Nonradial singular solutions for elliptic equations with exponential nonlinearity

1.   Introduction

We are interested in singular solutions of the following equation with exponential nonlinearity:

where $R > 0$ and $B_R = \{x \in \mathbb{R}^N \; (N \geq 3): \; |x| < R\}$ is a ball.

By a singular solution of (1.1) we mean that $u \in C^2 (B_R \backslash \{0\})$ and 0 is a nonremovable singular point of $u$.

It is easily known that (1.1) admits a (trivial) radial singular solution:

We are mainly concerned with nonradial singular solutions of (1.1) in this paper.

When $N = 2$, by using the moving plane method, the authors of ^[1] proved that every solution of

has the form

For $n > 0$, symmetry and uniqueness results were obtained in ^[2] for the solutions of the following problem:

If $n = 1$, problem (1.4) reduces to (1.3); also, classification of solutions of (1.4) can be found in ^[1]. Under the condition that $n \geq 2$ is an integer, the authors of ^[2] showed that problem (1.4) admits radial and nonradial solutions, but, when $n > 0$ is not an integer, problem (1.4) only has radial solutions. Note that, for each $n > 0$, if $u(x)$ is a solution of (1.4), we can perform the following transformation:

we see that $v(x)$ satisfies the following equation

The results in ^[2] imply that (1.5) admits a family of radial and nonradial singular solutions.

The asymptotic behavior of singular solutions of the problem given by

where $D_1 \subset \mathbb{R}^2$ is the unit disc, was studied in ^[3]. The authors of ^[3] obtained that if $u \in C^2 (D_1 \backslash \{0\})$ is a singular solution of (1.6), then there is $\alpha > -2$ such that

In a recent paper ^[4], the authors continued the study in ^[3] and obtained asymptotic expansions up to arbitrary orders for $u(x)$ as $|x| \to 0$.

Under the condition that $N \geq 2$, the structure of finite Morse index solutions of the equation

was studied in ^[5,6,7]. In particular, under the condition that $N = 3$, the asymptotic behavior at $x = 0$ of solutions $u$ with $|x|^2 e^u \in L^\infty (\mathbb{R}^3)$ of the equation

was classified in ^[8]. For the case that $N = 3$, if we write

where $(|x|, \theta) \in (0, \infty) \times S^2$ denotes the spherical coordinates in $\mathbb{R}^3 \backslash \{0\}$, we find that $\Theta (\theta)$ must satisfy

on $S^2$ where $\Delta_{\mathbb{S}^{2}}$ is the Laplace-Beltrami operator on $(S^2, g_0)$ and $g_0$ is the standard round metric. It means that the Gaussian curvature of the metric $g = e^{\Theta}g_0$ on $S^2$ is $\frac{1}{2}$. This and related equations have been studied for more than three decades. Chang and Yang^[9] and Onofri^[10] described all regular solutions of (1.9). Specifically, axially symmetric solutions of (1.9) can be written explicitly as $\Theta (\theta) = \log 2- 2 \log(\sqrt{c^2+1}-c \cos \theta)$, where $c \in \mathbb{R}$ is constant and $\theta \in [0, \pi]$ is the geodesic distance from the north pole of $S^2$. Hence,

is a one-parameter family of non-radial singular solutions of (1.8).

Recently, singular solutions in different settings have also been studied in ^[11] and ^[12]. The authors of ^[12] obtained the existence and asymptotic behavior of singular solutions to quasilinear elliptic inequalities with nonlocal terms. Moreover, by using mini-max and asymptotic approximation methods, the existence of positive singular solutions to the planar logarithmic Choquard equation with exponential nonlinearity was established in ^[11].

In this paper, we study singular solutions of (1.1) in $B_R \subset \mathbb{R}^N \; (N \geq 3)$. We are interested in not only the asymptotic behavior of singular solutions of (1.1) at $x = 0$, but also the existence of nonradial singular solutions of (1.1). The structure of nonradial singular solutions of the equation

remains largely open. Motivated by the main ideas in ^[13], the authors of ^[14,15] obtained infinitely many nonradial singular solutions of (1.10) of the following form given $4 \leq N \leq 10$:

where $\Theta (\theta)$ is a non-constant solution of the equation

on $S^{N-1}$, where $\Delta_{\mathbb{S}^{N-1}}$ is the Laplace-Beltrami operator on $(S^{N-1}, g_0)$. They constructed infinitely many axially symmetric non-constant classical solutions of (1.12). The only singular solutions to (1.10) known so far are the (trivial) radial singular solution $U_s (x)$ and the solutions given in (1.11). It is clear that they are also the singular solutions to (1.1).

We will construct a new type of singular solutions of (1.1) in the following form:

for some $\epsilon > 0$.

Our main result is as follows.

Theorem 1.1. For any $R > 0$, Eq (1.1) admits infinitely many nonradial singular solutions $u(x)$ of the following form:

where

It is known from Theorem 1.1 that the parameter $\epsilon$ in (1.13) is $\sigma_+^{(2)}$.

To prove Theorem 1.1, we firstly study the detailed asymptotic behavior at $x = 0$ of a prescribed singular solution $u$ of

with the form (1.13), where $B: = B_1 = \{x \in \mathbb{R}^N \; (N \geq 3): \; |x| < 1\}$ is the unit ball. Then, the infinitely many nonradial singular solutions of the form (1.14) can be constructed by using the asymptotic expansion and introducing suitable Hölder spaces.

This paper is organized as follows. In Section 2, we obtain asymptotic expansions near the isolated singular point $x = 0$ of solutions of (1.15). In Section 3, we establish weighted Hölder spaces and invertible operators related to our Equation (1.15). In Section 4, we construct infinitely many singular solutions of (1.1) and show that the singular solutions that we have constructed are non-radial singular solutions.

2.   Asymptotic expansion of a prescribed singular solution $u \in C^2 (B \backslash \{0\})$ of (1.15) that satisfies (1.13)

We will establish asymptotic expansions of the singular solution $u \in C^2 (B \backslash \{0\})$ of (1.15) such that (1.13) is satisfied.

