed to be nonnegative, see ^{[1,2,3,17,18,21,22,23]}. However, in this article, we do not assume that the nonlinear terms are nonnegative, $f, \; g\in C([r_1, \; r_2]\times \mathbb{R}^2\times \mathbb{R}^+, \mathbb{R})$. Using Leray-Schauder fixed point theorem, we prove the main results in the case of $f$ and $g$ superlinear or sublinear.

As usual, writing $r = |x|$, BVP (1.1) becomes the ordinary differential equation system boundary value problem

By discussing BVP (1.2) we will obtain radial solution of BVP (1.1).

Our main results are as follows:

Theorem 1.1. Let $f, \; g:[r_1, \; r_2]\times\mathbb{R}\times\mathbb{R}\times\mathbb{R}^{+}\to\mathbb{R}$ be continuous. If $f$ and $g$ satisfy the following conditions:

(F0) for any $M > 0$, there exists a positive monotone nondecreasing continuous function $G_M:[0, +\infty]\rightarrow (0, +\infty)$ satisfying

such that

where $r\in [r_1, r_2], \; |u|\le M, \; |v|\le M, \; \xi, \; \eta\in \mathbb{R}^+$;

(F1) there exist positive constants $a, \; b, \; c, \; d\ge 0$, satisfying $\frac{{r_2}^{N-1}}{{r_1}^{N-1}}\bigg(\frac{(r_1-r_2)^2}{2}(a+b)+c+d \bigg) < 1$ and $e > 0$, such that

where $(r, u, v)\in [r_1, r_2]\times \mathbb{R}\times \mathbb{R}, \; \xi, \; \eta\in \mathbb{R}^+$. Then BVP (1.1) has at least one radial solution.

Remark 1.1. Condition (F1) allows $f(r, \; u, \; v, \; \xi)$ and $g(r, \; u, \; v, \; \eta)$ to grow superlinearly with respect to $u, \; v, \; \xi, \; \eta$, while the Nagumo-type condition (F0) restricts $f(r, \; u, \; v, \; \xi)$ and $g(r, \; u, \; v, \; \eta)$ to grow at most quadratically with respect to $\xi$ and $\eta$, respectively.

Next we give the uniqueness condition.

Theorem 1.2. Let $f, \; g:[r_1, \; r_2]\times\mathbb{R}\times\mathbb{R}\times\mathbb{R}^{+}\to\mathbb{R}$ be continuous. If $f$ and $g$ satisfy (F0) and the following condition:

(F2) there exist positive constants $a, \; b, \; c, \; d\ge 0$, satisfying $\frac{{r_2}^{N-1}}{{r_1}^{N-1}}\bigg(\frac{(r_1-r_2)^2}{2}(a+b)+c+d \bigg) < 1$, such that

where $(r, u_i, v_i)\in [r_1, r_2]\times \mathbb{R}\times \mathbb{R}, \; \xi_i, \eta_i\in \mathbb{R}^+, \; i = 1, \; 2$. Then BVP (1.1) has a unique radial solution.

The main innovations of this article are as follows: First, the nonlinearities are sign-changing. Second, we replace the previous independent conditions with the correlation conditions of $f$ and $g$, which can better reflect the characteristics of the equations. Finally, as far as we know, there are few articles discussing the elliptic equation system of the nonlinear terms with gradient term, and this article is one of them.

In Section 2, we will present some preliminaries. The proofs of Theorems 1.1 and 1.2 are based on the Leray-Schauder fixed point theorem, which will be given in Section 3.

2.   Preliminaries

Let $I = [r_1, \; r_2]$. $C(I)$ denote the Banach space of all continuous function on $I$ with norm $\|u\|_{C} = \max_{t\in I}|u(t)|$. $C^1(I)$ denote the Banach space of all 1-order continuous differentiable function on $I$ with norm $\|u\|_{C^1} = \max_{t\in I}\{\|u\|_C, \; \|u'\|_C\}$. $L^{2}(I)$ denote the Hilbert space composed of all Lebesgue square integrable functions on $I$ with inner product $(u, \; v) = \int_0^1u(t)v(t)dt$, and its inner product norm is $\|u\|_{2} = (\int_0^1|u(t)|^2dt)^{\frac{1}{2}}$. Let $H^{1}(I) = \{u\in C(I):u$ be absolutely continuous on $I$, and $u'\in L^{2}(I)\}$.

Let $X$ and $Y$ be Banach spaces with norms $\|\cdot\|_{X}$, $\|\cdot\|_{Y}$, respectively. $X\times Y$ denotes the product space of $X$ and $Y$, forming the Banach space with norm $\|(x, \; y)\| = \max\{\|x\|_{X}, \; \|y\|_{Y}\}$.

