Positive radial solutions for the problem with Minkowski-curvature operator on an exterior domain

1.   Introduction

In this paper, we are concerned with the problem with Minkowski-curvature operator on an exterior domain

where $0\leq\gamma < 1$, $B^c = \{x\in \mathbb{R}^N: |x| > R\}$ is a exterior domain in $\mathbb{R}^N$, $N > 2$, $R > 0$, the function $f:[0, \infty)\to (0, \infty)$ is a continuous function such that $\lim\limits_{s\to\infty}\frac{f(s)}{s^{\gamma+1}} = 0$ and $\lambda > 0$ is a parameter. Assume that

(K1) $K\in C([R, \infty), (0, \infty))$ is such that $\int_R^\infty r K(r)dr < \infty$.

The prescribed mean curvature equation

with some boundary value condition is related to problems on a flat Minkowski space with a Lorentzian metric in differential geometry and the theory of classical relativity, see R. Bartnik and L. Simon ^[1], C. Gerhardt ^[6] and A. E. Treibergs ^[15].

If $\Omega$ is a strictly convex bounded domain in $\mathbb{R}^N$ with $C^2$ boundary $\partial\Omega$, there are some classic papers on the existence of solutions of Eq (1.2) with Dirichlet, Neuuman or periodic boundary conditions using the fixed point theory or topological methods (mostly bifurcation technique), see Bereanu, Jebelean and Mawhin ^[2,3], Bereanu, Jebelean and Torres ^[4], Ma, Gao and Lu ^[10], Obersnel, Omari and Rivetti ^[12] and the references therein.

The presence in the existing literature of very few results about the (1.2) in a exterior domain is in sharp contrast with the wide number of works that are available in the bounded domains {setting}. The likely reasons are that the singular coefficient and singular weight will occur in the problems considered on the exterior of a ball and the concave-convex properties of solutions are uncertain owing to the influence of the mean curvature operator and coefficient function, see ^[16].

Yang, Lee and Sim ^[16] concerned with the existence of nodal radial solutions of the following problem

where $g: \mathbb{R}\to\mathbb{R}$ a continuous and odd function satisfying $g(s)s > 0$ for $s\neq0$. As the main assumption on the nonlinearity they required $g(0) = 0$, which guarantees that $0$ is the trivial solution of the problem (1.3). Therefore, the bifurcation from the trivial solution can be considered in ^[16].

Although Yang, Lee and Sim ^[16] obtained the existence result of the radial solutions of the problem with mean curvature operator on the exterior domain of a ball; it is worth noticing, however, that the application to problem (1.1) of the method described in ^[16] is not feasible as nonlinearity in (1.1) is singular at $0$.

Existence of positive radial solutions of the prescribed mean curvature problem (1.1) with singular nonlinearity on the exterior domain of a ball have not been introduced yet as far as the authors know.

The novelty of this paper is twofold: we develop a method of sub-super solutions for a singular problem with mean curvature operator, and provide a existence result of the positive radial solutions for problem (1.1) which is a prescribed mean curvature problem with singular nonlinearity on the exterior domain of a ball.

In order to study the radial solutions of (1.1), we transform problem (1.1) into the one-dimensional problem via consecutive transformations $r = |x|$ and $t = (\frac{r}{R})^{-(N-2)}$ as follows

where $\varphi(y) = \frac{y}{\sqrt{1-y^2}}$ with $y\in(-1, 1)$, $p$ and $h$ can be obtained as

Notice that $p(t) > 0, p'(t)\leq0, \ t\in[0, 1]$ and

since $K$ satisfies $\int^\infty_R rK(r)dr < \infty$ (see ^[16]). We assume that

(F1) $f(s) > 0$ for all $s\geq0$;

(F2)$\lim\limits_{s\to \infty}\frac{f(s)}{s^{\gamma+1}} = 0$.

Theorem 1.1. Let (F1), (F2) and (K1) hold. Then problem (1.1) has at least one positive radial solution for all $\lambda > 0$.

