Qualitative properties of stable solutions to some supercritical problems

1.   Introduction

In this paper, we consider the Lane-Emden equation

and the equation

The structures of the positive solutions of (1.1) and (1.2) have been studied intensively in the last several years. When $n = 3$, (1.1) arises in the stellar structure in astrophysics. When $n = 4$, (1.1) is relevant to the famous Yang-Mills equations. When $n = 2$, (1.2) is an interesting problem in differential geometry and is known as the "Prescribing Gaussian Curvature" problem.

For Eq (1.1), the Sobolev exponent

plays a central role in the solvability question. In the subcritical case $1 < p < p_{s}(n)$, it was established by Gidas and Spruck in their celebrated work ^[1] that (1.1) has no positive solution. If $p = (n+2)/(n-2)$, then (1.1) is a special case of the Yamabe problem in conformal geometry. In ^[2], using the asymptotic symmetry technique, Caffarelli, Gidas and Spruck were able to classify all the positive solutions of (1.1) for $n\geq 3$. They showed that any positive solutions of (1.1) can be written in the form

where $\lambda > 0$ and $x_{0}$ is some point in $\mathbb{R}^{n}$. In ^[3], Chen and Li proved the same result for (1.1) by applying the moving plane method. In $n = 2$, Eq (1.2) is also classified in ^[3] under the additional assumption that

It is proved in ^[3] that if $u$ is a solution of (1.2) such that (1.3) holds, then

for some $\lambda > 0$ and some point $x_{0}\in\mathbb{R}^{2}$.

In the supercritical case $p > p_{s}(n)$, it is more difficult to classify the positive solutions of (1.1). The first result in this direction was given by Zou in ^[4]. It was proved in ^[4] that if $p_{s}(n) < p < p_{s}(n-1)$ and if $u$ is a positive solution of (1.1) with algebraic decay rate $2/(p-1)$ at infinity, then $u$ is radially symmetric about some point $x_{0}\in\mathbb{R}^{n}$. In ^[5], Guo generalized Zou's result to $p\geq p_{s}(n-1)$ by assuming that

Moreover, it is showed in ^[5] that (1.4) is a necessary and sufficient condition for a positive solution of (1.1) to be radially symmetric about some point. The analogous result for second order equation (1.2) is considered in ^[6]. It is proved in ^[6] that if $n\geq4$ and if $u\in C^{2}(\mathbb{R}^{n})$ is an entire solution of (1.2), then $u$ is radially symmetric about some point $x_{0}\in\mathbb{R}^{n}$ if and only if

If we focus on radial solutions, then the structure of positive solutions of (1.1) has been completely classified in ^[7]. They showed that for any $a > 0$, (1.1) admits a unique positive radial solution $u = u_{a}(r)$ with $u_{a}(0) = a$. Moreover, no two positive radial solutions of (1.1) can intersect each other when $p > p_{JL}(n)$, where $p_{JL}(n)$ is the exponent given by

Another important topic is the classification of stable solutions. In general, a solution of the semilinear equation

with $f$ be a Lipschitz function is called stable if

One of the most interesting questions concerning stable solutions is the following De Giorgi's conjecture.

Conjecture: Let $u$ be a bounded solution of the equation

such that $\frac{\partial u}{\partial x_{n}} > 0$. Then the level sets of $u$ are hyperplanes, at least if $n\leq 8$.

De Giorgi's conjecture was proved in dimension $n = 2$ by Ghoussoub and Gui in ^[8]. For $n = 3$, this is proved by Ambrosio and Cabré in ^[9]. Savin proved in ^[10] that for $4\leq n\leq 8$, the above conjecture is true under the additional limit condition that

For $n > 9$, a counterexample is constructed in ^[11]. The conjecture is still open for dimensions $4\leq n\leq 8$ without the additional assumption (1.5).