Let $v(x) = u(x)-U_s (x)$. Then $v(x)$ satisfies

Making the following transformations:

we see from (2.1) that $w(t, \theta)$ satisfies

We write (2.2) in the following forms

and

where

Moreover, (1.13) implies that

Define a linearized operator

Obviously, ${\mathcal L}$ can decouple into infinitely many ordinary differential operators, i.e.,

for $k = 0, 1, 2, \ldots$, where $\lambda_k$ is the $k$-th eigenvalue of

and $\lambda_k = k(N-2+k)$ with the following multiplicity:

The $\{Q_1^k (\theta), \ldots, Q_{m_k}^k (\theta)\}$ with $\|Q^k_j\|_{L^2 (S^{N-1})} = 1$ denotes the basis of the eigenspace $H_k (S^{N-1}) \subset L^2 (S^{N-1})$ corresponding to $\lambda_k$. Then two roots of characteristic polynomial of (2.7) are as follows:

For $k = 0$, we have from (2.9) that

For $k = 1$,

Then,

and

For $k \geq 2$, the fact that $k(N-2+k) > 2 (N-2)$ implies that

we see from (2.9) that

It is clear that

Proposition 2.1. Assume that $N \geq 3$ and $u = u(r)$ is a radial solution of (1.15) that satisfies

Then

Proof. Bt applying the following transformations:

by (2.3), $w (t)$ satisfies the following ordinary diifferential equation (ODE):

where

Note that $w(t) = O(e^{\epsilon t})$ for $t$ near $-\infty$. Therefore, for $3 \leq N \leq 9$,

where $|B_1| = |B_2| = 1/\gamma$, $|f(w(t))| = O(e^{2 \epsilon t})$,

Since $w(t) \to 0$ as $t \to -\infty$, we obtain from (2.16) that $A_1 = A_2 = 0$ and

and

Substituting (2.18) into (2.17), we see that

We can do the same process to obtain that $w(t) \equiv 0$ for $t \leq -\frac{4 \beta}{\epsilon}$. Since $w(t)$ satisfies the ODE in (2.15), then $w(t) \equiv 0$ for $t \in (-\infty, 0)$.

For $N = 10$, we see that

where $|B_1| = |B_2| = 1$. Note that $\tau = -\frac{(N-2)}{2} < 0$. Since $w(t) \to 0$ as $t \to -\infty$, we see that $A_1 = A_2 = 0$. Arguments similar to those in the proof for the case of $3 \leq N \leq 9$ imply that $w(t) \equiv 0$ for $t \in (-\infty, 0)$.

For $N \geq 11$, we see that

where $|B_1| = |B_2| = \Big|\frac{1}{\sigma_-^{(0)}-\sigma_+^{(0)}} \Big|$. Note that $\sigma_+^{(0)} < 0$ and $\sigma_-^{(0)} < 0$. Since $w(t) \to 0$ as $t \to -\infty$, we see that $A_1 = A_2 = 0$. Arguments similar to those in the proof for the case of $3 \leq N \leq 9$ imply that $w(t) \equiv 0$ for $t \in (-\infty, 0)$. This completes the proof of this proposition.

Lemma 2.2. Assume that $N \geq 3$ and $u \in C^2 (B \backslash \{0\})$ is a singular solution of (1.15) that satisfies (1.13). Defining $w(t, \theta) = u(x)-U_s (x)$ and $t = \ln r$, it follows that $w(t, \theta) = O(e^{\epsilon t})$ for $t \in (-\infty, -1]$, and

Proof. Let

Then $w_j^k (t)$ satisfies the following equation:

where

Note that

Since $\mathcal{F}(w) = O(w^2)$ and $w(t, \theta) = O(e^{\epsilon t})$, we see that

On the other hand, it follows from (2.23) that for $k \geq 2$, $T \ll -1$ and $t < T$,

where

Since $w_j^k (t) \to 0$ as $t \to -\infty$, we have that $A_{j, 2}^k = 0$. Moreover,

and

Then,

and for small enough $\delta > 0$,

with constants $C > 0$ and $C_\delta > 0$ being dependent on $\delta$ and independent of $(j, k)$.

For $k = 1$, $T \ll -1$ and $t < T$,

where

Note that $\sigma_+^{(1)} \leq 0$ and $\sigma_-^{(1)} < 0$. Since $w_j^1 (t) \to 0$ as $t \to -\infty$, we have that $A_{j, 1}^1 = A_{j, 2}^1 = 0$,

and

Note that

Similarly, we have

Then,

Notice that

and

since $\mathcal{F}(w) = O(w^2)$. We also know that

since

Let $[W(t)]^2 = \sum_{k = 2}^\infty \sum_{j = 1}^{m_k} (w_j^k (t))^2$. We have that, if $4 \epsilon \neq 2 \sigma_+^{(2)}-\delta$,

Now we show that, for $t < T \ll -1$,

In fact, two cases can occur: (ⅰ) $4 \epsilon \geq [2 \sigma_+^{(2)}-\delta]$ and (ⅱ) $4 \epsilon < [2 \sigma_+^{(2)}-\delta]$.

It suffices to consider that $4 \epsilon > [2 \sigma_+^{(2)}-\delta]$ in the case (ⅰ). So

Set

Then, if $|T|$ is large,

where $\sigma_-^{(2)} < 0$, $\sigma_+^{(2)} > 0$. Since $K_1 (t) \to 0$ and $K_2 (t) \to 0$ as $t \to -\infty$, then for $t < T$,

Substituting (2.31) into (2.30), we have

It follows from arguments similar to those in ^[16,17] that, for $t < T$ (enlarge $|T|$ if necessary),

where $\epsilon_T = C e^{2 \epsilon T}$ (i.e., $C$ is independent of $\epsilon$). On the other hand, (2.27) and (2.28) imply that

where $4 \epsilon > 2 \sigma_+^{(2)}-\delta$. Therefore, for $t < T$ (i.e., $T$ is sufficiently negative),

and

Using (2.35), we obtain

Choosing $\delta$ sufficiently small such that $\delta < 2 \epsilon-\epsilon_T$, then

