Introduction

In this paper, we consider the following semilinear elliptic equation

$$\Delta u+f(\vert x\vert,u)=0 \mathbb{R}^{n} (n\geq 3).$$ (1)

Since we are only interested in positive radial solutions, we study the following problem :

$$u^{\prime \prime }+\frac{n-1}{r}u^{\prime }+f(r,u)=0,$$ (2)

$$u>0 (r>0) u\rightarrow 0 r\rightarrow \infty,$$ (3)

where $r>0,$ $r=\vert x\vert$, $x\in \mathbb{R}^{n}$. If $f(r,u)>0$ for small $u>0$, then $r^{n-2}u(r)$ is increasing for large $r>0$(see the proof of Lemma 2.1), and therefore the solutions of (2)-(3) can be classified into two types:

(R)

if $r^{n-2}u(r)\rightarrow C<\infty (r\rightarrow \infty)$ for some $C>0$, then $u(r)$ is called a rapidly decaying solution,

(S)

if $r^{n-2}u(r)\rightarrow \infty (r\rightarrow \infty)$, then $u(r)$ is called a slowly decaying solution.

Many authors studied the non-existence or existence and uniqueness of positive radial solutions of (2)-(3) under suitable structure conditions on $f(r,u)$ (see [Ni], [NS], [Pan1], [LN], [SZ], [Li], [Pan2] and the references therein). The case $f(r,u)=u^{p}$ has been extensively studied. Among them, it is known that if $n/(n-2)<p<(n+2)/(n-2)$, there exists only one slowly decaying positive solution $u$ near infinity, namely $u(r)=\lambda r^{-\alpha}$ with $\alpha=2/(p-1), \lambda=\{\alpha(n-2-\alpha)\}^{1/(p-1)}$ (see e.g. [SZ]). For the case $f(r,u)=u^{p}+u^{q}, n/(n-2)<p<(n+2)/(n-2), p<q$, several authors studied existence and uniqueness of slowly decaying solutions of (2)-(3). In particular, Qi and Lu[QL] proved the existence and uniqueness of slowly decaying solution of (2)-(3) near infinity under the assumptions $n/(n-2)<p<(n+2)/(n-2), p<q$. Furthermore, they showed that if we impose the additional condition $q\le (n+2)/(n-2)$, then the slowly decaying solution can be extended on $(0,\infty)$ as a singular solution, i.e. $\lim_{r\to0}u(r)=+\infty$. See also [Pan1],[LN],[SZ] for some previous results on this problem. For the case $q>(n+2)/(n-2)$, the classification of the slowly decaying solution $u$ as $r\to0$ is difficult and has been an open problem. Very recently, partial results in this direction was obtained by R.Bamón, I.Flores and M.del Pino (see [BFP] for the details).
The purpose of this paper is to show uniqueness and existence of slowly decaying solutions near infinity for general nonlinearity $f(r,u)$ which satisfies certain structure conditions, including the typical one $f(r,u)=u^{p}+K(r)u^{q}$, but $K(r)$ is not necessarily bounded. Furthermore, we investigate the sufficient condition on $f(r,u)$ to make a slowly decaying solution singular at $r=0$.
Throughout this paper, we assume that the constants $p$ and $q$ satisfy the following relations:

$$\frac{n}{n-2}<p<\frac{n+2}{n-2},\, p<q.$$

We also use the notation :

$$\alpha=\frac{2}{p-1},\, \sigma=(q-p)\alpha,\, \lambda=\{\alpha(n-2-\alpha)\}^{\frac{1}{p-1}}.$$

We assume the following conditions for $f(r,u)$.

(A-1)

$f(r,u)=0$ for $u\le 0$ and $f(r,u)$ is continuous on $(0,\infty )\times (0,\infty )$ and locally Lipschitz continuous with respect to $u.$

(A-2)

There exist positive constants $C_{0}$ and $\delta$ such that $f(r,u)\geq C_{0}u^{p}$ holds for $u\in(0,\delta)$.

(A-3)

There exists a function $K(r)$ such that $f(r,u)=u^{p}+f_{1}(r,u)$,
$\vert f_{1}(r,u)\vert\leq K(r)u^{q}$for $u\in(0,\delta)$ and $K(r)=O(r^{l})$ ( $r\rightarrow \infty$) for some $l<\sigma$.

(A-4)

There exists a function $\widetilde{K}(r)$ such that $\vert\{f_{1}(r,u)\}_{u}\vert \leq \widetilde{K}(r)u^{q-1}$ for
$u\in(0,\delta)$ and $\widetilde{K}(r)=O(r^{l})\ (r\to\infty)$ for some $l<\sigma$.

First, we state the main result on the uniqueness of slowly decaying positive solutions near infinity.

Theorem 1.1
Suppose that $(A-1) \sim (A-4)$ hold. Then any slowly decaying solution of $(2)-(3)$, if it exists, satisfies that for any $\epsilon>0$

$$r^{\alpha }u-\lambda =o(r^{-(\sigma -l-\epsilon )}) r\to\infty$$ (4)

and there exists at most one slowly decaying solution of $(2)-(3)$ on $[r_{0},\infty )$ for any $r_{0}>0$.