For Eq (1.1), there are also some results concerning stable solutions. In ^[12], Liouville type results for solutions with finite Morse index were established. By making a delicate use of the classical Moser iteration method, Farina was able to classify finite Morse index solutions in his seminal paper ^[13]. It was proved in ^[13] that if $u\in C^{2}(\mathbb{R}^{n})$ is a stable solution of (1.1) with $1 < p < p_{JL}(n)$, then $u\equiv 0$. Moreover, (1.1) admits a smooth positive, bounded, stable and radial solution for $n\geq 11, p > p_{JL}(n)$. Actually, it was showed in ^[13] that the radial solutions considered in ^[7] are stable when $n\geq 11, p > p_{JL}(n)$. The results in ^[13] also have a lot of generalizations, we refer to ^{[14,15,16,17,18,19]}. As for the classification of the stable solutions of (1.2), it was proved in ^[20] that for $1\leq n\leq 9$, there is no stable solution $u\in C^{2}(\mathbb{R}^{n})$ of (1.2).

In spite of the above mentioned results, there are some intriguing problems which are still open. In ^[21], the authors proposed the following conjecture, which is a natural extension of De Giorgi type conjecture:

Conjecture: Let $n\geq 11, p_{JL}(n) < p < p_{JL}(n-1)$, then all stable solutions to (1.1) must be radially symmetric around some point.

Remark 1.1. When $p > p_{JL}(n-1)$, (1.1) has a positive stable solution which is not radially symmetric. Indeed, let $u$ be a positive radial stable solution of the equation

for $p > p_{JL}(n-1)$ (see ^[13]), then $u$ can also be viewed as a stable solution of the equation

But it is obvious that this solution is not radially symmetric in $\mathbb{R}^{n}$.

Our first objective in this paper is to give some partial results toward the above conjecture. To state our results in a more precise way, let us first introduce some new exponents. Let $\gamma$ be a constant such that

we define

Let

and let

We consider

as a function with respect to the variable $p$. Let $S$ be the set

and let $p_{*}$ be the infimum of $S$. We define

With the help of these numbers, we can give the statement of our first result.

Theorem 1.2. Let $n\geq 11, p_{JL}(n) < p < p_{cs}(n)$, where $p_{cs}(n)$ is defined in (1.10). Let $u$ be a positive stable solution of (1.1) such that $u$ is even symmetric with respect to the planes $\{x_{i} = 0\}, i = 1, 2, \cdots n$, then $u$ is radially symmetric with respect to the origin.

In Theorem 1.2, we need the assumption that $u$ is a positive solution of (1.1). But under suitable conditions, we can show that stable solutions of (1.1) do not change sign. For our purpose, we use $p_{1}(n)\leq p_{2}(n)\leq p_{3}(n)$ to denote the three numbers such that

We define

Theorem 1.3. Let $n\geq 11, p_{JL}(n) < p < p_{si}(n)$, where $p_{si}(n)$ is defined in (1.12). Let $u$ be a stable solution of Eq (1.1), then $u$ does not change sign. If $u$ is a axially symmetric stable solution of (1.1), then $u$ does not change sign when $p_{JL}(n) < p < p_{JL}(n-1)$.

Remark 1.4. By using MATLAB, we can give some examples for $p_{cs}(n)$ and $p_{si}(n)$.

For Eq (1.2), we have the following result.

Theorem 1.5. Let $u$ be a smooth stable solution of Eq (1.2) for $n = 10$, then $u$ is radially symmetric with respect to some point in $\mathbb{R}^{n}$.

Remark 1.6. If $n\geq 11$, then Eq (1.2) has a smooth stable solution which is not radially symmetric. For more discussions, we refer to ^[22].

The rest of the paper will be organized as follows. In section 2, we consider rigidity results on the unit sphere for some second order equations. Rigidity results on compact manifolds have been considered by several authors, see for instance ^{[1,23,24,25,26,27]}. We point out that in the above papers, the proof of the rigidity results depends heavily on the classical Bochner formula. In section 3, we first use a monotonicity formula to study the qualitative properties of solutions. Then, by combing the rigidity results and the qualitative properties of solutions, we can verify the assumption of Theorem 1.1 in ^[5] and obtain the symmetry properties of stable solutions. In section 4, we give the prove of Theorem 1.5.