For the case of a single equation, given $h\in C(I)$, we consider the linear boundary value problem (LBVP)

Lemma 2.1. If $h\in C(I)$, then the solution of LBVP (2.1) satisfies

Proof. Set $u\in C^2(I)$ is the solution of LBVP (2.1), then from the Hölder inequality, we have

The proof of Lemma 2.1 is completed. □

Given $(h_1, \; h_2)\in C(I)\times C(I)$, we consider the linear boundary value problem corresponding to BVP (1.2)

Lemma 2.2. For every $(h_1, \; h_2)\in C(I)\times C(I)$, LBVP (2.2) has a unique solution $(u, \; v): = S(h_1, \; h_2)\in C^2(I)\times C^2(I)$. Moreover, the solution operator $S:C(I)\times C(I)\rightarrow C^1(I)\times C^1(I)$ is a completely continuous linear operator.

Proof. The case of a single space is known, see ^[24] Lemma 2.1. We give the proof of the solution operator is completely continuous in product space.

Set

By direct computing we have

We define a function $G:\; I\times I\rightarrow \mathbb{R}^+$ by

Then $G\in C(I\times I)$. We verify that $G(r, \; s)$ is the Green function of the LBVP (2.2), namely

is the unique solution of LBVP (2.2). By the above and the definition of $G$, we have

By differentiating, we get that

Hence, we see that $(u(r), \; v(r))$ is a solution of LBVP (2.2) by direct calculation. By the maximum principle, LBVP (2.2) has only one solution. From (2.4)–(2.6), we see that the solution operator $S: C(I)\times C(I)\rightarrow C^1(I)\times C^1(I)$ is a completely continuous linear operator.

The proof of Lemma 2.2 is completed. □

Lemma 2.3. Let $f, \; g:[r_1, \; r_2]\times\mathbb{R}\times\mathbb{R}\times\mathbb{R}^{+}\rightarrow \mathbb{R}$ be continuous and satisty (F0). For all $M > 0$, there exist constants $M_1 = M_1(M) > 0, \; M_2 = M_2(M) > 0$, such that if the solution $(u, \; v)$ of BVP (1.2) satisfis $\|(u, \; v)\|_C\le M$, then we have

Proof. Set $M > 0$. By (1.3), there exist constants $M_1, \; M_2 > 0$, such that

Let $(u, \; v)\in C^2(I)\times C^2(I)$ is a solution of BVP (1.2) which satisfies $\|(u, \; v)\|_C\le M$, the following proof that $\|(u', \; v')\|_C\le \max\{M_1, \; M_2\}$. Suppose $(u'(r), \; v'(r))$ is not equal to $0$, then there exists $t_0\in (r_1, \; r_2)$ and $t_1\in I, \; t_0\ne t_1$, such that $(u'(t_0), \; v'(t_0)) = (0, \; 0), \; \|(u', \; v')\|_C = \max\{|u'(t_1)|, \; |v'(t_1)|\} > 0$. There are eight cases as follows:

$1)\; u'(t_{1}) > 0, \; v'(t_{1}) > 0, \; t_{0} < t_{1}$;

$2)\; u'(t_{1}) > 0, \; v'(t_{1}) < 0, \; t_{0} < t_{1}$;

$3)\; u'(t_{1}) < 0, \; v'(t_{1}) > 0, \; t_{0} < t_{1}$;

$4)\; u'(t_{1}) < 0, \; v'(t_{1}) < 0, \; t_{0} < t_{1}$;

$5)\; u'(t_{1}) > 0, \; v'(t_{1}) > 0, \; t_{1} < t_{0}$;

$6)\; u'(t_{1}) > 0, \; v'(t_{1}) < 0, \; t_{1} < t_{0}$;

$7)\; u'(t_{1}) < 0, \; v'(t_{1}) > 0, \; t_{1} < t_{0}$;

$8)\; u'(t_{1}) < 0, \; v'(t_{1}) < 0, \; t_{1} < t_{0}$.

We only prove case 1), other cases are similar. Set

then $s_1 < t_1$, and $(u'(s_1), \; v'(s_1)) = (0, \; 0)$. When $r\in (s_1, \; t_1]$, we have $u'(r) > 0, \; v'(r) > 0$. Hence,

Hence, for all $r\in [s_1, \; t_1]$, we have

Integrating both sides of this inequality on $[s_1, \; t_1]$, and variable substitution $\rho = |u'(r)|, \; \sigma = |v'(r)|$, we obtain that

By (2.7) it follows that

Similarly, it can be obtained

Therefore,

The proof of Lemma 2.3 is completed. □

Theorem 2.1. (Leray-Schauder fixed point theorem) ^[26,27] Let $E$ be a Banach space, $A: E\times E\rightarrow E\times E$ be a completely continuous mapping. If the solution set of the equation

is bounded in $E\times E$, then $A$ has a fixed point.