Remark 1.1
Theorem 1.1 is an extension of [QL], [SZ]. Actually, in [QL] they asserts existence and uniqueness for more general nonlinearity $f(r,u)=f(u)$ with $\lim\limits_{u\to0}\frac{f(u)-u^{p}}{u^{q}}=1$ or $f(r,u)=u^{p}+K(r)u^{q}$ with a bounded function $K(r)$. However, their proof, even for the uniqueness, is based on the higher order asymptotic expansion of slowly decaying solutions. We suspect that such higher order asymptotic expansion can not be obtained if there is an oscillation in the higher order terms of $f(u)$ or $K(r)$ (cf.Theorem 1.2 and 1.3). The method of [QL] cannot be applied for $f(r,u)$ satisfying (A-1)$\sim$(A-4) in general.

We show the estimate (4) by the argument in [QL] and prove Theorem 1.1 by the method in [SZ] by using the estimate (4).
To show existence of slowly decaying solutions, we need the higher order asymptotic expansion for the solution $u$, since we use the contraction mapping principle as in [QL].

Theorem 1.2
Let $f(r,u)=u^{p}+r^{l}u^{q}$ $(l<\sigma ),n/(n-2)<p<(n+2)/(n-2)$, $q>p$, $\sigma =(q-p)\alpha$, $\alpha =2/(p-1).$ Then there exists a unique slowly decaying solution on $(r_{0},\infty )$ for sufficiently large $r_{0}>0.$

Theorem 1.2 is an extension of Theorem 1 in [QL]. Moreover, although we follow the strategy of [QL], we simplify the proof slightly (see Lemma 4.1 and Remark 4.1). For more general nonlinearity, we have the following theorem.

Theorem 1.3
Let $f(r,u)=u^{p}+K(r)g(u)$. Suppose $g(u)=u^{q}(\sum_{n=0}^{\infty}b_{n}u^{n}) \\ (u\to 0)$ and $K(r)=r^{l}\sum_{n=0}^{\infty}\alpha_{n}/r^{n} \ (r\to\infty), \ l<\sigma$. Furthermore, we assume $\sigma-l$ and $\alpha\in\mathbb{Q}$. Then there exists a unique slowly decaying solution on $(r_{0},\infty )$ for sufficiently large $r_{0}>0.$

In these two cases, the precise asymptotic expansion of the slowly decaying solution will be given in details in section 4 (see Theorem 4.1 and 4.2). See also Remark 4.2 for more general statements. Although basically we follow the strategy of [QL], we simplify the procedure slightly and give the proof in details.
Next, we consider the following problem; when we have a slowly decaying positive solution $u$ on $[r_{0},\infty )$,to what extent we can extend the solution $u$ backward into the region $(0,r_{0})$. The behavior can be classified as follows. Instead of (A-1),(A-2), we impose a slightly stronger condition:

(A-1)'

$f(r,u)=0$ for $u\le 0$ and $f(r,u)$ is continuous on $[0,\infty)\times [0,\infty)$ and locally Lipschitz continuous with respect to $u$.

(A-2)'

$f(r,u)\ge 0$ on $(0,\infty )\times (0,\infty )$ and there exist positive constants $C_{0}$ and $\delta$ such that $f(r,u)\ge C_{0}u^{p}$ holds for any $u\in(0,\delta)$.

Proposition 1.1
Suppose that (A-1)' and (A-2)' hold. Then the behavior of the solution of (2)-(3) on $(0,\infty)$ is classified into three types:

1. singular solution which satisfies $u(r)\rightarrow +\infty$ as $%$f(r,u)=u^{p}$ r\rightarrow +0.$
2. regular solution which satisfies $u(r)\rightarrow c^{\prime }$ as $%$n/(n-2)<p<(n+2)/(n-2)$ r\rightarrow +0$ for some $c^{\prime }>0.$
3. 0-hit solution which satisfies $u(r_{1})=0$ for some $r_{1}>0$.

Furthermore if $f(r,u)$ satisfies the certain additional structure condition,by using Pohozaev identity (see Section 4), we can extend the slowly decaying solutions of (2)-(3) backward into $\left( 0,+\infty \right)$ as a singular solution. To state the result precisely, instead of (A-3), we must impose a stronger condition.

(A-3)'

There exist a function $L(r)$ and a constant $\delta_{2}$ such that $f(r,u)=u^p+f_{1}(r,u)$, $\vert f_{1}(r,u)\vert\leq L(r)u^q$ for $u\in(0,\delta_{2})$, $\vert r \partial f_{1}/\partial r\vert \leq ML(r)u^q$ for some $M>0$, and $L(r)=O(r^l)\ (r\to\infty)$ for some $l<\sigma$.

Theorem 1.4
Suppose that (A-1)',(A-2)',(A-3)' hold and $f(r,u)$ satisfies

$$nF(r,u)-\frac{n-2}{2}uf(r,u)+rF_{r}(r,u)\ge 0 \text { and } \not\equiv 0,\ (r,u)\in(0,\infty)\times(0,\delta)$$ (5)

for any $\delta>0$, where $F(r,u)=\int_0^{u}f(r,t)dt$. Then any positive solution of $(2)-(3)$, if it exists on $[r_{0},\infty )$ for some $r_{0}>0$, can be extended to $(0,\infty)$ and satisfies

$$u(r)\rightarrow \infty (r\rightarrow +0).$$ (6)

Theorem 1.4 is an extension of Theorem 3 in [QL]. The classification problem for slowly decaying solutions on $(0,\infty)$ is widely open (see [BFP]).