Energy minimizing solutions to slightly subcritical elliptic problems on nonconvex polygonal domains

1.   Introduction

In this paper we study asymptotic profile of energy minimizing solutions to the Lane-Emden-Fowler equation

as $\varepsilon > 0$ goes to zero. Here $\Omega \subset \mathbb{R}^n$ $(n \geq 3)$ is a bounded polygonal domain and $p = \frac{n+2}{n-2}$ is the critical exponent. In the seminar papers Han ^[1] and Rey ^[2], the asymptotic behavior of energy minimizing solutions to (1.1) was obtained for smooth bounded domains $\Omega$.

The asymptotic behavior was first studied by Atkinson and Peletier ^[3] when $\Omega$ is the unit ball in $\mathbb{R}^3$ using an ODE argument. The result was revisited by Brezis and Peletier ^[4] by applying PDE methods. Extensions to smooth bounded domains were obtained by Han ^[1] and Rey ^[2]. The asymptotic behavior have been studied by a lot of researchers for nonlinear elliptic equations with various settings (see e.g., ^{[5,6,7,8,9,10,11,12,13,14,15,16]}) and we note that most of the results have been obtained for elliptic problems on bounded smooth domains.

Pistoia and Rey ^[17] showed that as for problem (1.1) posed on a specific nonsmooth bounded domain constructed by Flucher-Garroni-Müller ^[18], the maximum point of $u_{ \varepsilon}$ may approach to the boundary point as $\varepsilon \rightarrow 0$. By the way, we mention that the arguments of Han ^[1] and Rey ^[2] work straight-forwardly for convex bounded domains, which may not be non-smooth. In fact a key part in the analysis of Han ^[1] and Rey ^[2] is that the maximum point of $u_{ \varepsilon}(x)$ is uniformly away from the boundary $\partial \Omega$ by showing that the solutions $u_{ \varepsilon}(x)$ are uniformly bounded for $\varepsilon > 0$ and $x$ near the boundary $\partial \Omega$ by the moving plane argument. If $\Omega$ is a smooth nonconvex domain, Han ^[1] obtained the unfirom boundedness by using the Kelvin transform to (1.1) on balls which touch the domain $\Omega$ by the boundary $\partial \Omega$. However, the argument is difficult to apply when $\Omega$ is not smooth.

Given this result, a natural question is that can we extend the result of Han ^[2] and Rey ^[2] to certain class of nonsmooth convex domains? In this paper, we show that the results of Han ^[1] and Rey ^[2] to nonconvex polygonal domains. The following is the main result of this paper.

Theorem 1.1. For $n \geq 3$ we let $\Omega \subset \mathbb{R}^n$ be a bounded polygonal domain. Assume that $\{u_{ \varepsilon}\}_{ \varepsilon > 0}$ is a set of solutions to (1.1) such that

where $S_n = \left[\pi n (n-2) \Gamma (n/2)/ \Gamma (n)\right]^{-1}$ is the best Sobolev constant in $\mathbb{R}^n$. Then the family of solutions $\{u_{ \varepsilon}\}_{ \varepsilon > 0}$ are uniformly bounded near the boundary, i.e., there are costants $\delta > 0$ and $C > 0$ independent of $\varepsilon > 0$ such that

Given the boundary estimates of Theorem 1.1, one may apply standard argument to deduce the following result ^[1].

Theorem 1.2. For $n \geq 3$ we let $\Omega \subset \mathbb{R}^n$ be a bounded polygonal domain. Assume that $\{u_{ \varepsilon}\}_{ \varepsilon > 0}$ is a set of solutions to (1.1) such that (1.2) holds. Then, there exists a point $x_0 \in \Omega$ such that, up to a subsequence,

● The solution $u_{ \varepsilon}$ converges to $0$ in $C^1 (\Omega \setminus \{x_0\})$.

● $\nabla R(x_0) = 0$, where $R(x) = H(x, x)$.