So for $t < T$,

Moreover, we choose $0 < \delta < \min\{2\epsilon-\epsilon_T, 2\sigma_+^{(3)}-2\sigma_+^{(2)} \}$; then,

It is known from (2.36) and $\delta < 2\epsilon-\epsilon_T$ that

By (2.26) and $g_j^1 (t) = O(e^{\sigma_+^{(2)} t})$, we obtain

Similarly,

We obtain from (2.37)–(2.40) that

For any fixed $(t, \theta) \in (-\infty, T-1) \times \mathbb{S}^{N-1}$, by applying the interior $L^\infty$-estimate to (2.4) with $(t-1, t+1) \times \mathbb{S}^{N-1}$, we obtain from (2.41) and (2.24) that

where $C > 0$ is independent of $t$. Note that we can also use arguments similar to those in the proof of ^[18] to obtain

Defining

it follows that $v(r, \theta)$ satisfies

where $R = e^{T-1}$. For any $x_0 \in B_{R} \backslash \{0\}$, denote $r_0 = |x_0| > 0$ and $\Omega = B_{r_0/2} (x_0)$. Consider (2.44) to be a linear equation in $\Omega$ as in Lemma 5.1 and Theorem 5.1 of ^[18] with

where $Q = Q(v) > 0$. Then, (2.41) implies that there is a positive constant (independent of $r_0$)

such that

In particular, we have

Hence, (2.43) follows for $t \in (-\infty, T-1)$.

For the case of $4 \epsilon = 2\sigma_+^{(2)}-\delta$, we may choose $\delta'$ a little larger than $\delta$ such that $0 < \delta < \delta'$ and $4 \epsilon > 2 \sigma_+^{(2)}- \delta'$. By similar arguments, we can prove (2.41) and (2.43).

For the case (ⅱ), by $\mathcal{F}(w) = O(e^{2 \epsilon t})$, $\sum_{k = 2}^\infty \sum_{j = 1}^{m_k} (g_j^k (s))^2 = O(e^{4 \epsilon t})$ and $4 \epsilon < 2\sigma_+^{(2)}-\delta < 2\sigma_+^{(2)}$, we can obtain

Then

Together with (2.27) and (2.28), we know that

Arguments similar to those in the proof of (2.43) imply that

where $M: = M(w) > 0$. As a consequence,

Then (2.26) implies that

Similarly,

Therefore,

Note that

In what follows, there are also two cases: (a) $8 \epsilon \geq [2 \sigma_+^{(2)}-\delta]$ and (b) $8 \epsilon < [2 \sigma_+^{(2)}-\delta]$.

For the case (a), using (2.49) and arguments similar to those in the proof of (i), we can obtain (2.43).

The case (b) implies that $\mathcal{F}(w) = O(e^{4\epsilon t})$. Then

This, (2.47) and (2.48) imply that

and

Therefore,

Then we have

and

Similarly, we still consider two cases: $10 \epsilon \geq [2 \sigma_+^{(2)}-\delta]$ and $10 \epsilon < [2 \sigma_+^{(2)}-\delta]$; then, we obtain (2.43).

The proof of this lemma is complete.

Theorem 2.3. Assume that $N \geq 3$ and $u \in C^2 (B \backslash \{0\})$ is a singular solution of (1.15) that satisfies (1.13). Defining $w(t, \theta) = u(x)-U_s (x)$ and $t = \ln r$, there is a positive number sequence $\{\mu_k\}_{k \geq 1}$, strictly increasing and converging to $\infty$ with

such that for any positive integer $n \gg 1$ and any $(t, \theta) \in (-\infty, -1) \times S^{N-1}$,

where

and $M_{k \ell}$ is a nonnegative integer depending on $N, k, \ell$; $a_{k \ell i}$ is constant and $Q_i (\theta)$ is a linear combination of $\{Q_1^i (\theta), Q_2^i (\theta), \ldots, Q_{m_i}^i (\theta)\}$. Especially, for $k = 1$,

where $a_{102}$ is a constant.

Proof. By using the starting estimate (2.22), constructing the index set $\mathcal{I}$, and examining the equation of $w$, the expansion of $w (t, \theta)$ can be established via similar arguments to those in Theorem 1.1 of ^[19].

Let $\{\rho_k \}_{k \geq 1}$ be positive strictly increasing and converging to $\infty$:

Also, let $\mathbb{Z}_+$ be the collection of nonnegative integers. Define the index set ${\mathcal{I}}$ by

Set

and

Assume that ${\mathcal I}_{{\tilde \rho}}$ is given by a strictly increasing sequence $\{{\tilde \rho}_k\}_{k \geq 1}$ with ${\tilde \rho}_1 = 2 \rho_1$. There may be identical elements in $\mathcal{I}_\rho$ and $\mathcal{I}_{{\tilde \rho}}$.

For ${\tilde \rho}_k \in \mathcal{I}_{{\tilde \rho}}$, there are nonnegative integers $n_1, \ldots, n_{i_1}$ such that

The collections of nonnegative integers $n_1, \ldots, n_{i_1}$ that satisfy (2.60) are finite. Set

Arrange $\mathcal{I}$ as follows:

Note that if $\rho_1 < {\tilde \rho}_1 < \rho_2$, we choose $i_1 = 1$ and $l_1 = 1$ and the arrangement of (2.62) becomes $\rho_1 < {\tilde \rho}_1 < \rho_2 < \ldots$. Similarly, if $\rho_{i_k+1} \leq {\tilde \rho}_{l_k+1} < \rho_{i_k+2}$ for some $k \geq 1$, define $i_k+1 = i_{k+1}$ and $l_k+1 = l_{k+1}$. We do not consider the multiplicity of $\rho_k$ here, since all terms containing $e^{\rho_k t}$ in the expansions of $w(t, \theta)$ can be combined as one term. We know

where $\mathcal{F}(w) = \sum_{k = 2}^\infty b_k w^k$ for $|w| < {\hat \epsilon}$ with some sufficiently small ${\hat \epsilon} > 0$; also, the expansion of $\mathcal{F}(w)$ consists of terms including $\sum_{\ell = 0}^{I(k)} \Big(\sum_{i = 0}^{{\tilde M}_k} c_{k \ell i} Q_i (\theta) \Big) t^\ell e^{{\tilde \rho}_k t}$ for ${\tilde \rho}_k \in \mathcal{I}_{{\tilde \rho}}$ in (2.59).

