Non-radial solutions to a bi-harmonic equation with negative exponent

TL;DR

This paper establishes the existence of non-radial entire solutions to a specific bi-harmonic equation with a negative exponent in three-dimensional space, addressing an open question in the field.

Contribution

It proves the existence of non-radial solutions to a bi-harmonic PDE with negative exponent, solving an open problem posed by McKenna and Reichel.

Findings

Existence of non-radial solutions for q>1

Answers an open question in PDE theory

Extends understanding of bi-harmonic equations with negative exponents

Abstract

We prove the existence of non-radial entire solution to $$\Delta^2 u+u^{-q}=0\quad\text{in }\mathbb{R}^3,\quad u>0,$$ for $q>1$. This answers an open question raised by P. J. McKenna and W. Reichel (E. J. D. E. \textbf{37} (2003) 1-13).

Equations

$${\Delta }^{2}u+u^{-q}=0\text{in }{\mathbb{R}}^3,u>0,$$

$${\Delta }^{2}u=\frac{n-4}{2}{u}^{\frac{n+4}{n-4}}\text{in }{\mathbb{R}}^n,u>0.$$

$$\underset{|x|\to +\infty }{\lim }\frac{u(x)}{|x|}=C>0,$$

$$\underset{|x|\to +\infty }{\lim }\frac{u(x)}{|x{|}^2}=C>0.$$

$$u(x)=\frac{1}{8\pi }{\int }_{{\mathbb{R}}^3}\frac{|x-y|}{u^q(y)}dy+\gamma ,$$

$$(-\Delta {)}^{n}u+u^{-(4n-1)}=0\text{ in }{\mathbb{R}}^{2n-1},n\ge 2,$$

$$u(x)=c_n{\int }_{{\mathbb{R}}^{2n-1}}\frac{|x-y|}{u^{4n-1}(y)}dy,$$

$${\Delta }^{2}w+w^{-q}=0\text{in }{\mathbb{R}}^n,w>0$$

$$\beta :=\frac{1}{8\pi }{\int }_{{\mathbb{R}}^3}{u}^{-q}dx<+\infty .$$

$$u(x)=(\beta +o(1))|x|+\sum _{i\in {\mathcal{I}}_1}{a}_i{x}_i^2+\sum _{i\in {\mathcal{I}}_2}{b}_i{x}_i+c,o(1)\stackrel{|x|\to \infty }{\to }0,$$

$${\mathcal{I}}_1,{\mathcal{I}}_2\subseteq \{1,2,3\},{\mathcal{I}}_1\cap {\mathcal{I}}_2=\varnothing ,a_i>0\text{ for }i\in {\mathcal{I}}_1,|b_i|<\beta \text{ for }i\in {\mathcal{I}}_2,c>0.$$

$$\underset{|x|\to \infty }{\mathrm{liminf}}\frac{u(x)}{|x{|}^2}={\kappa }_1\text{and }\underset{|x|\to \infty }{\mathrm{limsup}}\frac{u(x)}{|x{|}^2}={\kappa }_2.$$

$$\underset{|x|\to \infty }{\mathrm{liminf}}\frac{u(x)}{|x|}\in (0,\infty )\text{and }\underset{|x|\to \infty }{\mathrm{limsup}}\frac{u(x)}{|x{|}^2}=\kappa .$$

$${\int }_{{\mathbb{R}}^3}{u}^{-q}dx<+\infty ,$$

$$(-\Delta {)}^{\frac{n}{2}}u=e^{nu}\text{mbox}in{\mathbb{R}}^n,{\int }_{{\mathbb{R}}^n}{e}^{nu}dx<+\infty .$$

$$v(x):=\frac{1}{8\pi }{\int }_{{\mathbb{R}}^3}\frac{|x-y|-|y|}{u^q(y)}dy,w:=u-v.$$

$${\int }_{B_R^c}\frac{dx}{u^q(x)}<8\pi \epsilon ,$$

$$v(x)\ge \frac{1}{8\pi }{\int }_{B_R}\frac{|x|-2|y|}{u^q(y)}dy-\frac{1}{8\pi }{\int }_{B_R^c}\frac{|x|}{u^q(y)}dy\ge (\beta -2\epsilon )|x|-C(R).$$