● We have

● We have

Here $G$ denotes the Green's function and $H$ is the regular part of $G$ (see Section 2 for the detail).

In order to prove Theorem 1.1 we assume that contrary that the maximum point $x_{ \varepsilon}$ approaches to the boundary. Under this assumption, we shall deduce a contradiction from the following Pohozaev type identity on an annulus centered at the blow up point; $1 \leq j \leq$n,

where $x_{ \varepsilon} \in \Omega$ is the maximum point of $u_{ \varepsilon}$ and $d_{ \varepsilon} = {\rm{dist}}(x_{ \varepsilon}, \partial \Omega_{ \varepsilon})/4$.

In fact we shall prove Theorem 1.1 for more general domain $\Omega$ satisfying the following assumption.

Assumption D. Consider a sequence of points $\{x^k\}_{k \in \mathbb{N}}$ in the domain $\Omega$ such that ${\bf d}_k: = {{dist}}(x^k, \partial \Omega)$ goes to zero as $k \rightarrow \infty$. Take $z^k \in \partial \Omega$ such that $|x^k -z^k| = {\bf d}_k$. Let $\Omega_k: = \frac{1}{{\bf d}_k} (\Omega -z^k)$. Note that we have $0 \in \Omega_k$, and also $\frac{1}{{\bf d}_k} (x^k - z^k) \in S^{n-1}$. Thus we can find a rotation $R_k : \mathbb{R}^n \rightarrow \mathbb{R}^n$ such that

Then, the domain $D_k : = R_k \Omega_k$ converges to an infinite star-shaped domain $\mathbb{P} \subsetneq \mathbb{R}^n$.

It is not difficult to see that any bounded polygonal domain $\Omega$ satisfies the above assumption. Under the above assumption we will obtain the following result on the regular part $H$ of the Green's function.

Theorem 1.3. For $n \geq 3$ we let $\Omega \subset \mathbb{R}^n$ be a bounded open domain satisfying Assumption D. Then, for any sequence of points $\{y^k\}_{k \geq 1}$ in $\Omega$ such that $\lim_{k \rightarrow \infty} {\bf d}_k = 0$, where ${\bf d}_k: = {{dist}}(y^k, \partial \Omega)$, there exists a constant $c > 0$ and $N \in \mathbb{N}$ such that, for $k \geq N$ we have

If $\Omega$ is smooth, then the result of Theorem 1.3 was proved in Rey ^[19] by applying the Maximum principle. To obtain the above inequality for the nonsmooth domains, we shall rescale the function ${H}$ in a suitable way and investigate its limit.

This paper is organized as follows. In Section 2, we are concerned about the properties of Green's function. Also we show that a sequence of the minimal energy solutions blows up as $\varepsilon \rightarrow 0$ and that the blow up point does not approach to the boundary too fast in some sense (see Lemma 2.2). In Section 3, we will obtain a sharp estimate of the function $u_{ \varepsilon}$ on an annulus centered at the blow up point. In Section 4, we prove Theorem 1.1. In Section 5, we give a proof of Theorem 1.2. Section 6 is devoted to prove Theorem 1.3.

Notations.

Here we list some notations which will be used throughout the paper.

- $C > 0$ is a generic constant that may vary from line to line.

- For $k \in \mathbb{N}$ we denote by $B^k (x_0, r)$ the ball $\{ x \in \mathbb{R}^{k}: |x-x_0| < r \}$ for each $x_0 \in \mathbb{R}^k$ and $r > 0$.

- For $x \in \Omega$ we denote by ${\rm{dist}}(x, \partial \Omega)$ the distance from $x$ to $\partial \Omega$ and we denote ${\bf d}(x): = {\rm{dist}}(x, \partial \Omega)$.

- For a domain $D \subset \mathbb{R}^n$, the map $\nu = (\nu_1, \cdots, \nu_n): \partial D \to \mathbb{R}^n$ denotes the outward pointing unit normal vector on $\partial D$.

- $dS$ stands for the surface measure.