Define

according to the arrangement in (2.62). To ensure that $\{\mu_k\}_{k \geq 1}$ is a strictly increasing sequence of positive constants, when $\rho_{i_1} = {\tilde \rho}_1$, define $\mu_{i_1} = \rho_{i_1}$ and $\mu_{i_1+1} = {\tilde \rho}_2$. In this case, an extra power of $t$ term corresponding to $\mu_{i_1}$ may appear in the expansion of $w(t, \theta)$. Similarly, make the same choices of $\mu_k$ for the cases ${\tilde \rho}_{l_1} = \rho_{i_1+1}$, $\rho_{i_2} = {\tilde \rho}_{l_1+1}$, ${\tilde \rho}_{l_2} = \rho_{i_2+1}$, etc. As a consequence, for any positive integer $n \gg 1$ and any $(t, \theta) \in (-\infty, -1) \times S^{N-1}$,

where

and $M_{k \ell}$ is a nonnegative integer that is dependent on $N, k, \ell$, $a_{k \ell i}$ is constant and $Q_i (\theta)$ is in the span of $Q_1^i (\theta), Q_2^i (\theta), \ldots, Q_{m_i}^i (\theta)$. Especially, for $k = 1$,

The proof of this theorem is complete.

3.   Linearized equations and inverse of the operator

We will introduce the appropriate weighted Hölder spaces and obtain the inverse of the operator $\mathcal{L}$ on those spaces, where $\mathcal{L}$ is given in (2.4). We use some ideas from ^[20], where the authors derived singular solutions to the following equation:

Fix a $t_0 < 0$. For a nonnegative integer $i$, $\alpha \in (0, 1)$, and $\mu \in \mathbb{R}$, define

and

where $[\cdot]_{C^\alpha}$ is the usual Hölder semi-norm.

Definition 3.1. The collection of functions $v$ in $C^i ((-\infty, t_0] \times \mathbb{S}^{N-1})$ with a finite norm $\|v\|_{C_\mu^{i, \alpha} ((-\infty, t_0] \times \mathbb{S}^{N-1})}$ is the weighted Hölder space $C_\mu^{i, \alpha} ((-\infty, t_0] \times \mathbb{S}^{N-1})$.

For $\mu > 0$ and some $g \in C_\mu^{0, \alpha} ((-\infty, t_0] \times \mathbb{S}^{N-1})$, to consider the linear equation given by

we introduce a boundary condition on $t = t_0$ such that

has a bounded inverse. However, since signs of coefficients of zero order terms are inappropriate, we cannot directly apply the maximum principle to the following Dirichlet boundary-value problem

Lemma 3.1. Let $\mu > 0$, $g \in C_\mu^0 ((-\infty, t_0] \times \mathbb{S}^{N-1})$, and $\varphi \in C^0 (\mathbb{S}^{N-1})$. Then, there is at most one solution $v \in C_\mu^2 ((-\infty, t_0] \times \mathbb{S}^{N-1})$ of (3.2).

Proof. Assume that $g = 0$, $\varphi = 0$ and $v \in C_\mu^2 ((-\infty, t_0] \times \mathbb{S}^{N-1})$ is a solution of (3.2). For each $k \geq 0$, define

So $\mathcal{L}_k (v_k) = 0$ on $(-\infty, t_0)$ and $v_k (t_0) = 0$. This implies that $v_k$ is a linear combinations of the basis of Ker$(\mathcal{L}_k)$. In particular, for $k = 0$,

and for $k \geq 1$,

where $c_k^1$ and $c_k^2$ are constants for $k = 0, 1, 2, \ldots$. By the assumption, we have the following for any $t \in (-\infty, t_0)$:

Hence, $v_k = 0$ for $k = 0$ and $k = 1$. Note that

$\sigma_+^{(1)} \leq 0$ and $\sigma_-^{(1)} < 0$ for $N \geq 3$. Moreover, $v_k (t) = c_k^1 e^{\sigma_+^{(k)} t}$ for $k \geq 2$, which decays exponentially as $t \to -\infty$ (note that $c_k^2 = 0$ since $\sigma_-^{(k)} < 0$ for $k \geq 2$). Since $v_k (t_0) = 0$, we can directly obtain $c_k^1 = 0$ and $v_k (t) \equiv 0$ for $k \geq 2$. In conclusion, $v_k = 0$ for all $k \geq 0$, i.e., $v \equiv 0$.

Lemma 3.2. Let $\alpha \in (0, 1)$, $\mu > 0$, $g \in C_\mu^{0, \alpha} ((-\infty, t_0] \times \mathbb{S}^{N-1})$, and $\varphi \in C^{2, \alpha} (\mathbb{S}^{N-1})$. Suppose that $v \in C_\mu^{2, \alpha} ((-\infty, t_0] \times \mathbb{S}^{N-1})$ is a solution of (3.2). Then

where $C$ is a positive constant that is only dependent on $N, \alpha, \mu$ and is independent of $t_0$.

Proof. Using similar arguments to that of Lemma 2.5 of ^[20], consider two cases:

(ⅰ) $t < t_0-2$. We have

where $C$ is a positive constant that is independent of $t$. We estimate $[g]_{C^\alpha ([t-2, t+2] \times \mathbb{S}^{N-1})}$, by setting $(t_1, \theta_1)$, $(t_2, \theta_2) \in [t-2, t+2] \times \mathbb{S}^{N-1}$ with $(t_1, \theta_1) \neq (t_2, \theta_2)$. There are two cases: $|t_1-t_2| \leq 2$ and $|t_1-t_2| > 2$.

When $|t_1-t_2| \leq 2$, choose $t' \in [t-1, t+1]$ such that $t_1, t_2 \in [t'-1, t'+1]$ is satisfied. Then,

So,

We multiply both sides by $e^{-\mu t}$ and take the supremum over $t \in (-\infty, t_0-2)$. The following holds

where $C$ is a positive constant that is independent of $t_0$.