Example 1.1. Consider the problem (1.1) with

and

Obviously, $\lim\limits_{s\to0}\frac{f(s)}{s^\gamma} = \infty$, i.e. nonlinearity $\frac{f(s)}{s^\gamma}$ is singular at $0$. It is easy to check that $K$ and $f$ satisfies

and $(F1)$. According to Theorem 1.1, the problem (1.1) has at least one positive radial solution for all $\lambda > 0$.

As a by-product of the proof of Theorem 1.1, we also proved the following theorem.

Theorem 1.2. Assume that $h\in H$. Then for any fixed $M > 0$, the problem

have a unique positive solution $w\in C([0, 1], [0, \infty))\cap C^1((0, 1], \mathbb{R})$.

Remark 1.1. For the existence of solutions of the problems with a elliptic operator on an unbounded domain, see Dai et al. ^[5], Iaia ^[7,8], Ko et al. ^[9] and Ma et al. ^[11] and the references therein.

2.   An auxiliary problem

Motivated by ^[13], we consider the perturbation problem

where $n\geq0$ is any constant, $M > 0$ is a fixed constant, $p, h$ are defined by (1.5).

Lemma 2.1. For each fixed constant $n\geq0$, problem $(2.1)_n$ has at most one positive solution.

Proof. Assume on the contrary that the functions $u_1$ and $u_2$ are all the positive solutions of $(2.1)_n$ and $u_1\not\equiv u_2$ on $[0, 1]$. Then there exists $t_0\in(0, 1]$ such that $u_1(t_0)\neq u_2(t_0)$. Without loss of generality, we assume that $u_1(t_0) > u_2(t_0)$. Then there exists an interval $[a, b]\subset[0, 1]$ such that $b$ is the point of positive maximum of $u_1(t)-u_2(t)$ on $[a, b]$ and

Integrating both sides of the equation in $(2.1)_n$ over $[t, b]\subset[a, b]$, we can obtain

Integrating both sides of the above equalities from $a$ to $b$, we have

By a simple calculation, we can obtain

This is a contradiction. The proof of Lemma 2.1 is complete.□

Let

Now, we consider the following problem

where $v\in D_n$.

Lemma 2.2. For each fixed $n > 0$ and each $v\in D_n$, $(2.2)_n$ has a unique solution $u\in C[0, 1]\cap C^1(0, 1]$ satisfying $u(t)\geq n$ on $[0, 1]$.

Proof. Define $A: D_n\to D_n$ by

Obviously, $\varphi^{-1}(y) = \frac{y}{\sqrt{1+y^2}}$ and $\varphi^{-1}(y)\leq y$ for any $y\in \mathbb{R}^+$. Then for any $v\in D_n$,

and $(Av)(t)$ is continuous on $[0, 1]$. This suggests that $Av\in D_n$ and $A$ is well defined.

Let $u = Av$, then $u\in C[0, 1]\cap C^1(0, 1]$, $u(0) = n, u'(1) = 0$ and

It is easy to verify

Therefore, $u(t)$ is a solution of problem $(2.2)_n$. The proof of the uniqueness of solution of $(2.2)_n$ is similar to that of Lemma 2.1.□

From the definition of mapping $A$, we can easily get the following lemma.

Lemma 2.3. Let $A:D_n\to D_n$ be the mapping defined by $(2.3)_n$. Then, for any $v_1, v_2 \in D_n$ with $v_1(t)\leq v_2(t)$ on $[0, 1]$,

We can verify that $A:D_n\to D_n$ is a compact continuous mapping. For any fixed $n > 0$, the problem $(2.1)_n$ has at least one solution $u(t, n)\geq n, t\in[0, 1]$, by Schauder fixed point theorem, see ^[13]. Furthermore, the uniqueness of the solution $u(t, n)$ of $(2.2)_n$ is guaranteed by Lemma 2.1. Applying the method of similarity to the proof of Lemma 2.1, we can obtain the following result.