- $|S^{n-1}| = 2\pi^{n/2}/\Gamma(n/2)$ denotes the Lebesgue measure of $(n-1)$-dimensional unit sphere $S^{n-1}$.

2.   Preliminary results

In this section we obtain preliminary results for a sequence of the solutions $\{u_{ \varepsilon}\}_{ \varepsilon > 0}$ satifsying (1.2). For this purpose, we first recall Green's function $G$ of the Laplacian $-\Delta$ on $\Omega$ with the Dirichlet boundary condition. It is divided into a singular part and a regular part as

where $c_n = {1}/{(n-2)|S^{n-1}|}$ and the regular part $H: \Omega \times \Omega \rightarrow \mathbb{R}$ is the function such that

Let ${\bf d}(x) = {\rm{dist}}(x, \partial \Omega)$ for $x \in \Omega$. Take a small constant $\delta > 0$.

We take a value $\lambda_{\epsilon} > 0$ and a point $x_{\epsilon} \in \Omega$ such that

Now we recall the sharp Sobolev embedding

If we replace the function $f$ by $(-\Delta)^{-1/2} f$ in the above inequality, we find the Hardy-Littlewood-Sobolev inequality:

We let $K$ denote Green's function of the Laplacain on $\mathbb{R}^n$, i.e.,

The estimate (2.5) is then written as

For given a domain $Q \subset \mathbb{R}^n$ we denote by $K_{Q}: Q \times Q \rightarrow \mathbb{R}$ Green's function of the Laplacian $(-\Delta)^{1/2}$ on domain $Q$ with the Dirichlet zero boundary condition, i.e., for the solution $u \in H^1 (\Omega)$ to the problem

with $f \in L^2 (\Omega)$ admits the representation

Then, it is a classical fact that for any open subset $Q \subset \mathbb{R}^n$ with $Q \neq \mathbb{R}^n$, we have

Here we remark that $(-\Delta)^{1/2}$ is defined by the spectral decomposition of $(-\Delta)$ on domain $\Omega$.

Lemma 2.1. The value $\lambda_{ \varepsilon} > 0$ defined in (2.3) satisfies $\lim_{ \varepsilon \rightarrow 0} \lambda_{ \varepsilon} = \infty$.

Proof. In order to prove the lemma, we assume the contrary. Then there is a subsequence $\{ \varepsilon_{k}\}_{k \in \mathbb{N}}$ such that $\lim_{k \rightarrow \infty} \varepsilon_k = 0$ and $\sup_{k \in \mathbb{N}} \lambda_{\epsilon_k} < \infty$. This implies that the solutions $\{u_{\epsilon_k}\}_{k \in \mathbb{N}}$ are uniformly bounded in $C^{1, \alpha}(\Omega)$ for some $\alpha \in (0, 1)$ by the standard regularity theory applied to (1.1). Up to a subsequence, the solution $u_{\epsilon_k}$ converges in $C^1 (\Omega)$ to a function $u_0 \in C^1 (\Omega)$, and taking $k \rightarrow \infty$ in the formula

we find

and so

On the other hand, by taking the limit $k \rightarrow \infty$ in (1.2) we get

Let us set $w_0 : \Omega \rightarrow \overline{\mathbb{R}_{+}}$ by $w_0 (x) = (-\Delta_{\Omega})^{1/2} u_0 (x)$ for $x \in \Omega$. Then $u_0 (x) = (-\Delta_{\Omega})^{-1/2} w_0 (x)$ for $x \in \Omega$ and so we have

We extend the function $w_0$ to set $W_0 : \mathbb{R}^n \rightarrow \overline{\mathbb{R}_{+}}$ by

Then, using the inequality (2.6) and (2.8) we obtain the following estimate

However, this contradicts to the optimality of the constant $S_n$ of the inequality (2.5). Therefore it should hold that $\lim_{\epsilon \rightarrow 0} \lambda_{\epsilon} = \infty$. The lemma is proved. □

For each $\varepsilon > 0$ we set $\Omega_{\epsilon}: = \lambda_{\epsilon}(\Omega-x_{\epsilon})$ and normalize the solution $u_{ \varepsilon}$ as follows

so that

and $\max_{x \in \Omega_{\epsilon}} \{ U_{\epsilon}(x)\} = 1 = U_{\epsilon}(0).$ In the next lemma, we obtain an estimate for the distance between the maximum point of the solutions and the boundary $\partial \Omega$.