(ⅱ) $t_0-2 \leq t \leq t_0$. From the boundary Schauder estimate, we see that

Similarly,

Combining (3.5) and (3.6), (3.4) holds.

Then, by arguments similar to those in ^[20,21,22], we obtain the $L^\infty$ estimates of solutions on finite cylinders to (3.2) with a 0 boundary value.

Lemma 3.3. Let $\mu > \rho_1$ and $\mu \neq \rho_k$ for $k \geq 1$, $T$ and $t_0$ be constants with $t_0 \leq 0$ and $T-t_0 \leq -4$, and $g \in C^0 ([T, t_0] \times \mathbb{S}^{N-1})$. Suppose that $v \in C^2 ([T, t_0] \times \mathbb{S}^{N-1})$ satisfies the following:

and $\int_{\mathbb{S}^{N-1}} v(t, \theta) Q_k (\theta) d \theta = 0$ for $k = 0, 1, \cdots, K$, where $K$ is the largest integer satisfying that $\rho_{K-1} < \mu$. Then,

where $C$ is a positive constant dependent only on $N, \mu$ and independent of $T$ and $t_0$.

Proof. Note that $\rho_1 = \sigma_+^{(2)}$.

Let the sequences $\{T_i\}$, $\{t_i\}$, $\{v_i\}$ and $\{g_i\}$ with $t_i \leq 0$ and $T_i-t_i \leq -4$, satisfy the following:

and

There exists $t_i^* \in (T_i, t_i)$ that satisfies

Set

The following holds:

for any $(t, \theta) \in [T_i-t_i^*, t_i-t_i^*] \times \mathbb{S}^{N-1}$,

and

Assume the following for some $\tau_- \in \mathbb{R}^{-} \cup \{-\infty\}$ and $\tau^+ \in \mathbb{R}^{+} \cup \{\infty\}$:

From (3.10) we have

and hence

Since ${\tilde v}_i = 0$ on $\{T_i-t_i^*\} \times \mathbb{S}^{N-1}$, we have

This implies that $T_i-t_i^*$ remains bounded away from zero. Similar arguments imply that $t_i-t_i^*$ is bounded away from zero. As a consequence, $0 \in (\tau_-, \tau_+)$. Let

Moreover, ${\tilde g}_i \to 0$ in every compact set of $(\tau_-, \tau_+)$. So the following holds:

and

where $\tau_* = \tau_-$ or $\tau_+$ if it is finite.

Let

Then $\mathcal{L}_k ({\hat v}_k) = 0$; hence, ${\hat v}_k$ is the linear combination of the basis of Ker$(\mathcal{L}_k)$. We now take $k \geq 2$ with $\rho_k > \mu$. Then

where $c_{k+1}^1$ and $c_{k+1}^2$ are constants. From (3.13), we obtain the following for any $t \in (\tau_-, \tau_+)$:

When $\tau_+ = \infty$, $c_{k+1}^1 = 0$ and hence ${\hat v}_{k+1} (t) = c_{k+1}^2 e^{\sigma_-^{(k+1)} t}$. When $\tau_+$ is finite, $\lim_{t \to \tau_+} {\hat v}_{k+1} (t) = 0$ by (3.14). Similarly, when $\tau_- = -\infty$, ${\hat v}_{k+1} (t) = c_{k+1}^1 e^{\rho_{k} t} = c_{k+1}^1 e^{\sigma_+^{(k+1)} t}$. When $\tau_-$ is finite, $\lim_{t \to \tau_-} {\hat v}_{k+1} (t) = 0$ by (3.14). Thus,

Since $\rho_k > \mu > 0$, it follows that $\lambda_{k+1} > 2N$ for each $k$. This implies that $\lambda_{k+1}-2 (N-2) > 0$ for each $k$. Therefore, ${\hat v}_{k+1} = 0$ for each $k$. By the assumption, we have

provided that $\rho_{k-1} < \mu$. In conclusion, ${\hat v}_k = 0$ for any $k \geq 0$; hence, ${\hat v} \equiv 0$. This is a contradiction.

Lemma 3.4. Let $\alpha \in (0, 1)$, $\mu > \rho_K$ for some $K \geq 1$, and $g \in C^{0, \alpha}_\mu ((-\infty, t_0] \times \mathbb{S}^{N-1})$ with $g(t, \cdot) \in \mathit{\mbox{span}} \{ Q_0, Q_1, \ldots, Q_{K+1} \}$ for $t \leq t_0$. Then, there is a unique solution $v \in C^{2, \alpha}_\mu ((-\infty, t_0] \times \mathbb{S}^{N-1})$ of (3.1) with $v(t, \cdot) \in \mathit{\mbox{span}} \{ Q_0, Q_1, \ldots, Q_{K+1} \}$ for $t \leq t_0$. Furthermore, $g \mapsto v$ is linear, and

where $C$ is a positive constant that is only dependent on $N, \alpha, \mu$ and independent of $t_0$.

Proof. For $k = 0, 1, \ldots, K+1$, define

Then,

and

Consider the following ODE:

Suppose that there is a solution $v_k \in C^{2, \alpha}_\mu ((-\infty, t_0])$ of (3.17) and

where $C$ is a constant that depends only on $N, \alpha, \mu$ and independent of $t_0$.

If $k = 0$, it is known from Section 2 that Ker$(\mathcal{L}_0)$ encompasses $e^{\tau t} \cos \gamma t$ and $e^{\tau t} \sin \gamma t$ with

provided that $3 \leq N \leq 9$. Ker$(\mathcal{L}_0)$ encompasses $e^{\sigma_+^{(0)} t}$ and $t e^{\sigma_+^{(0)} t}$ with $\sigma_+^{(0)} = \sigma_-^{(0)} < 0$ provided that $N = 10$. Ker$(\mathcal{L}_0)$ encompasses $e^{\sigma_+^{(0)} t}$ and $e^{\sigma_-^{(0)} t}$ with $\sigma_+^{(0)} < 0$, $\sigma_-^{(0)} < 0$ provided that $N \geq 11$. If $k = 1$, Ker$(\mathcal{L}_1)$ encompasses $e^{\sigma_+^{(1)} t}$ and $e^{\sigma_-^{(1)} t}$ with $\sigma_+^{(1)} \leq 0$, $\sigma_-^{(1)} < 0$ provided that $N \geq 3$.