Notation. In some situations, we will write a point $x\in\mathbb{R}^{n}$ as $x = (r, \theta)$, where $(r, \theta)$ is the spherical coordinates and $S^{n-1}\subset\mathbb{R}^{n}$ is the unit sphere. In the rest of the paper, $c$ will denote a positive constant which may vary from line to line.

2.   A second order equation on the unit sphere

In this section, we consider the equation

where

and $\Delta_{S^{n-1}}$ is the Laplace-Beltrimi operator on the unit sphere. In the rest of this section, we will always assume that $n\geq 11$. The main result in this section is the following.

Theorem 2.1. Let $\gamma$ be a constant such that

If $p_{JL}(n) < p < p_{cs}(n)$ with $p_{cs}(n)$ be the number defined in (1.9) and if $\phi$ is a positive solution of (2.1) such that (2.2) holds. Suppose

where $\Phi_{i}, i = 1, 2, \cdots, n$ are the eigenfunctions of the operator $-\Delta_{S^{n-1}}$ corresponding to the eigenvalue $n-1$, then $\phi$ is a constant solution of Eq (2.1). If $\{\phi-\beta^{\frac{1}{p-1}}\}$ has at least three connected components, then the same result holds.

In order to prove Theorem 2.1, we need several lemmas.

Lemma 2.2. Let $p_{JL}(n) < p < p_{JL}(n-1)$ and let $\phi\in H^{1}(S^{n-1})$ be a weak solution of (2.1) such that

for every $\psi\in H^{1}(S^{n-1})$, then $\phi\in C^{2}(S^{n-1})$.

Proof. We take $\psi = |\phi|^{\frac{\gamma-1}{2}}\phi$ into (2.2), where $\gamma$ is a positive constant which will be chosen later. Then

Multiplying the both sides of (2.1) by $|\phi|^{\gamma-1}\phi$ and integrating over $S^{n-1}$, we can get that

(2.4) is equivalent to

By combining (2.3) and (2.5) together, we can obtain that

It is easy to check that

when $1\leq\gamma < 2p+2\sqrt{p(p-1)}-1$. By applying Hölder's inequality, we have

It follows from (2.7) that $\int_{S^{n-1}}|\phi|^{p+\gamma}d\theta$ is finite. By formula (5.10) in ^[13], we know that there exists a constant $\gamma$ such that $(p+\gamma)/(p-1) > (n-1)/2$. Therefore, $|\phi|^{p-1}\in L^{q}(S^{n-1})$ for some $q > (n-1)/2$. The standard regularity results in ^[28] imply that $\phi\in C^{2}(S^{n-1})$.

Lemma 2.3. Let $p_{JL}(n) < p < p_{JL}(n-1)$ and let $\phi\in C^{2}(S^{n-1})$ be a positive solution of (2.1) such that (2.2) holds, then

where $\gamma$ is the constant used in the proof of Lemma 2.2 and $\alpha(p, \gamma, n)$ is the constant defined in (1.6).

Proof. Let $\theta_{0}\in S^{n-1}$ be a point such that $\phi(\theta_{0}) = \|\phi\|_{L^{\infty}(S^{n-1})} = \eta$. By taking suitable orthogonal transformation, we may assume that $\theta_{0}$ is the south pole. Let us introduce the following coordinates on $S^{n-1}$,

where $\xi\in[0, \pi), \xi_{1}\in [0, 2\pi), \xi_{k}\in [0, \pi)$ for $k = 2, 3, \cdots n-2$. The coordinate of the point $\theta_{0}$ is given by $(0, 0, \cdots, 0)$. By (2.1), we know that $\phi$ satisfies the equation

where $S^{n-2}$ is the unit sphere in $\mathbb{R}^{n-1}$ and $\Delta_{S^{n-2}}$ is the Laplace -Beltrami operator on $S^{n-2}$. We define

where $\omega_{n-2}$ is the area of $S^{n-2}$. It follows from (2.9) that $\hat{\phi}$ satisfies