$$|v(x)|\le \beta |x|\text{in }{\mathbb{R}}^3.$$

$$\underset{|x|\to \infty }{\lim }\frac{v(x)}{|x|}=\beta .$$

$${\Delta }^{2}w=0\text{ in }{\mathbb{R}}^3,w(x)\ge -\beta |x|,$$

$$w(x)=\sum _{i\in {\mathcal{I}}_1}{a}_i{x}_i^2+\sum _{i\in {\mathcal{I}}_2}{b}_i{x}_i+c_0,$$

$$u(x)=\frac{1}{8\pi }{\int }_{{\mathbb{R}}^3}\frac{|x-y|-|y|}{u^q(y)}dy+\sum _{i\in {\mathcal{I}}_1}{a}_i{x}_i^2+\sum _{i\in {\mathcal{I}}_2}{b}_i{x}_i+c.$$

$$u(x)\le C+|b_1{x}_1|+b_1{x}_1\text{on }\mathcal{C}:=\{x=(x_1,\bar{x})\in \mathbb{R}\times {\mathbb{R}}^2:|\bar{x}|\le 1\},$$

$${\Delta }^{2}v=-(v+P{)}^{-q},v+P>0.$$

$$v(x)=\frac{1}{8\pi }{\int }_{{\mathbb{R}}^3}|x-y|\frac{1}{(P(y)+v(y){)}^q}dy,$$

$$X:=\left\{v\in C^0({\mathbb{R}}^3):\Vert v{\Vert }_X<\infty \right\},\Vert v{\Vert }_X:=\underset{x\in {\mathbb{R}}^3}{\sup }\frac{|v(x)|}{1+|x|},$$

$$X_{ev}:=\left\{v\in X:v(x)=v(-x)\forall x\in {\mathbb{R}}^3\right\},\Vert v{\Vert }_{X_{ev}}:=\Vert v{\Vert }_X,$$

$$X_{rad}:=\left\{v\in X:v\text{ is radially symmetric}\right\},\Vert v{\Vert }_{X_{rad}}:=\Vert v{\Vert }_X.$$

Full text

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for $q>1$. This answers an open question raised by P. J. McKenna and W. Reichel (E. J. D. E. 37 (2003) 1-13).

1 Introduction

We consider the following bi-harmonic equation with negative exponent

[TABLE]

where $q>0$.

For $q=7$, problem (1) can be seen as a fourth order analog of the Yamabe equation (see [1, 5, 21]), namely

[TABLE]

In the recent past, radial solutions to equation (1) have been studied by many authors, especially the existence and asymptotic behavior:

Theorem A

([5, 6, 8, 11, 16, 20])

• i)

There is no entire solution to (1) for $0<q\leq 1$.

• ii)

If $u$ has exact linear growth at infinity, that is,

[TABLE]

then $q>3$. Moreover, for $q=7$, $u$ is given by $u(x)=\sqrt{\sqrt{1/15}+|x|^{2}}$, and is unique up to dilation and translations.

• iii)

For $q>3$ there exists radial solution with exact linear growth.

• iv)

For $q>1$ there exists radial solution with exact quadratic growth, that is,

[TABLE]

• v)

For $1<q<3$ there exists a radial solution $u$ such that $r^{-\frac{4}{q+1}}u(r)\to C(q)>0$ as $r\to\infty$ (the constant $C(q)$ is explicitly known).

• vi)

For $q=3$ there exists a radial solution $u$ such that $r^{-1}(\log r)^{-\frac{1}{4}}u(r)\to 2^{\frac{1}{4}}$ as $r\to\infty$.

It has been shown by Choi-Xu [5] that if $u$ is a solution to (1) with $q>4$, and $u$ has exact linear growth at infinity then $u$ satisfies the integral equation

[TABLE]

for some $\gamma\in\mathbb{R}$, and $\gamma=0$ if and only if $q=7$. In fact, every positive solution $u$ to

[TABLE]

with exact linear growth at infinity satisfies

[TABLE]

where $c_{n}$ is a dimensional constant, see [7], [17]. For the classification of solutions to the above integral equation we refer the reader to [12], [20].