Lemma 2.4. Let $u(t, n)$ be the unique solution of $(2.1)_n$. Then for $n_1 > n_2 > 0$, we have

The proof of Theorem 1.2. Let $\{n_j\}_{j = 1}^\infty$ be a decreasing sequence which converges to $0$. We known that $(2.1)_{n_j}$ has a unique solution $u(t, n_j)$. From Lemma 2.4, for each $j < k$,

Then there exists $u\in C[0, 1]$ such that

Claim. For any $t\in (0, 1]$, $u(t) > 0$.

Let $\delta = \max\limits_{0\leq t\leq 1}u(t, 1)$. From (2.4), we have

This suggests that for any $j = 1, 2, \cdots$

Passing to the limit, we have

From the Monotone convergence theorem (see ^[14]),

It is easy to verify that $u$ is a solution of $(2.1)_0$, i.e., $u$ is a solution of (1.6). The uniqueness of the solution $u$ is guaranteed by Lemma 2.1. □

3.   The method of sub-super solutions

In this section, we will develop a method of sub-super solutions for (1.4) which is a singular problem with mean curvature operator. By a subsolution of (1.4) we mean a function $\alpha\in C[0, 1]\cap C^1(0, 1]$ such that

and by a supersolution of (1.4) we mean a function $\beta\in C[0, 1]\cap C^1(0, 1]$ such that

Theorem 3.1. Suppose there exist a subsolution $\alpha$ and a supersolution $\beta$ of $(1.4)$ such that $0 < \alpha\leq\beta$ on $(0, 1)$, then $(1.4)$ has at least one positive solution $u$ satisfying $\alpha\leq u \leq \beta$ on $[0, 1]$.

Proof. Step 1. Construction of a modified problem. Take a sequence of subintervals of $(0, 1)$, say $\{I_j\}_{j = 1}^\infty$, such that

and $\bigcup\limits_{j = 1}^\infty I_j = (0, 1)$. For any $j = 1, 2, \cdots$, we are consider the problem

Let $\alpha_j = \min_{\bar I_j}\alpha$ and $\beta_j = \max_{\bar I_j}\beta$. Define $\bar g_j:I_j\times(0, \infty)\to \mathbb{R}^+$ by

Then we consider the modified problem

Obviously the restrictions of the functions $\alpha$ and $\beta$ on $I_j$ are the subsolution and supersolution of $(3.4)_j$, respectively. In other words, for any $t\in I_j$, $\alpha_j$ and $\beta_j$ satisfy

and

Step 2. Every solution $u_j$ of $(3.4)_j$ satisfies $\alpha\leq u_j \leq \beta$ on $I_j$. Let $w_j = (u_j-\alpha_j)\in C^1(\bar I_j)$ and $I^-_j = \{t\in I_j: u_j(t)\leq\alpha_j(t)\}$. By a calculation, we obtain

and

Summing up we obtain

The strict monotonicity of the function $y\mapsto \frac{y/p}{\sqrt{1-|y/p|^2}}$ yields $(u_j-\alpha_j)' = 0$ a.e. in $I^-_j$. Hence, $(u_j-\alpha_j)^- = 0$ and $u_j\geq\alpha_j$ on $I_j$. In a completely similar way we can obtain that $u_j\leq\beta_j$ on $I_j$.

Step 3. Problem $(3.3)_j$ has at least one solution $u_j$, with $\alpha\leq u_j \leq \beta$ on $I_j$. Define an operator $\mathcal{T}: C^1(\bar I_j)\to C^1(\bar I_j)$ which sends any function $v\in C^1(\bar I_j)$ onto the unique solution $u\in C^1(\bar I_j)$, of the problem

Similar to Section 2, the operator $\mathcal{T}$ is completely continuous. Clearly, $u_j$ is a solution of $(3.5)_j$ if and only if $u_j$ is a fixed point of $\mathcal{T}$. Since for any $t\in I_j\subset\subset(0, 1)$, there exist the constant $\varepsilon_2 > \varepsilon_1 > 0$ and $c = c(I_j, \varepsilon_1, \varepsilon_2) > 0$ such that $\varepsilon_1 < \min_{I_j}\alpha < \max_{I_j}\beta < \varepsilon_2$ and

and then

where $\mathcal{I}$ is the identity operator. By Step 2 we know that $u_j$ satisfies $\alpha\leq u_j\leq\beta$ on $I_j$ and hence it is a solution of $(3.4)_j$ as well.