Lemma 2.2. We have $\lim_{\epsilon \rightarrow 0} \lambda_{\epsilon} {\rm{dist}}(x_{\epsilon}, \partial \Omega) = \infty$.

Proof. We assume the contrary. Then, up to a subsequence, we have $\lim_{\epsilon \rightarrow 0} \lambda_{\epsilon} {\rm{dist}} (x_{\epsilon}, \partial \Omega) = l$ for some $l \in (0, \infty)$. This implies that the extended domain $\Omega_{\epsilon}$ converges to a infinite star-shaped domain $\mathbb{P}\subsetneq \mathbb{R}^n$ as $\varepsilon \rightarrow 0$. Also, the normalized functions $U_{\epsilon}$ converge to a nontrivial solution $\overline{U}$ in $C^2_{loc}(\mathbb{P})$ of the problem

and we know that $K_{\mathbb{P}} (x, y) < K(x, y)$ from (2.6). Then we can obtain a contradiction as in the proof of Lemma 2.1. Thus the result of the lemma is true. □

We set $d_{\epsilon} : = \frac{1}{4} {\rm{dist}}(x_{\epsilon}, \partial \Omega)$ and $N_{ \varepsilon} = d_{ \varepsilon} \lambda_{ \varepsilon}$. Then we see from Lemma 2.2 that

We remark that the fact $N_{ \varepsilon} \rightarrow \infty$ as $\varepsilon \rightarrow 0$ will be important in the proofs of Theorem 1.1. By Lemma 2.2 the domain $\Omega_{\epsilon}$ converges to $\mathbb{R}^n$ as $\epsilon$ goes to zero, and so the rescaled solution $U_{\epsilon}$ converges in $C_{loc}^2 (\mathbb{R}^n)$ to a solution $U$ of the problem

Then it is well-known that the function $U$ is equal to

where $\eta = \sqrt{n(n-2)}$. Next we recall the following result from Corollary 1 and Lemma 3 in ^[1].

Lemma 2.3 (^[1]). The value $\lambda_{ \varepsilon} > 0$ defined in (1.2) and the rescaled solution $U_{ \varepsilon}$ defined (2.9) satisfy the following.

(1) There is a constant $C > 0$ independent of $\epsilon > 0$ such that

(2) There exists a constant $C > 0$ such that

We end this section with a local version of the Pohozaev type identity for the problem (1.1).

Lemma 2.4. Let $1 \leq j \leq n.$ Suppose that $u \in C^2 (\Omega) \times C^2 (\Omega)$ is a solution of (1.1). Then, for any open smooth subset $D \subset \Omega$, we have the following identity.

where $D$ is an open subset of $\Omega$.

Proof. Multiplying (1.1) by $\frac{\partial u}{\partial x_j}$ we get $-\Delta u \frac{\partial u}{\partial x_j} = u^p \frac{\partial u}{\partial x_j}$. Integrating this over the domain $D$ and using an integration by part, we get

We use an integration by parts to get

The lemma is proved. □

3.   Estimates for $u_{ \varepsilon}$ on the annulus

This section is devoted to prove the following lemma regarding a sharp estimate for $u_{ \varepsilon}$ and its derivatives on the annulus $\partial B(x_{ \varepsilon}, 2d_{ \varepsilon})$.