By the Jensen's inequality, we can get that

Let $\xi_{1}$ be the first point such that $\hat\phi(\xi_{1}) = \frac{\eta}{2}$. It follows from (2.11) that $\hat{\phi}$ is strictly decreasing in $(0, \xi_{1})$. We will focus on the case $\xi_{1} < \frac{\pi}{2}$ since the case $\xi_{1}\geq\frac{\pi}{2}$ can be dealt with similarly. Let $\gamma$ be the constant used in the proof of Lemma 2.2. By (2.10), we can obtain that

This implies $\xi_{1}\geq\eta^{\frac{1-p}{2}}.$ By the above analysis, we can get that

By Lemma 2.2, we know that

We get from (2.12) and (2.13) that

Hence (2.8) holds.

Lemma 2.4. Let $\phi$ be a positive solution of (2.1) such that

where $\Phi_{i}, i = 1, 2, \cdots, n$ are the eigenfunctions of the operator $-\Delta_{S^{n-1}}$ corresponding to the eigenvalue $n-1$, then

Proof. We define

where $\overline{\phi}$ is given by

Then $\tilde{\phi}$ satisfies the equation

Multiplying the both sides of (2.17) by $\tilde{\phi}$ and using integration by part, we can get that

By (2.15) and the definition of $\tilde{\phi}$, we know that

By (2.18) and the Poincaré's inequality, we have

If $\tilde{\phi}\neq 0$, then

It follows that

Hence (2.16) holds.

Lemma 2.5. If $\phi$ is a positive solution of (2.1) such that $\{\phi-\beta^{\frac{1}{p-1}}\neq 0\}$ has at least three connected components, then

Proof. The equation (2.1) can be written as

Since $\{\phi-\beta^{\frac{1}{p-1}}\neq 0\}$ has at least three connected components, then there is a connected component $\Omega_{0}$ of $\{\phi-\beta^{\frac{1}{p-1}}\neq 0\}$ such that the area of $\Omega_{0}$ is less than $\frac{1}{3}\omega_{n-1}$. let $1_{\Omega_0}$ be the function defined by

Multiplying the both sides of (2.22) by $(\phi-\beta^{\frac{1}{p-1}})1_{\Omega_{0}}$ and using integration by part, we can get that

Let $\lambda_{1}(\Omega_{0})$ be the first eigenvalue of the eigenvalue problem

By (2.23) and the mean value theorem, we can get that

It follows from (2.24) that

Since the area of $\Omega_{0}$ is less than $\frac{1}{3}\omega_{n-1}$, where $\omega_{n-1}$ is the area of the unit sphere in $\mathbb{R}^{n}$. By using Schwartz symmetrization, we can get that

It follows from (2.25) and (2.26) that

Hence (2.21) holds.

Remark 2.6. We notice that

then

Lemma 2.7. Let $\overline{p}$ be a constant such that $p_{JL}(n) < \overline{p} < p_{JL}(n-1)$. There exists a positive constant $c$ such that if $\phi\in C^{2}(S^{n-1})$ is a nonconstant solution of (2.1) for $p_{JL}(n) < p < \overline{p}$, then

Proof. Suppose (2.28) does not hold, then there exists a sequence $\{\phi_{m}\}$ such that $\phi_{m}$ satisfies

and

Since $-\phi_{m}$ is also a solution of (2.29), without loss of generality, we can assume that

It follows from the proof of Lemma 2.2 that $\int_{S^{n-1}}\phi_{m}^{2}d\theta$ remains bounded. So (2.30) implies

By (2.8) and (2.32), we can get that there exist two constants $p_{0}$ and $c_{0}$ such that