Step 4. Complete the proof of result. We claim that, for fixed $k$, there exists $C_k > 0$ such that $\|u_j\|_{C^2(\bar I_k)}\leq C_k$ for all $j\geq k+1$. In fact, take $J_k$ such that $I_k\subset\subset J_k\subset\subset I_{k+1}$. Define $g_j(t) = h(t)\frac{f(u_j(t))}{(u_j(t))^\gamma}$. Then

Since $\{u_j\}_{j\geq k+1}$ are uniformly bounded on $\bar I_{k+1}$, we know that there exists $c_k > 0$ such that

This suggests that there exists a constant $C_k > 0$ such that $\|u_j\|_{C^2(\bar I_k)}\leq C_k$ for all $j\geq k+1$.

Since the embedding $C^2(\bar I_k)\hookrightarrow C^1(\bar I_k)$ is compact, for each $k$, sequence $\{u_j\}_{j = 1}^\infty$ has a subsequence, renamed $\{u_j\}^\infty_{j = 1}$, which converges to $u$ in $C^1(\bar I_k)$. This implies that $u\in C^1(\bar I_k)$ for every $k$. Consequently, $u\in C^1(0, 1)$. For any $k$, we have

for all $\phi\in C_0^\infty(I_k)$ and $j\geq k+1$. By taking the limit of the sequence converging in $C^1(\bar I_k)$,

for all $\phi\in C_0^\infty(I_k)$. Also, since $\alpha(t)\leq u_j(t)\leq\beta(t)$ for all $j$, we have $\alpha(t)\leq u(t)\leq \beta(t)$ for all $t\in(0, 1)$. Thus, from $\alpha(0) = \beta(0) = 0$ and $\alpha'(1) = \beta'(1) = 0$, we know that $u\in C[0, 1]\cap C^1(0, 1]$. This completes the proof of Theorem 3.1. □

4.   The proof of Theorem 1.1

Firstly, we construct a positive supersolution $\beta$ of $(1.4)$. Let $f^*(s) = \max\limits_{0\leq r\leq s}f(r)$. Obviously, $f^*(s)$ is nondecreasing and

since $\frac{f(s)}{s^{\gamma+1}}\to 0$ as $s\to\infty$. Then there exists a constant $M_\lambda\gg1$ such that

where $w$ is the unique positive solution of (1.6) with $M_\lambda$. Then

This suggests that $w$ is a positive supersolution of (1.4).

Next we construct a positive subsolution $\alpha$ of (1.4). For any fixed constant $m > 0$, the problem

has a solution $v_m$. We can verify that $v_m\to0$ as $m\to 0$. Since $\frac{f(s)}{s^\gamma}\to\infty$ as $s\to0$, for any fixed $\lambda > 0$, there exists a sufficiently small $m_\lambda \ll 1$ such that

where $v$ is the solution of $(4.1)_{m_\lambda}$. Then

This suggests that $v$ is a positive subsolution of (1.4) such that $v\leq w$ for sufficiently small $m_\lambda$.

From Theorem 3.1, (1.4) has a positive solution for all $\lambda > 0$. This completes the proof of Theorem 1.1.

Use of AI tools declaration

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

Acknowledgments

This work was supported by National Natural Science Foundation of China (No.12061064) and Shaanxi Fundamental Science Research Project for Mathematics and Physics (Grant No. 22JSY018).

Conflict of interest

All authors declare no conflicts of interest in this paper.