In [16] McKenna-Reichel proved the existence of non-radial solution to

[TABLE]

for $n\geq 4$. This was a simple consequence of their existence results to (4) in lower dimension. More precisely, if $u$ is a radial solution to (4) with $n\geq 3$ then $w(x):=u(x^{\prime})$ is a non-radial solution to $\Delta^{2}w+w^{-q}=0$ in $\mathbb{R}^{n+1}$, where $x=(x^{\prime},x^{\prime\prime})\in\mathbb{R}^{n}\times\mathbb{R}$. Then they asked whether in $\mathbb{R}^{3}$ non-radial positive entire solution exist. (See [Open Questions (1), [16]].)

We answer this question affirmatively. (See Theorem 1.2 below.) In fact we prove the following theorems.

Theorem 1.1

Let $u$ be a solution to (1) for some $q>1$. Assume that

[TABLE]

Then, up to a rotation and translation, we have

[TABLE]

where

[TABLE]

Theorem 1.2

Let $q>1$. Then for every $0<\kappa_{1}<\kappa_{2}$ there exists a non-radial solution $u$ to (1) such that

[TABLE]

Theorem 1.3

Let $q>7$. Then for every $\kappa>0$ there exists a non-radial solution $u$ to (1) such that

[TABLE]

The non-radial solutions constructed in Theorem 1.2 also satisfies the following integral condition

[TABLE]

for $q>\frac{3}{2}$. Note that McKenna-Reichel’s non-radial example has infinite $L^{1}$ bound: $\int_{\mathbb{R}^{n+1}}w^{-q}dx=+\infty$.

The existence of infinitely many entire non-radial solutions with different growth rates for the conformally invariant equation $\Delta^{2}u+u^{-7}=0$ in $\mathbb{R}^{3}$ is in striking contrast to other conformally invariant equations $-\Delta u=u^{\frac{n+2}{n-2}}$ in $\mathbb{R}^{n}$, $n\geq 3$ and $(-\Delta)^{m}u=u^{\frac{n+2m}{n-2m}}$ in $\mathbb{R}^{n},\ n>2m$. In both cases all solutions are radially symmetric with respect to some point in $\mathbb{R}^{n}$, see [2], [4], [13] and [18].

Our motivation in the proof of Theorems 1.2-1.3 come from a similar phenomena exhibited in the following equation

[TABLE]

It has been proved that for $n\geq 4$ problem (9) admits non-radial entire solutions with polynomial growth at infinity, see [3], [9], [10], [14], [15], [19] and the references therein. It is surprising to see that conformally invariant equations with negative powers share similar phenomena.

In the remaining part of the paper we prove Theorems 1.1-1.3 respectively. We also give a new proof of $iii)$-$iv)$ of Theorem A, see sub-section 2.1.

2 Proof of the theorems

We begin by proving Theorem 1.1.

Proof of Theorem 1.1 Let $u$ be a solution to (1)-(5). We set

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Fixing $\varepsilon>0$ and $R=R(\varepsilon)>0$ so that

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one gets

[TABLE]

Using that $||x-y|-|y||\leq|x|$, form (10), we obtain

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Combining these estimates we deduce that

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It follows that $w$ satisfies

[TABLE]

and hence, $w$ is a polynomial of degree at most $2$, see for instance [14, Theorem 5]. Indeed, up to a rotation and translation, we can write

[TABLE]

where $\mathcal{I}_{1},\mathcal{I}_{2}$ are two disjoint (possibly empty) subsets of $\{1,2,3\}$, $a_{i}\neq 0$ for $i\in\mathcal{I}_{1}$, $b_{i}\neq 0$ for $i\in\mathcal{I}_{2}$ and $c_{0}\in\mathbb{R}$. Therefore, up to a rotation and translation, we have

[TABLE]

Now $u>0$ and $|v(x)|\leq\beta|x|$ lead to $a_{i}>0$ for $i\in\mathcal{I}_{1}$, $|b_{i}|\leq\beta$ for $i\in\mathcal{I}_{2}$ and $c=u(0)>0$.

In order to prove that $|b_{i}|<\beta$ we assume by contradiction that $|b_{i_{0}}|=\beta$ for some $i_{0}\in\mathcal{I}_{2}$. Up to relabelling we may assume that $i_{0}=1$. Then

[TABLE]

a contradiction to (5).