Lemma 3.1. Assume that $\{u_{ \varepsilon}\}_{ \varepsilon > 0}$ is a sequence of solutions to (1.1) of type $(ME)$ and that $\lim_{ \varepsilon \rightarrow 0} d_{ \varepsilon} = 0$. Then, for $x \in \partial B(x_{ \varepsilon}, 2d_{ \varepsilon})$ we have the estimates

and

Here the value $A_U$ is defined as

In addition, the $o$-notation is uniform with respect to $x \in \partial B(x_{ \varepsilon}, 2d_{ \varepsilon})$, i.e., it holds that

Proof. Since $u_{\epsilon}$ is a solution to (1.1), we have

Given the estimate (2.14) we apply the dominated convergence theorem to find

Using this and noting that $G(x, x_{ \varepsilon}) = O(|x-x_{ \varepsilon}|^{-(n-2)}) = O (d_{ \varepsilon}^{-(n-2)})$ for $x \in \partial B(x_{ \varepsilon}, 2d_{ \varepsilon})$, we find

where we also used that

due to the fact that $\frac{2}{p-1} = \frac{n}{p+1}$ and (2.13). Similarly, we may deduce

Therefore, in order to prove (3.1), we only need to estimate the last term of (3.4) as $o (d_{ \varepsilon}^{-(n-2)} \lambda_{ \varepsilon}^{-\frac{n}{p+1}})$ and its derivatives as $o (d_{ \varepsilon}^{-(n-1)} \lambda_{ \varepsilon}^{-\frac{n}{p+1}})$. For this aim, we decompose it into three parts as follows:

where

We shall show that $I_1 (x)$, $I_2 (x)$, and $I_3 (x)$ are estimated as $o \left(d_{ \varepsilon}^{-(n-2)} \lambda_{ \varepsilon}^{-\frac{n}{p+1}}\right)$ and their derivatives $\nabla I_1 (x)$, $\nabla I_2 (x)$, and $\nabla I_3 (x)$ are estimated as $o \left(d_{ \varepsilon}^{-(n-1)} \lambda_{ \varepsilon}^{-\frac{n}{p+1}}\right)$.

Estimate of $I_1$. Since $x \in \partial B(x_{ \varepsilon}, 2d_{ \varepsilon})$, we have $|x-y| \geq d_{\epsilon}$ for $y \; \in B(x_{\epsilon}, d_{\epsilon})$, and so

Combining this with the mean value formula yields

for all $y \in B(x_{ \varepsilon}, d_{ \varepsilon})$. Applying this and (2.14) we may estimate $I_1$ as follows:

Using (2.13) and that $\frac{2p}{p-1} - (n+1) < - \frac{n}{p+1}$ we find that $I_1 (x) = o (d_{ \varepsilon}^{-(n-2)} \lambda_{ \varepsilon}^{-\frac{n}{p+1}}).$ By the same way along with the second inequality of (3.7), we can obtain the estimate

Estimate of $I_2$. For $y \in B(x_{\epsilon}, 4d_{\epsilon}) \setminus B(x_{\epsilon}, d_{\epsilon})$ we use the estimate (2.14) and (2.13) to find

Noting that

we have

Combining the first estimate of (3.11), (3.10) and (3.9) in (3.6) yields

Due to the fact that $p = \frac{n+2}{n-2}$ the above estimate gives the estimate $I_2 (x) = o\left(\lambda_{ \varepsilon}^{-\frac{n}{p+1}}d_{ \varepsilon}^{-(n-2)}\right).$ Similarly, using the second estimate of (3.11), we obtain

Estimate of $I_3$. Since $|x-x_{\epsilon}| = 2d_{\epsilon}$, we have the following estimates

Applying the first inequality of (3.12), we have

Using (2.14) we deduce

Combining the above two estimates, we arrive at the following estimate

where the fact that $(n-2)p > n$ was also used. Similarly, using the second estimate of (3.12), then we get

Finally, gathering the above estimates on $I_1$, $I_2$, and $I_3$, we finally get

and

The lemma is proved. □

4.   The proof of Theorem 1.1

This section is devoted to prove Theorem 1.1.