We now consider $k = 0$ and $k = 1$. For $k = 0$, we see that

where

We only consider $v_0(t)$ for $N \geq 11$ and $v_1(t)$. The other cases for $v_0 (t)$ can be demonstrated similarly since $\tau < 0$ and $\sigma_+^{(0)} < 0$ in these cases. Let

where

Direct calculation implies the following for $t \leq t_0$:

Set

where

and

Then,

Then we have

and hence, for $t \leq t_0-1$,

Therefore, for $t \leq t_0-1$,

Combining (3.21), (3.22) and (3.23), (3.18) holds for $k = 0$ with $N \geq 11$ and $k = 1$.

For $2 \leq k \leq K+1$, since $e^{-\mu t} |g_k (t)| \leq C$ and $\mu > \rho_K$, we can also define

Note that, at this time, $\sigma_+^{(k)} > 0$ and $\sigma_-^{(k)} < 0$ for $k = 2, \ldots, K+1$ and $\rho_k = \sigma_+^{(k+1)}$. By arguments similar to those in the proof of (3.18) for $k = 0$ with $N \geq 11$ and $k = 1$, we obtain (3.18) for $2 \leq k \leq K+1$.

With the solution $v_k$ of (3.17) for $k = 0, 1, \ldots, K+1$, we set

Then, $\mathcal{L} v = g$ and, by (3.18), we have

Then the extra requirement $v(t, \cdot) \in$ span$\{Q_0, Q_1, \ldots, Q_{K+1}\}$ implies the uniqueness of $v$.

Lemma 3.5. Let $\alpha \in (0, 1)$, $\mu > \rho_1$ and $\mu \neq \rho_k$ for any $k \geq 1$; also, $g \in C_\mu^{0, \alpha} ((-\infty, t_0] \times \mathbb{S}^{N-1})$, with $\int_{\mathbb{S}^{N-1}} g(t, \cdot) Q_{k} (\theta) d \theta = 0$ for any ${k} = 0, 1, \ldots, K, K+1$, where $K$ is the largest integer such that $\rho_K < \mu$, and $t \leq t_0$. Then, there exists a unique solution $v \in C^{2, \alpha}_\mu ((-\infty, t_0] \times \mathbb{S}^{N-1})$ of (3.1) with $v = 0$ on $\{t_0\} \times \mathbb{S}^{N-1}$. Moreover,

where $C$ is a positive constant that is dependent on $N, \alpha$ and $\mu$ and independent of $t_0$.

Proof. Assume that $T \leq t_0-4$. We claim that there is a solution $v_T \in C^{2, \alpha}([T, t_0] \times \mathbb{S}^{N-1})$ to the following problem:

The problem described by (3.26) can be written as follows:

where $\tau = N-2$. Consider the energy function

Set

Then, for any $\phi \in \Gamma$,

since $\lambda_{k+1} \geq \lambda_2 = 2N$ with $\mu > \rho_k \geq \rho_1$. Hence, for any $v \in H_0^1((T, t_0) \times \mathbb{S}^{N-1})$ with $v(t, \cdot) \in \Gamma$ for any $t \in (T, t_0)$, we have

The inequality $2N-2 (N-2) > 0$ implies that $\mathcal{G}_T$ is coercive and weakly lower semi-continuous. Then there is a minimizer $v_T$ of $\mathcal{G}_T$ in the following statement:

So $v_T$ is a solution of (3.26) that satisfies $v_T (t, \cdot) \in \Gamma$ for any $t \in (T, t_0)$.

We obtain that by Lemma 3.3,

where $C$ is a positive constant that is dependent on $N, \mu$ and independent of $T$ and $t_0$. Fix $T_0 < t_0$. By the interior and boundary Schauder estimates in $[t_0+T_0, t_0] \times \mathbb{S}^{N-1} \subset [t_0+T_0-1, t_0] \times \mathbb{S}^{N-1}$ and $v_T (t_0, \theta) = 0$, there exists a subsequence $v_T$ that converges to a $C^{2, \alpha}$-solution $v$ of (3.1) in $[t_0+T_0, t_0] \times \mathbb{S}^{N-1}$ satisfying $v = 0$ on $\{t_0\} \times \mathbb{S}^{N-1}$, as $T \to -\infty$. Then, we can obtain that $v_T$ converges to a $C^{2, \alpha}$-solution $v$ of (3.1) in $(-\infty, t_0] \times \mathbb{S}^{N-1}$ such that the following is satisfied: $v = 0$ on $\{t_0\} \times \mathbb{S}^{N-1}$,

or

Combining (3.28) and (3.4) with $\varphi = 0$, (3.25) holds.

Theorem 3.6. Let $\alpha \in (0, 1)$, $\mu > \rho_1$, $\mu \neq \rho_k$ for any $k \geq 1$ and $g \in C^{0, \alpha}_\mu ((-\infty, t_0] \times \mathbb{S}^{N-1})$. Then (3.1) admits a solution $v \in C^{2, \alpha}_\mu ((-\infty, t_0] \times \mathbb{S}^{N-1})$ and

where $C$ is a positive constant that is dependent on $N, \alpha, \mu$, and independent of $t_0$. Also, $g \mapsto v$ is linear.

Proof. Assume that $K \geq 1$ is the largest integer with $\rho_K < \mu$. Define

Then $v_1 \in C_\mu^{2, \alpha} ((-\infty, t_0] \times \mathbb{S}^{N-1})$ is a solution of

as in Lemma 3.4. By Lemma 3.5, let $v_2 \in C_\mu^{2, \alpha} ((-\infty, t_0] \times \mathbb{S}^{N-1})$ be the unique solution of the following problem:

Then $v = v_1+v_2$ is a solution of (3.1) satisfying that $v(t_0, \theta) = v_1 (t_0, \theta) = \sum_{k = 0}^{K+1} v_k (t_0) Q_k (\theta)$, where $v_k (t)$ for $k = 0, 1, \dots, K, K+1$ is given in Lemma 3.4.