We conclude the proof. $\square$

Now we move on to the existence results. We look for solutions to (1) of the form $u=v+P$ where $P$ is a polynomial of degree $2$. Notice that $u=v+P$ satisfies (1) if and only if $v$ satisfies

[TABLE]

In particular, if $P\geq 0$, and $v$ satisfies the integral equation

[TABLE]

then $v$ satisfies (11). Thus, we only need to find solutions to (12) (or a variant of it), and we shall do that by a fixed point argument. Let us first define the spaces on which we shall work:

[TABLE]

[TABLE]

[TABLE]

The following proposition is crucial in proving Theorem 1.2.

Proposition 2.1

Let $P$ be a positive function on $\mathbb{R}^{3}$ such that $P(-x)=P(x)$ and for some $q>0$

[TABLE]

Then there exists a function $v\in X_{ev}$ satisfying $\min_{\mathbb{R}^{3}}v=v(0)=0$,

[TABLE]

and

[TABLE]

Moreover, if $P$ is radially symmetric then there exists a solution to (13) in $X_{rad}$.

Proof.

Let us define an operator $T:X_{ev}\to X_{ev}$, $v\mapsto\bar{v}$, (In case $P$ is radial we restrict the operator $T$ on $X_{rad}$. Notice that $T(X_{rad})\subset X_{rad}$.) where

[TABLE]

We proceed by steps.

Step 1 $T$ is compact.

Using that $||x-y|-|y||\leq|x|$ we bound

[TABLE]

Differentiating under the integral sign one gets

[TABLE]

We let $(v_{k})$ be a sequence in $X_{ev}$. Then $\bar{v}_{k}:=T(v_{k})$ is bounded in $C^{1}_{loc}(\mathbb{R}^{3})$. Moreover, up to a subsequence, for some $c_{i}\geq 0$ with $i=0,1$, we have

[TABLE]

We rewrite (14) (with $v=v_{k}$ and $\bar{v}=\bar{v}_{k}$) as

[TABLE]

It follows that

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Using that $||x-y|-|x||\leq|y|$ we bound

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This implies that

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Since

[TABLE]

up to a subsequence,

[TABLE]

for some $I\in X_{ev}$. This proves Step 1 as $T$ is continuous.

Step 2 $T$ has a fixed point in $X_{ev}$.

It follows form (15) that there exists $M>0$ such that $T(X_{ev})\subset\mathcal{B}_{M}\subset X_{ev}$. In particular, $T(\bar{\mathcal{B}}_{M})\subset\mathcal{B}_{M}$. Hence, by Schauder fixed point theorem there exists a fixed point of $T$ in $\mathcal{B}_{M}$.

Step 3 $\lim_{|x|\to\infty}\frac{\bar{v}(x)}{|x|}=\frac{1}{8\pi}\int_{\mathbb{R}^{3}}\frac{dy}{(P(y)+|v(y)|)^{q}}=:\alpha(P,v).$

Step 3 follows from

[TABLE]

Step 4 If $v$ is a fixed point of $T$ then $v\geq 0$.

Differentiating under the integral sign, from (14) one can show that the hessian $D^{2}\bar{v}$ is strictly positive definite, and hence $\bar{v}$ is strictly convex. Moreover, using that $(P+|v|)$ is an even function, one obtains $\nabla\bar{v}(0)=0$. This leads to

[TABLE]

We conclude the proposition. ∎

In the same spirit one can prove the following proposition.

Proposition 2.2

Let $P$ be a positive even function on $\mathbb{R}^{3}$ such that for some $q>0$

[TABLE]

Then there exists a positive function $v\in X_{ev}$ satisfying

[TABLE]

Proof of Theorem 1.2 Let $q>1$ and $0<\kappa_{1}<\kappa_{2}$ be fixed. For every $\varepsilon>0$ let $v_{\varepsilon}\in X_{ev}$ be a solution of (13), that is,

[TABLE]

where

[TABLE]

We claim that for every multi-index $\beta\in\mathbb{N}^{3}$ with $|\beta|=2$

[TABLE]

where

[TABLE]

For $|\beta|=2$, differentiating under the integral sign, from (17), we obtain

[TABLE]

where

[TABLE]

Since $q>1$ we have $|D^{\beta}v_{\varepsilon}|\leq C$ on $B_{2}$. For $|x|\geq 2$ we bound