Proof of Theorem 1.1. Let $d_{\epsilon} = {\rm{dist}}(x_{\epsilon}, \partial \Omega)$. In view of (2.9), it is enough to show that $\inf_{ \varepsilon > 0} d_{ \varepsilon} > 0$. For this purpose, with a view to a contradiction, we assume the contrary that $d_{\epsilon} \rightarrow 0$ as $\epsilon \rightarrow 0$ in a subsequence.

We use the notation $\lambda_{ \varepsilon}$ and $N_{ \varepsilon} = {d_{ \varepsilon}}{\lambda_{ \varepsilon}}$ defined in (2.3) and (2.11). Then we recall from Lemma 2.2 that we have $N_{\epsilon} \rightarrow \infty$. Let us set $D_{\epsilon} = B(x_{\epsilon}, 2d_{\epsilon})$ for each $1 \leq j \leq n$ and we define the values $L_{\epsilon}^j$ and $R_{\epsilon}^j$ by

Applying Lemma 2.4 to $u_{ \varepsilon}$ with $D = D_{\epsilon}$, we find that

In what follows, we proceed to obtain sharp estimates of the values of $L_j^{\epsilon}$ and $R_j^{\epsilon}$, which will lead to a contradiction.

First, we compute $L_{\epsilon}^j$ using the expression (3.2) as follows.

where we have set

In order to compute the value of $I(2d_{\epsilon})$, we first notice that $I(r)$ is independent of $r > 0$. Indeed, it follows from that $-\Delta_x G(x, x_{\epsilon}) = 0$ for $x \in A_{r}: = B(x_{ \varepsilon}, 2d_{ \varepsilon})\setminus B(x_{ \varepsilon}, r)$ for each $r \in (0, 2d_{ \varepsilon})$, and an integration by parts performed as follows:

which means that $I(r)$ is constant on $(0, 2d_{\epsilon}]$. Therefore we can evaluate $I(2d_{ \varepsilon})$ by computing the following limit;

Thanks to the oddness of the integrand, we have

Also, since $-\Delta_x H(x, x_{ \varepsilon}) = 0$ holds for $x \in B(x_{ \varepsilon}, 2d_{ \varepsilon})$, we may proceed as in (4.2) to get

Using the above estimates, we complete the estimation as follows.

Plugging this into (4.1) shows that

Now, we take $j \in \{1, \cdots, n\}$ such that $\left| \frac{\partial H}{\partial x_j}(x_{ \varepsilon}, x_{ \varepsilon}) \right| \geq \frac{C}{d_{ \varepsilon}^{n-1}}$, which is guaranteed by Theorem 1.3. Injecting this into (4.3) we have

Next we shall find an upper bound of $R_j^{\epsilon}$. Applying (2.14) we have

Using this we estimate

which yields

Now we combine (4.4) and (4.6) to get

Since $N_{\epsilon}$ goes to infinity as $\epsilon \rightarrow 0$, the above inequality yields that

which is equivalent to $p \leq \frac{n}{n-2}$. However this contradicts to the fact that $p = \frac{n+2}{n-2}$. Thus the assumption $d_{\epsilon} \rightarrow 0$ cannot hold, and so $\inf_{ \varepsilon > 0} d_{ \varepsilon} > 0$. The proof is completed. □

5.   Proof of Theorem 1.2

In this section we provide a proof of Theorem 1.2.

Proof of Theorem 1.1. From the result of Theorem 1.1, we know that the maximum point $x_{ \varepsilon}$ of the solution $u_{ \varepsilon}$ are uniformly away from the boundary $\partial \Omega$. Therefore, up to a subsequence, the point $x_{ \varepsilon}$ converges to an interior point $x_0 \in \Omega$. By Lemma 2.3 we know the first statement of the theorem holds.

We may easily deduce the version of Lemma 3.1 under the assumption that $x_{ \varepsilon}$ converges to an interior point $x_0$. Indeed, it is direct to deduce from (3.4) that

for $x \in \Omega \setminus \{x_0\}$. Thus we have

Proposition 5.1. We have $\lim_{ \varepsilon \rightarrow 0} \lambda_{ \varepsilon}^{ \varepsilon} = 1$.