Remark 3.7. Theorem 3.6 implies that the bound of

is independent of $t_0$.

4.   Nonradial singular solutions of (1.1)

In what follows, singular solutions of (1.1) will be constructed.

We set

Then $w$ satisfies

if $\mathcal{N} (w) = 0$ in $(-\infty, 0) \times \mathbb{S}^{N-1}$. This also implies that $u(x) = U_s (x)+w(\ln |x|, \theta)$ is a solution of (1.15) in $B \backslash \{0\}$.

Theorem 4.1. Let $U_s (x)$ be given as in (1.2), the index set $\mathcal{I}_\rho$ be given as in (2.58), and $\mu > \rho_1$ with $\mu \not \in \mathcal{I}_\rho$. Suppose that ${\hat w} \in C^{2, \alpha} ((-\infty, 0] \times \mathbb{S}^{N-1})$ satisfies

and for $(t, \theta) \in (-\infty, 0] \times \mathbb{S}^{N-1}$,

where $C$ is a positive constant. Then, there exist $t_0 < 0$ and a solution $w \in C^{2, \alpha} ((-\infty, t_0] \times \mathbb{S}^{N-1})$ of the equation in (4.2) such that the following is satisfied for $(t, \theta) \in (-\infty, t_0) \times \mathbb{S}^{N-1}$:

where $C$ is a positive constant.

Proof. The proof consists of 4 steps.

Step 1. We claim that there is $z \in C_\mu^{2, \alpha} ((-\infty, t_0] \times \mathbb{S}^{N-1})$ such that

Rewrite this equation as

and

where

Defining $\mathcal{T}$ by

we prove that $\mathcal{T}$ is a contraction on some ball in $C_\mu^{2, \alpha} ((-\infty, t_0] \times \mathbb{S}^{N-1})$, for some $t_0 < 0$ with $|t_0|$ large. We set

Step 2. We claim that $\mathcal{T}$ maps $\Gamma_{B, t_0}$ to itself, for some fixed $B$ and any $t_0$ with $|t_0|$ sufficiently large, namely, for any $z \in C_\mu^{2, \alpha} ((-\infty, t_0] \times \mathbb{S}^{N-1})$ with $\|z\|_{C_\mu^{2, \alpha} ((-\infty, t_0] \times \mathbb{S}^{N-1})} \leq B$, we have that $\mathcal{T} (z) \in C_\mu^{2, \alpha} ((-\infty, t_0] \times \mathbb{S}^{N-1})$ and $\|\mathcal{T}(z)\|_{C_\mu^{2, \alpha} ((-\infty, t_0] \times \mathbb{S}^{N-1})} \leq B$.

First, it follows from (4.4) that

Next, set

Then, $P(z) = z E(z)$. Take any $z \in C_\mu^{2, \alpha} ((-\infty, t_0] \times \mathbb{S}^{N-1})$ with $\|z\|_{C_\mu^{2, \alpha} ((-\infty, t_0] \times \mathbb{S}^{N-1})} \leq B$ where $B$ will be determined later. Because

where $\epsilon$ is increasing such that $\epsilon (t) \to 0$ as $t \to -\infty$ and

Then, for $t \leq t_0$,

and hence,

By Theorem 3.6, we have

where $C$, $C_1$ and $C_2$ are constants that are independent of $t_0$. Choose $B \geq 2 C C_1$ and $t_0$ with $|t_0|$ sufficiently large such that $C C_2 (\epsilon (t_0)+B e^{\mu t_0}) \leq 1/2$. Then,

Step 3. We claim that $\mathcal{T}: \; \Gamma_{B, t_0} \to \Gamma_{B, t_0}$ is a contraction, i.e., for any $z_1, z_2 \in \Gamma_{B, t_0}$,

for some constant $\kappa \in (0, 1)$.

Note that

and

By (4.11), we have

Then,

By (4.12), we have the following for any $t \leq t_0$,

and hence

By Theorem 3.6, we obtain

We derive (4.13) by choosing $t_0$ with $t_0$ sufficiently negative.

Step 4. The contraction mapping principle implies that there exists $z \in C_\mu^{2, \alpha} ((-\infty, t_0] \times \mathbb{S}^{N-1})$ such that $\mathcal{T} (z) = z$. Then $z \in C_\mu^{2, \alpha} ((-\infty, t_0] \times \mathbb{S}^{N-1})$ is a solution of (4.6) and hence $w = {\hat w}+z$ is a solution of (4.2).

We call a function ${\hat w}$ that satisfies (4.3) and (4.4) an approximate solution to (4.2) with order $\mu$.

Lemma 4.2. Assume that $Q_k (\theta)$ is the combination of $Q_1^k (\theta), \ldots, Q_{m_k}^k (\theta)$. Then,

Proof. The proof is similar to that of Lemma 2.4 in ^[19]. We omit the details here.

Proposition 4.3. Let the index sets $\mathcal{I}_\rho$ and $\mathcal{I}_{{\tilde \rho}}$ be respectively given as in (2.58) and (2.59) and $\mu > \rho_1$ with $\mu \not \in \mathcal{I}_\rho \cup \mathcal{I}_{{\tilde \rho}}$. Let $\eta$ be a solution of $\mathcal{L} (\eta) = 0$ on $\mathbb{R} \times \mathbb{S}^{N-1}$ such that $\eta(t, \cdot) \to 0$ as $t \to -\infty$ uniformly on $\mathbb{S}^{N-1}$. Therefore for some $t_0 < 0$, there exists a smooth function ${\tilde \eta}$ on $(-\infty, t_0] \times \mathbb{S}^{N-1}$ such that ${\hat w} = \eta+{\tilde \eta}$ satisfies (4.3) and (4.4).

