A Pohožaev type identity and its application to uniqueness of positive radial solutions of Brezis-Nirenberg problem on an annulus

Abstract

We study the uniqueness of positive radial solutions of the Brezis-Nirenberg problem

$$\{\begin{array}{cc}\mathrm{\Delta }u(x)+\lambda u(x)+u{(x)}^p=0,\hfill & x\in A_{a,b},\hfill \\ u(x)=0,\hfill & x\in \partial A_{a,b},\hfill \end{array}$$

where $n\ge 3$, $b>a>0$, $0<\lambda <{\lambda }_1$, $p>(n+2)/(n-2)$ and ${\lambda }_1$ is the first eigenvalue of −Δ under the Dirichlet boundary condition on $A_{a,b}=\{x\in {\mathbb{R}}^n|a<|x|<b\}$. In particular, in the case $n=3$, we completely solve the problem without any additional assumption, and in the case $n\ge 4$, we show the uniqueness result under $0<{\lambda }_1-\lambda \ll 1$. These results are obtained through a kind of Pohožaev identity.

Introduction

We consider the Brezis-Nirenberg problem on an annulus

$$\{\begin{array}{cc}\mathrm{\Delta }u(x)+\lambda u(x)+u{(x)}^p=0,\hfill & x\in A_{a,b},\hfill \\ u(x)=0,\hfill & x\in \partial A_{a,b},\hfill \end{array}$$

where $n\in \mathbb{N}$ with $n\ge 3$, $b>a>0$, $A_{a,b}=\{x\in {\mathbb{R}}^n|a<|x|<b\}$, $p>1$, $\lambda <{\lambda }_1$ and ${\lambda }_1$ is the first eigenvalue of −Δ under the Dirichlet boundary condition on $A_{a,b}$. It is known that if $\lambda \ge {\lambda }_1$, problem (1) has no positive solution. In the case, a domain is an open ball (replacing $A_{a,b}$ with an open ball), any positive solutions of (1) are radially symmetric [10]. Different from the case domain is an open ball, non-radial positive solutions of problem (1) may exist; see Coffman [4] for the case $n=2,\lambda =-1$ and $p>1$, Li [14] for the case $n\ge 4,\lambda =-1$ and $1<p<(n+2)/(n-2)$, Byeon [2] for the case $n=3,\lambda =0$ and $1<p<5$. In these results, it is assumed that a is sufficiently large and $b=a+c$, where $c>0$ is some constant. On the other hand, the problem of uniqueness of positive radial solution of (1) is also the subject of interest. This paper treats the uniqueness of positive radial solution of (1). Since we are concerned with the uniqueness of positive radial solutions of (1), we study the uniqueness of solutions of the problem

$$\{\begin{array}{cc}\hfill & u_{rr}(r)+\frac{n-1}{r}{u}_r(r)+\lambda u(r)+u{(r)}^p=0,u(r)>0,r\in (a,b),\hfill \\ \hfill & u(a)=u(b)=0.\hfill \end{array}$$

We know that problem (2) has a solution. The uniqueness of solutions of (2) has been studied by several researchers; see [1], [5], [6], [8], [9], [13], [15], [16], [17], [18], [19], [20], [21], [22], [23], [24]. Ni-Nussbaum [18, Theorem 1.2] showed the uniqueness of solutions of (2) in the case $n\ge 2$, $\lambda =0$ and $p>1$. From [18, Theorem 1.4], we see that it has a unique solution if $n\ge 3$, $\lambda >0$ and $1<p\le n/(n-2)$. Yadava [22, Theorems 1.1 and 1.2] obtained the uniqueness in the case $n\ge 3$, $\lambda <0$ and $p\ge (n+2)/(n-2)$ and in the case $n\ge 3$, $\lambda >0$ and $1<p\le (n+2)/(n-2)$. Tang [21, Theorem 1] showed the uniqueness in the case $n\ge 3$, $\lambda <0$ and $p>1$. Felmer-Martínez-Tanaka [8, Theorem 1.1] obtained the uniqueness in the case $n=2$, $\lambda <0$ and $p>1$.

Recently, Yao-Li-Chen [24, Theorem 1.1] essentially obtained the following comprehensive result. Although [24, Theorem 1.1] has different appearances and it does not include the case $\lambda \le 0$, it states almost the same and its proof seems to work for the case $\lambda \le 0$.

Theorem A

Let $n\in \mathbb{N}$ with $n\ge 2$ and $b>a>0$. If one of the conditions

• $n=2$, $\lambda <{\lambda }_1$ and $p>1$,

• $n\ge 3$, $\lambda <{\lambda }_1$ and $1<p\le (n+2)/(n-2)$,

• $n\ge 3$, $\lambda \le 0$ and $p>(n+2)/(n-2)$,

• $n\ge 3$, $0<\lambda <{\lambda }_1$, $p>(n+2)/(n-2)$, and $r_{⁎}\notin (a,b)$, where

  $$r_{⁎}=\sqrt{\frac{2((n-2)p+n-4)((n-2)p-(n+2))}{\lambda (p-1){(p+3)}^2}},$$

holds, then problem (2) has a unique positive solution.

For the reader's convenience, we give the proof of Theorem A in Appendix A. From Theorem A, we see that an additional assumption $r_{⁎}\notin (a,b)$ is assumed only in the case $n\ge 3$, $0<\lambda <{\lambda }_1$ and $p>(n+2)/(n-2)$, so if we can remove the assumption $r_{⁎}\notin (a,b)$ in the case $n\ge 3$, $0<\lambda <{\lambda }_1$ and $p>(n+2)/(n-2)$, the uniqueness of solutions of problem (2) is completely solved. We note that even in the case $r_{⁎}\in (a,b)$, there are some results which ensure the uniqueness of solutions of problem (2), although they need another assumption instead of $r_{⁎}\in (a,b)$. For example, Korman [13, Theorem 3.3] obtained the following.

Theorem B

Let $n\in \mathbb{N}$ with $n\ge 3$, $b>a>0$, $\lambda <{\lambda }_1$ and $p>1$. If

$${\left(\frac{b}{a}\right)}^{n-2}\le 2n-3$$

then problem (2) has a unique positive solution.

In this paper, we show that we can remove the assumption $r_{⁎}\notin (a,b)$ in some restricted cases of $n\ge 3$, $0<\lambda <{\lambda }_1$ and $p>(n+2)/(n-2)$. Our results are the following.

Theorem 1

Let $n=3$, $b>a>0$, $p>5$ and $0<\lambda <{\lambda }_1$. Then problem (2) has a unique positive solution.

Theorem 2

Let $n\in \mathbb{N}$ with $n\ge 4$, $b>a>0$ and $p>(n+2)/(n-2)$. Then there exists $\delta >0$ such that if ${\lambda }_1-\delta <\lambda <{\lambda }_1$ then problem (2) has a unique positive solution.

We note the theorems above will be proved through a kind of Pohožaev identity, see Proposition 1 in Section 2, which differs from the ones in [19], [20].