Proof. Let $v_{ \varepsilon} = (x-x_0) \cdot \nabla u_{ \varepsilon} + ({2 \over p-1- \varepsilon_n}) u_{ \varepsilon}$. Then it satisfies

Therefore we have

By (5.1) we have

Taking $\varepsilon \rightarrow 0$ we have

where the last equality is derived in [1, pg. 170]. The right hand side of (5.2) is equal to

Using this we have

Injecting (5.3) and (5.4) into (5.2) we get

This implies that $\lambda_{ \varepsilon} \leq C \varepsilon^{-\frac{2n}{p-1- \varepsilon}} \leq C \varepsilon^{-\frac{4n}{p-1}}$ for any small $\varepsilon > 0$. Therefore we have

The lemma is proved. □

Given the result of Proposition 5.1, we deduce from (5.1) that

where we used $p = \frac{n+2}{n-2}$ in the first equality. This proves the third statement of Theorem 1.2. Next, taking $D = B(x_0, r)$ and $u = u_{ \varepsilon}$ in (2.15), we have

By (2.9) we have $u_{ \varepsilon} (x) \leq \lambda_{ \varepsilon}^{-\frac{(n-2)}{2}}$ for $x \in \partial B(x_0, r)$ we have

Using this and (5.1) we take limit $\varepsilon \rightarrow 0$ in (5.7) to get

which yields the second statement of the theorem. Finally, given the result of Proposition 5.1, we get from (5.5) that

This proves the last statement of the theorem. The proof is finished. □

6.   The proof of Theorem 1.3

We prove the second main theorem of this paper.

Proof of Theorem 1.3. Consider a sequence of points $\{x^k\}_{k \in \mathbb{N}}$ in the domain $\Omega$ such that ${\bf d}_k: = {\rm{dist}}(x^k, \partial \Omega)$ goes to zero as $k \rightarrow \infty$. Take $z^k \in \partial \Omega$ such that $|x^k -z^k| = {\bf d}_k$. Let $\Omega_k: = \frac{1}{{\bf d}_k} (\Omega -z^k)$. Note that we have $0 \in \Omega_k$, and also $\frac{1}{{\bf d}_k} (x^k - z^k) \in S^{n-1}$. Thus we can find a rotation $R_k : \mathbb{R}^n \rightarrow \mathbb{R}^n$ such that

Then, by Assumption D. the domain $D_k : = R_k \Omega_k$ converges to an infinite star-shaped domain $\mathbb{P} \subsetneq \mathbb{R}^n$. To prove the estimate (1.4) we set the function $W_{k} : D_{k} \rightarrow \mathbb{R}$ for each $k \in \mathbb{N}$ by

Let $G_{k}$ be Green's function of $-\Delta$ on $D_k$ with the Dirichlet boundary condition. For each $y \in \mathbb{R}^{n}_{+}$ we denote $y^* = (y_1, \cdots, y_{n-1}, -y_n)$ for $y = (y_1, \cdots, y_n) \in \mathbb{R}^{n}_{+}$. We consider the function ${H}_0:\overline{\mathbb{P}} \times \overline{\mathbb{P}} \rightarrow \mathbb{R}$ satisfying

Here $c_n$ is the value defined in (2.1). Now we set $W_0 : \overline{\mathbb{P}} \rightarrow \mathbb{R}$ by $W_0 (z) : = {H}_0 (z, e_n)$. Then we have the following result.

Lemma 6.1. As ${{\bf d}_k} \rightarrow 0$, the function $W_{k}$ converges to $W_0$ in $C^1 (B(e_n, 1/4))$.