Proof. Choose $\phi$ with $|\phi| < {\hat \epsilon}$ on $(-\infty, 0) \times \mathbb{S}^{N-1}$, where ${\hat \epsilon} > 0$ is a sufficiently small number. Then, a simple computation yields

Therefore,

Assume $K \geq 1$ and ${\tilde K}$ represent the largest corresponding integer with $\rho_K < \mu$ and ${\tilde \rho}_{{\tilde K}} < \mu$, respectively. There is no function that converges to 0 as $t \to -\infty$ in Ker$\mathcal{L}_0$ and Ker$\mathcal{L}_1$; for $k \geq 2$, $\psi_k^+ (t) = e^{\sigma_+^{(k)}}$ and $\psi_k^- (t) = e^{\sigma_-^{(k)} t}$ in Ker$\mathcal{L}_k t$. Any term $e^{\rho_k t}$ with $k > K$ in $\eta$ will produce the term $e^{{\tilde \rho}_l t}$ with ${\tilde \rho}_l > \mu$ in $\mathcal{N}(\eta)$; set

where $c_k$ denotes constants.

For the following case:

we will show that there ${\tilde \eta_0}, {\tilde \eta}_1, \ldots, {\tilde \eta}_{{\tilde K}}$ exists in succession such that, for any $i = 0, 1, \ldots, {\tilde K}$,

Set $\phi = \eta$. From (4.15) and $\mathcal{L}(\eta) = 0$, we obtain

where $n_1, \ldots, n_{i_1}$ are nonnegative integers and $a_{n_1 \ldots n_{i_1}}$ is a constant. By the definition of $\mathcal{I}_{{\tilde \rho}}$, $n_1 \rho_1+\ldots+n_{i_1} \rho_{i_1}$ is some ${\tilde \rho}_k$. Hence, by Lemma 4.2,

where ${\tilde M}_k$ is defined as in (2.61) and $a_{km}$ is constant. We see that

where ${\tilde \rho}_1 = 2 \rho_1$. Then ${\tilde \eta}_0 = 0$.

Assume that (4.18) holds for $0, 1, \ldots, i-1$. Set

where $c_{im}$ is constant. This implies that

where $a_{km}$ is different from that in (4.19). The induction hypothesis implies that

Choose ${\tilde \eta}_i$ such that

So (4.18) holds for $i$. For $m = 0, 1, \ldots, {\tilde M}_i$, solve

Similar to that in Lemma 3.4, there is a formula for $c_{im} e^{{\tilde \rho}_i t}$ in terms of $a_{im} e^{{\tilde \rho}_i t}$. For $0 < \rho_m < {\tilde \rho}_i$, the expression is in the form of (3.24). For $\rho_m > {\tilde \rho}_i$, the expression is similar. If $m = 0$ or $1$, the expression is (3.19) or (3.20). Therefore, define

where $c_{km}$ is a constant. We obtain

The estimate of $\nabla \mathcal{N}(\eta+{\tilde \eta})$ is similar.

Then for the general case, $\rho_k$ can be some ${\tilde \rho}_i$. We only need to modify (4.21). When $\rho_m = {\tilde \rho}_i$, there exist constants $c_{i0m}$ and $c_{i1m}$ such that

By iteration, there are more powers of $t$. Therefore, there exist constants denoted by $c_{ijm}$ with $j = 0, 1, \ldots, J+1$ such that

In conclusion, instead of (4.22), apply

where $c_{kjm}$ is a constant.

Proof of Theorem 1.1

Theorem 4.1 and Proposition 4.3 imply that we can obtain $t_0$ and a solution $w \in C^{2, \alpha} ((-\infty, t_0] \times \mathbb{S}^{N-1})$ of the equation in (4.2) such that, for $\mu > \rho_1$ with $\mu \not \in \mathcal{I}_\rho \cup \mathcal{I}_{{\tilde \rho}}$ and any $(t, \theta) \in (-\infty, t_0) \times \mathbb{S}^{N-1}$,

Let $r_0 = e^{t_0}$. Then $0 < r_0 < 1$. Since $u(x) = U_s (x)+w(\ln |x|, \theta)$, we add easily see that

On the other hand, for any $R > 0$, we see that ${\tilde u}(y): = u(x)+2 \ln (R^{-1} r_0)$, $y = R r_0^{-1} x$ satisfies the following equations:

and

This implies that ${\tilde u}$ is a singular solution of (1.1).

Now, suppose that ${\tilde u}$ is non-radial. It suffices to prove that $u$ is non-radial in $B_{r_0} \backslash \{0\}$. Suppose that $u(x) = u(|x|)$. Then from Proposition 2.1 we have

Moreover, we can easily see from (4.24) and (4.25) that

This implies that $u$ and ${\tilde u}$ are non-radial singular solutions of (1.15) in $B_{r_0} \backslash \{0\}$ and (1.1) in $B_R \backslash \{0\}$, respectively.

Since $w(t, \theta)$ depends on the parameter $\mu$, for each $\mu > \rho_1$, different coefficients $c_k$ of $\eta (t, \theta)$ in (4.16) can determine infinitely many ${\tilde u}_\mu$ and $t_0$ may change. When restricting all coefficients of $\eta (t, \theta)$ in (4.16) in a bounded interval, there is a minimal $t_0 < 0$. Since $K$ of $\eta (t, \theta)$ in (4.16) depends on $\mu$, we can obtain infinitely many ${\tilde u}(y)$ by choosing a sequence of parameters $\mu > \rho_1$ with $\mu \to \infty$. Hence, a family of nonradial singular solutions of (1.1) can be constructed.

5.   Conclusions

In this manuscript, infinitely many nonradial singular solutions have been constructed for the equation

where $B_R = \{x \in \mathbb{R}^N \; (N \geq 3): \; |x| < R\}$.

Use of AI tools declaration

The authors declare that they have not used artificial intelligence tools in the creation of this article.

Acknowledgments

The research was supported by the 2024 Training Program for College Students Innovation and Entrepreneurship in Anhui University (No. X20240024). The third author was supported by the National Key R$&$D Program of China (No. 2020YFA0713100), the National Natural Science Foundation of China (No. 12201001), and the Natural Science Foundation of Universities of Anhui Province (No. KJ2020A0009).

Conflict of interest

The authors declare that there is no conflicts of interest.