Moreover, $c_{0}$ is a constant solution of (2.1) for $p = p_{0}$. Therefore,

We get from (2.31) that

Therefore,

It follows from (2.34) that $c_{0}$ is not zero. Let

then $\lim_{m\rightarrow +\infty}\psi_{m} = 0$ and $\psi_{m}$ satisfies the equation

It is easy to verify that

for some positive constant $c$ independent of $m$. We define

then $v_{m}$ satisfies

Since

and

By standard elliptic estimates, we know that there exists a nontrivial function $v_{\infty}$ such that $v_{m}\rightarrow v_{\infty}$ in $H^{1}(S^{n-1})$. Moreover, $v_{\infty}$ satisfies the equation

Then we deduce that $v_{\infty}$ is a nontrivial eigenfunction of $-\Delta_{S^{n-1}}$ corresponding to the eigenvalue $(p_{0}-1)\beta_{0}$. On the other hand, it is easy to see that

and $p_{JL}(n) > (n+1)/(n-3)$ when $n\geq 11$. Therefore, $(p_{0}-1)\beta_{0}$ can not be an eigenvalue of $-\Delta_{S^{n-1}}$. By combining these two facts together, we obtain a contradiction. Next, we can give some estimates about the constant $c$ in Lemma 2.7.

Lemma 2.8. Let $p_{JL}(n) < p < p_{JL}(n-1)$ and let $\phi$ be a positive solution of (2.1) such that

then the constant $c$ in Lemma 2.7 can be estimated by $c_{s}(p, n)$, where $c_{s}(p, n)$ is defined by (1.8).

Proof. Multiplying the both sides of (2.1) by $\phi$ and integrating over $S^{n-1}$, we can get that

We take $\psi = \phi$ into (2.2), then

By (2.38) and (2.39), we can get that

By the Poincaré's inequality, we know that

It follows from (2.40) and (2.41) that

In order to estimate the constant $c$ in Lemma 2.7, we need to give a lower bound for $\int_{S^{n-1}}|\nabla_{S^{n-1}}\phi|^{2}d\theta$. Since we have assumed that

then there exists a point $\theta_{0}$ such that $\phi(\theta_{0}) = \beta(p, n)$. By taking suitable orthogonal transformation, we may assume that $\theta_{0}$ is the south pole. We use the coordinates used in the proof of Lemma 2.2. By (2.1), we know that $\phi$ satisfies Eq (2.9). We define

then $\hat{\phi}$ satisfies (2.10) and (2.11). Let $\xi_{1}$ be the first point such that

We know from (2.11) that

We will assume that $\xi_{1} < \frac{\pi}{2}$ since the case $\xi_{1} < \frac{\pi}{2}$ can be dealt with similarly. By (2.10), we can get that

We deduce that

Let

By (2.1) and the Jensen's inequality, we can get that $\overline{\phi}\leq\beta^{\frac{1}{p-1}}$. Therefore,

It follows from the Poincaré's inequality that

Therefore,

By the above analysis, we know that (1.8) holds.

Lemma 2.9. Let $\phi$ be a positive solution of (2.1) such that (2.2) holds. If

then $\phi$ is a constant when $p_{JL}(n) < p < p_{cs}(n)$, where $p_{cs}(n)$ is defined by (1.9).

Proof. By (2.38) and (2.39), we have

Let $\phi$ be a nonconstant solution of (2.1) satisfying (2.2), we know from Lemma 2.7 that $\phi$ satisfies (2.28). By combining (2.28) and (2.45) together, we can get that

It follows from (2.46) that

when $p_{JL}(n) < p < p_{cs}(n)$. Since we have assumed that $\phi$ is a nonconstant solution of $(2.1)$, this is a contradiction.

Proof of Theorem 2.1:. This result follows from Lemma 2.4, Lemma 2.5 and Lemma 2.9.