3.   Proofs of the main results

Proof of Theorem 1.1. We known LBVP (2.2) has a unique solution $(u, \; v)\in C^2(I)\times C^2(I)$ by Lemma 2.2

where $G(r, \; s)$ defined by (2.3). We make integral operator $A: C^1(I)\times C^1(I)\rightarrow C^1(I)\times C^1(I)$ as follows:

then, $A$ is a completely continuous linear operator. The solution of BVP (1.2) is equivalent to the fixed point of $A$. Next we prove that $A$ has fixed point. We consider the equation

Let $(u, \; v)\in C^1(I)\times C^1(I)$ be the solution of (3.1), then, $(u, \; v)\in C^2(I)\times C^2(I)$ satisfies the equations

Multiply both sides of the first formula of Eq (3.2) by $u(r)$, and multiply both sides of the second formula by $v(r)$. Then, add the two formulas together, by condition (F1) we have

Multiply both sides of the above formula by $r^{N-1}$, we have

By integrating on $I$, by Lemma 2.1 we have

namely,

Hence,

Then,

For all $r\in I$, we have

namely,

Similarly, it can be obtained

Therefore,

By condition (F0), we have

Hence, $\lambda f$ and $\lambda g$ satisfy condition (F0). By Lemma 2.3, there exist constants $M_1 = M_1(M) > 0$ and $M_2 = M_2(M) > 0$, such that

Therefore,

Hence, the solution set of the Eq (3.1) is bounded in $C^1(I)\times C^1(I)$. By the Leray-Schauder fixed point, we know that $A$ has fixed point $(u, \; v)\in C^1(I)\times C^1(I)$. By the definition of $A$, $(u, \; v)$ is a solution of BVP (1.2), namely, $(u(|x|), \; v(|x|))$ is a radial solution of BVP (1.1).

The proof of Theorem 1.1 is completed. □

Proof of Theorem 1.2. First, we prove that $(F2)\Rightarrow (F1)$. For all $(r, \; u, \; v)\in I\times \mathbb{R}\times \mathbb{R}, \; \xi, \; \eta \in \mathbb{R}^+$, we take $u_2 = u, \; v_2 = v, \; \xi_2 = \xi, \; \eta_2 = \eta, \; u_1 = v_1 = \xi_1 = \eta_1 = 0$ in (F2). Set

By condition (F2), we have

Let

we have

where $(r, \; u, \; v)\in I\times \mathbb{R}\times \mathbb{R}, \; \xi, \; \eta \in \mathbb{R}^+$ and $\frac{(r_1-r_2)^2}{2}(a_1+b_1)+c_1+d_1 = \frac{\frac{(r_1-r_2)^2}{2}(a+b)+c+d+1}{2} < 1$.

Hence, $f$ and $g$ satisfy condition (F1), by Theorem 1.1, BVP (1.1) has at least one radial solution.

Next, we prove the uniqueness. Set $(u_1, \; v_1), \; (u_2, \; v_2)\in C^2(I)\times C^2(I)$ are the solution of BVP (1.1), then

Subtract the first formula of Eq (3.4) and the first formula of Eq (3.3), we get

Similarly, it can be obtained

Multiply both sides of Eq (3.5) by $u_2(r)-u_1(r)$, and multiply both sides of Eq (3.6) by $v_2(r)-v_1(r)$. Then, add the two formulas together, by condition (F2), for all $r\in I$, we have

Multiply both sides of the above formula by $r^{N-1}$, we have

By integrating on $I$, by Lemma 2.1 we have

namely,

Hence,

namely $u_2'-u_1' = 0, v_2'-v_1' = 0$, then, $u_2-u_1 = C_1, v_2-v_1 = C_2$, where $C_1, C_2$ are constants. From the boundary conditions, $C_1 = C_2 = 0$, namely, $u_2 = u_1, v_2 = v_1$. Thus, BVP (1.1) has a unique radial solution.

The proof of Theorem 1.2 is completed. □

Example 3.1. Consider the elliptic boundary value problem

The corresponding nonlinear term of Eq (3.7) are

It is easy to see that $f$ and $g$ are quadratic growth with respect to $\xi$ and $\eta$ respectively, satisfying condition (F0). We next verify that $f$ and $g$ satisfy condition (F1), take $r_1 = \frac{1}{2}, r_2 = 1, a = 1+\varepsilon, b = 3+\varepsilon, c = d = 0, e = \frac{1}{2\varepsilon}$. When $\varepsilon < 2$, we have $\frac{(r_1-r_2)^2}{2}(a+b) < 1$, $f$ and $g$ satisfy

Thus, $f(r, u, v, \xi)$ and $g(r, u, v, \eta)$ satisfy condition (F1). By Theorem 1.1, BVP (3.7) has at least one radial solution.

4.   Conclusions

It is well known that elliptic equations arises in many different areas of applied mathematics and physics, for instance, incineration theory of gases, solid state physics, variational methods and optimal control. Due to the appearance of the gradient term in the nonlinearity, the equation system has no variational structure and the variational method cannot be applied to it directly. Therefore, we given existence and uniqueness results of radial solution in the case of $f$ and $g$ superlinear or sublinear, we replace the previous independent conditions with the correlation conditions of $f$ and $g$. In this paper, we just consider the existence of solutions. However, the properties of the solution have not been fully discussed.

Use of AI tools declaration

The authors declare they have not used Artificial Intelligence (AI) tools in the creation of this article.

Acknowledgments

The authors are most grateful to the editor Professor and anonymous referees for the careful reading of the manuscript and valuable suggestions that helped improve an earlier version of this article.

This work was supported by National Natural Science Foundations of China (No.12061062, 11661071).

Conflict of interest

All authors declare that they have no competing interests.