[TABLE]

[TABLE]

[TABLE]

This proves (18). Since $v_{\varepsilon}(0)=|\nabla v_{\varepsilon}(0)|=0$, by (18), we have

[TABLE]

Therefore, for some $\varepsilon_{k}\downarrow 0$ we must have $v_{\varepsilon_{k}}\to v$ in $C^{3}_{loc}(\mathbb{R}^{3})$ for some $v$ in $\mathbb{R}^{3}$, where $v$ satisfies

[TABLE]

Hence, $u=v+P_{0}$ is a solution to (1). Moreover, as $v$ satisfies (22), we have

[TABLE]

This completes the proof.

$\hfill\square$

Proof of Theorem 1.3 Let $q>7$ be fixed. Then for every $\varepsilon>0$ there exists a positive solution $v_{\varepsilon}$ to (16) with

[TABLE]

Setting $u_{\varepsilon}:=v_{\varepsilon}+P_{\varepsilon}$ one gets

[TABLE]

Since $c_{q}:=\frac{1}{2}-\frac{3}{q-1}>0$ for $q>7$, from (26), one obtains

[TABLE]

which implies that $2c_{q}u_{\varepsilon}(0)<4$, that is, $u_{\varepsilon}(0)\leq C$. Therefore, by (23)

[TABLE]

Hence, differentiating under the integral sign, from (23)

[TABLE]

Thus, $(u_{\varepsilon})_{0<\varepsilon\leq 1}$ is bounded in $C^{1}_{loc}(\mathbb{R}^{3})$. This yields

[TABLE]

for some $\delta>0$. Using this, and recalling that $q>4$, we deduce

[TABLE]

Therefore, for some $\varepsilon_{k}\downarrow 0$, we have $u_{\varepsilon_{k}}\to u$, where $u$ satisfies

[TABLE]

We conclude the proof. $\hfill\square$

2.1 A new proof of $iii)$-$iv)$ of Theorem A

Proof of $iii)$ Let $q>3$ be fixed. Then by Proposition 2.1, for every $\varepsilon>0$, there exists a radial function $u_{\varepsilon}$ satisfying

[TABLE]

Since $u_{\varepsilon}$ is radially symmetric, one has (see Eq. (3.3) in [5])

[TABLE]

for some $\delta>0$. Therefore, as $q>3$

[TABLE]

which gives

[TABLE]

As $u_{\varepsilon}(0)=1$, one would get

[TABLE]

Thus, the family $(u_{\varepsilon})_{0<\varepsilon\leq 1}$ is bounded in $C^{1}_{loc}(\mathbb{R}^{3})$. Hence, for some $\varepsilon_{k}\downarrow 0$ we have $u_{\varepsilon_{k}}\to u$ where $u$ satisfies

[TABLE]

Finally, as before, we have

[TABLE]

This completes the proof of $iii)$. $\square$

Proof of $iv)$ Let $q>1$ be fixed. Then by Proposition 2.1, for every $\varepsilon>0$, there exists a non-negative radial function $v_{\varepsilon}$ satisfying

[TABLE]

The rest of the proof is similar to that of Theorem 1.2. $\square$

In the spirit of [5, Lemma 4.9] we prove the following Pohozaev type identity.

Lemma 2.3** **(Pohozaev identity)

Let $u$ be a positive solution to

[TABLE]

for some non-negative polynomial $P$ of degree at most $2$ and $q>4$. Then

[TABLE]

Proof.

Differentiating under the integral sign, from (25)

[TABLE]

Multiplying the above identity by $u^{-q}(x)$ and integrating on $B_{R}$

[TABLE]

Integration by parts yields

[TABLE]

Since $q>4$ and $u(x)\geq\delta|x|$ for some $\delta>0$ and $|x|$ large

[TABLE]

Writing $x=\frac{1}{2}((x+y)+(x-y))$, and setting

[TABLE]

we get

[TABLE]

Notice that $F(x,y)=-F(y,x)$. Hence,

[TABLE]

and

[TABLE]

where the last equality follows from $|x|u^{-q}(x)\in L^{1}(\mathbb{R}^{3})$. Combining these estimates and taking $R\to\infty$ in (27) one gets (26). ∎