Remark 2.10. In this section, we always assume that $\phi$ is positive. But we can prove that if $\phi$ is a solution of (2.1) depends only on the variable $\xi$, then $\phi$ does not change sign. The proof of this fact will be given in the appendix.

Remark 2.11. It is proved in ^[29] that if $n\geq 4$ and $(n+1)/(n-3) < p < p_{JL}(n-1)$, then (2.1) has a nonconstant positive solution.

Remark 2.12. By Lemma 1 in ^[30], we have the following Hardy type inequality,

The equation (2.1) has a singular solution which is given by

Suppose $\phi_{*}$ satisfies (2.2), then

If $p = p_{JL}(n-1)$, then

Let us define

then

If $p > (n-1)/(n-5)$, then $g'(p) < 0$. Therefore, the singular solution $\phi_{*}$ satisfies (2.2) if $p\geq p_{JL}(n-1)$.

3.   Qualitative properties of stable solutions

In this section, we give the proof of Theorem 1.2 and Theorem 1.3.

Lemma 3.1. Let $n\geq 11, p_{JL}(n) < p < p_{si}(n)$, where $p_{si}(n)$ is defined in (1.12). If $\phi$ is a nontrivial solution of (2.1) such that (2.2) holds, then $\phi$ does not change sign.

Proof. We assume that $\phi$ change sign. Without loss of generality, we can assume that there exists a connected component $\Omega_1$ of $\{\phi > 0\}$ such that $\lambda_{1}(\Omega_{1})\geq n-1$, where $\lambda_{1}(\Omega_{1})$ is the first eigenvalue of the eigenvalue problem

Multiplying the both sides of (2.1) by $\phi$ and integrating over $\Omega_1$, we can get that

We take $\psi = u1_{\Omega_1}$ into (2.2), where $1_{\Omega_1}$ is the function defined by

Then

By (3.1) and (3.2), we know that

It follows that if $p_{JL}(n) < p < p_{si}(n)$, then $\phi$ vanishes identically on $\Omega_1$. Since we have assumed that $\phi > 0$ on $\Omega_1$, this is a contradiction.

Proof of Theorem 1.3: We consider the transform

Since $u$ satisfies (1.1), then $w$ is a bounded solution of the equation

We set

By (3.4), we get that

By the estimates in ^[19], we can get that $\partial_{t}w, \partial_{tt}w, |\nabla_{S^{n-1}}|$ are uniformly bounded. Integrating (3.6) from $-s$ to $s$, we find

for some constant $c$ independent of $s$. Let $s$ tend to $+\infty$ in (3.7), then

Similar to the proof of Theorem 1.4 in ^[1], we can obtain that

For any sequence $\{t_{k}\}$ such that $t_{k}\rightarrow \infty$ as $k\rightarrow \infty$, we consider the translation of $w$ defined by $w_{k}(t, \theta) = w(t+t_{k}, \theta)$. Then there exist a subsequence $\{w_{l_{k}}(t, \theta)\}$ and a function $w_{\infty}(t, \theta)$ such that $w_{l_{k}}(t, \theta)\rightarrow w_{\infty}(t, \theta)$ in $C^{2}([-1, 1]\times S^{n-1})$. By (3.8) and the dominated convergence theorem, we know that there exists a function $\phi(\theta)$ such that $w_{\infty}(t, \theta) = \phi(\theta)$. Moreover, $\phi$ is a solution of (2.1) such that (2.2) holds. If $\phi = 0$, then $\lim_{t\rightarrow +\infty}E(w)(t) = 0$. But we also have $\lim_{t\rightarrow -\infty}E(w)(t) = 0$ since $u$ is regular at the origin. It follows easily that $w\equiv 0$. Since we have assumed that $u$ is a nontrivial solution, this is a contradiction. Therefore $\phi$ is not zero. If $\phi\neq 0$, we know from Lemma 3.1 that $\phi$ does not change sign. Suppose there exist two sequences $\{t_{k}\}$ and $\{\tilde{t}_{k}\}$ such that

and

then $\{u\neq 0\}$ has a bounded connected component. Without loss of generality, we can assume there exists a bounded connected component $\Omega_{-}$ such that $u < 0$ on $\Omega_{-}$. Then $u$ satisfies the equation