Proof. By definition (6.2) and (2.2), the function $W_{k}$ satisfies

Set the difference $R_{k} : \Omega_{k} \rightarrow \mathbb{R}$ by $R_{k} (x) = W_0 (x)- W_{k} (x)$ for $x \in \Omega_{k}$. Then, it suffices to show that $R_{k} \rightarrow 0$ in $C_{loc}^1 (\mathbb{P})$. By (6.4) and (6.3) we have

Let us prove the $C^0$ convergence of $\mathcal{R}_{k}$. Since $\mathcal{R}_{k}$ is harmonic in $\Omega_{k}$, we only need to show that

Take a large number $R > 0$. Then we have

We note that for $z\in \partial D_k$, using (6.1) we have

and for $z \in \partial \mathbb{P}$,

For fixed $R > 0$, we have

and we note that $\partial D_k \cap B_R$ is compact. Combining this fact with (6.6) and (6.7), we obtain

Thus,

Since $R > 0$ is arbitrary, we have

Combining the above two convergence results, we can deduce that $R_{k} (x) \rightarrow 0$ uniformly for $x \in B(e_n, 1/4)$. From (6.4) we know that $R_{k}$ is contained in $C^{1, \beta} (B(e_n, 1/4))$ uniformly in $k \in \mathbb{N}$ for some $\beta > 0$. Thus $R_{k}$ converges to a function $f$ in $C^1 (B(e_n, 1/4))$. In this paper we are concerned with the Lane-Emden-Fowler equation

where $\Omega \subset \mathbb{R}^n$ ($n \geq 3$) is a polygonal domain and $\varepsilon > 0$. We study the asymptotic behavior of minimal energy solutions as $\varepsilon > 0$ goes to zero. we have $f\equiv 0$ since $R_{k}$ converges to $0$ in $C^0 (B(e_n, 1/2))$. The lemma is proved. □

Lemma 6.2. We have $\frac{\partial}{\partial x_n} W_0 (e_n) \neq 0$.

Proof. Notice that $H_0 (x, y)$ satisfies

Since $H_0$ is the regular part of Green's function on $\mathbb{P}$, we have

For given $t > 0$ consider the function $f (x): = t^{n-2} H_0 (tx, te_n)$ defined on $\frac{1}{t} \mathbb{P} = \mathbb{P}$ which satisfies

This exaclty means that $f (x) = H_0 (x, e_n)$, and so $H_0 (x, e_n) = t^{n-2} H_0 (tx, t e_n)$. Combining this with the symmetric property (6.9), we have

Also we note that $H_0 (e_n, e_n) \neq 0$ by the maximum principle since $(-\Delta)H_0 = 0$ in $\mathbb{P}$ and $H_0 > 0$ on $\partial \mathbb{P}$. Combining this fact with (6.10) we deduce that $\frac{\partial}{\partial x_n} W_{0}(e_n) < 0$. The proof is finished. □

Now we are ready to finish the proof of Theorem 1.3. By Lemma 6.1, we know that $W_k (x)$ converges to $W_0 (x)$ in $C^1 (B(e_n, 1/4))$. Since $\left|\frac{\partial}{\partial x_n} W_0 (e_n)\right| > c > 0$, we conclude that for large $k \in \mathbb{N}$, we have $\left|\frac{\partial}{\partial x_n}W_k (e_n)\right| > c/2$. By definition of $W_k$ given in (6.2), we have

Therefore we may conclude that for large $k \in \mathbb{N}$,

which implies that

for $k \in \mathbb{N}$ large enough. The proof is finished. □

7.   Conclusions

In this paper, we study the energy minimizing solutions to slightly subcritical elliptic problems on nonconvex polygonal domains. The main part for the analysis is to exclude the possibility that the peak of the solution approaches the boundary of the domain as the moving plane method is difficult to apply directly for the nonconvex polygonal domain. To address this challenge, we make use of the Pohozaev identity and the Green's function to show that a contradiction aries when we assume that the solution blows up near the boundary.

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 the National Research Foundation of Korea (grant NRF-2021R1F1A1059671).

Conflict of interest

The authors declare no conflict of interest.