Since $u$ is a stable solution of (1.1), then $L = \Delta+p|u|^{p-1}$ satisfies the refined maximum principle (see ^[31]). Since

we get from the refined maximum principle that $u\geq0$ on $\Omega_{-}$. In view of the definition of $\Omega_{-}$, we get a contradiction. By the above arguments, we know that there exits a positive constant $R_{0}$ such that $u$ doesn't change sign on $\mathbb{R}^{n}\backslash B_{R_{0}}$. By applying the refined maximum principle again, we know that $u$ does not change sign.

If $u$ is axially symmetric, then the proof is essentially the same as the arguments above. The only difference is that we need to use remark 2.10 rather than Lemma 3.1 to show that there exits a positive constant $R_{0}$ such that $u$ doesn't change sign on $\mathbb{R}^{n}\backslash B_{R_{0}}$.

Proof of Theorem 1.2. Let $n\geq11, p_{JL}(n) < p < p_{cs}(n)$ and let $u$ be a positive stable solution of (1.1). Let us consider the transform

Since $u$ satisfies (1.1), then $w$ is a bounded solution of Eq (3.4). Let $\{t_{k}\}$ be a sequence such that $t_{k}\rightarrow \infty$ as $k\rightarrow \infty$. Similar to the arguments used in the proof of Theorem 1.3, we know that there exists a function $\phi\in H^{1}(S^{n-1})$ such that

Moreover, $\phi$ is a solution of (2.1) such that (2.2) holds. By Lemma 2.2, we have $\phi\in C^{2}(S^{n-1})$. If $u$ is even symmetric with respect to the $\{x_{i} = 0\}, i = 1, 2, \cdots n$, then

Since $p_{JL}(n) < p < p_{cs}(n)$, we know from Theorem 2.1 that $\phi$ is a constant function. In particular, we have

Since the sequence $\{t_{k}\}$ can be arbitrary, we conclude that

Since $p > p_{JL}(n)$, then $p > n/(n-4)$. By Theorem 4.4 in ^[5], we can get that

where

and

Moreover, for any integer $\tau\geq0$, we have $\nu(r, \theta)$ satisfies

uniformly in $C^{\tau}(S^{n-1})$, where $V$ equals either zero or a first eigenfunctions of the operator $-\Delta_{S^{n-1}}$. Since we have obtained the asymptotic expansion (3.11) which is good enough to apply the moving plane method, then the rest of the proof is essentially the same as the proof of Theorem 1.1 in ^[4].

4.   The proof of Theorem 1.5

In this section, we give the proof of Theorem 1.5, the proof is mainly based on the following observation.

Proposition 4.1. Let $n = 10$ and let $u$ be a smooth stable solution of Eq (1.2), then

In order to prove Proposition 4.1, we first recall a monotonicity formula.

Lemma 4.2. If $u$ is a solution of the equation (1.2), then

where

Moreover, if $u$ is a smooth stable solution of (1.1), then

Proof. The proof of (4.2) follows from a scaling argument which is similar to the proof Proposition 5.1 in ^[32]. The proof of (4.3) follows easily from the capacity estimates in ^[33].

With the help of Lemma 4.2, we can give the proof of Proposition 4.1.

proof of Proposition 4.1. The proof of Proposition 4.1 will consist of the following four steps.

Step 1: Let $\{\lambda_{k}\}$ be a sequence such that $\lim_{k\rightarrow +\infty}\lambda_{k} = +\infty$. For any $\lambda_k$, we define $u^{\lambda_k}(x) = u(\lambda_{k} x)+2\ln(\lambda_{k}).$ It is easy to check that $u^{\lambda_{k}}(x)$ is also a stable solution of (1.1). By the capacity estimates (see for instance ^[33]), we know that $u^{\lambda_{k}}\rightarrow u^{\infty}$ for some function $u^{\infty}\in H^{1}_{loc}(\mathbb{R}^{n})$. Moreover, $u^{\infty}$ is a stable solution of (1.1).

Step 2: For any $0 < R_{1} < R_{2} < +\infty$, by Lemma 4.2,

By the scaling invariance of $E$, we have

We use Lemma 4.2 again, then

Therefore,

It follows that there exists a function $\phi\in H^{1}(S^{n-1})$ such that $u^{\infty} = \phi-2\ln(r)$. Moreover, $\phi$ satisfies the equation

Step 3: For every $\delta > 0$, we choose a function $\eta_{\delta}\in C^{\infty}_{0}((\frac{\delta}{2}, \frac{2}{\delta}))$ such that $\eta_{\delta}\equiv 1$ in $(\delta, \frac{1}{\delta})$, and $r|\eta'_{\delta}(r)|\leq 4.$ For every $\psi\in H^{1}(S^{n-1})$, we define $\psi_{\delta} = r^{-\frac{n-2}{2}}\psi(\theta)\eta_{\delta}(r)$. For every $\psi\in H^{1}(S^{n-1})$, we define $\psi_{\delta} = r^{-\frac{n-2}{2}}\psi(\theta)\eta_{\delta}(r)$. Since $u^{\infty}$ is stable, we have

Therefore, $\phi$ satisfies

for every $\psi\in H^{1}(S^{n-1})$.

Step 4: We take $\psi = e^{\frac{\phi}{2}}$ into (4.9), then

Multiplying the both sides of (4.8) by $e^{\phi}$ and using integration by part, we have

If $n = 10$, then $(n-2)^{2}/4 = 2(n-2)$. By (4.10) and (4.11), we can get that

It follows from (4.12) that $\phi = \ln(16)$ is a constant. Since $\{\lambda_{k}\}$ can be arbitrary, we can obtain that proposition 4.1 holds.

Proof of Theorem 1.5. It follows from proposition 4.1 and Theorem 1.3 in ^[6].

Appendix 1: A Liouville type result

In this appendix, we prove the claim in remark 2.10. The proof is based on the the following result.

Proposition 4.3. Let $p\geq \frac{n+1}{n-3}$ and $(p-1)\mu\geq n-1$. If $\phi$ is a solution of the equation

where $B_{r}\subset\mathbb{R}^{n-1}$ is a ball and $0 < r < 1$, then $\phi = 0$.

Proof. Multiplying the both sides of (4.13) by $(\frac{2}{1+|x|^{2}})^{n-1}\phi$ and using integration by part, we can get that

Multiplying the both sides of (4.13) by $(\frac{2}{1+|x|^{2}})^{n-1}(x\cdot\nabla \phi)$ and using integration by part, we can get that

where

It follows that

Multiplying the both sides of (4.13) by $x\cdot\nabla(\frac{2}{1+|x|^{2}})^{n-1}\phi$ and using integration by part, we can get that

By some computations, we can get that

By (4.16) and (4.17), we have

We combine (4.14), (4.15) and (4.18) in the following way:

then

If $p\geq \frac{n+1}{n-3}$ and $(p-1)\mu\geq(n-1)$, then the left hand side of the last identity will become non-positive, therefore, Eq (4.13) has only trivial solution.

Corollary 4.4. If $p\geq \frac{n+1}{n-3}$ and if $\phi$ is a nontrivial solution of Eq (2.1) depends only on the variable $\xi$, here we use the coordinates in the proof of Lemma 2.3, then $\phi$ does not change sign.

Proof. If $\phi$ change sign, then there exists $0 < r < 1$ such that (4.13) has a nontrivial solution, this is a contradiction.

Acknowledgments

The research of J. Wei is partially supported by NSERC of Canada.

Conflict of interest

We declare that we have no financial and personal relationships with other people or organizations that can inappropriately influence our work.