Introduction

The structure of positive radial solutions for nonlinear elliptic equations have attracted much attention for these years. In particular, many interesting and beautiful results have been obtained concerning the structure of positive radial solutions on entire space ${\bf R}^n$ (see, e.g., the survey paper by Ni [8]). However, it is not straightforward to extend these results to boundary value problems on bounded domains.

In this paper, we consider the structure of solutions of the semilinear elliptic equation

$$\Delta u + Q(\vert x\vert)u^p = 0 \quad \mbox{ in } B,$$ (1.1)

where $p > 1$,

$$u^p = \left \{ \begin{array}{ll} \vert u\vert^p & \mbox{ i... ... \vspace{5pt}\\ 0 & \mbox{ if } u \leq 0, \end{array}\right.$$

$Q(\vert x\vert)$ is a given nonnegative function, and

$$B=\{x\in {\bf R}^n;\vert x\vert < 1\}, \quad n>2.$$

Our main concern is the existence and uniqueness of positive radial solutions of (1.1) under the Dirichlet boundary condition

$$u=0 \ \mbox{ on } \partial B.$$ (1.2)

Before studying the boundary value problem, we describe the result of [11] concerning the structure of positive radial solutions of (1.1) in the entire space ${\bf R}^n$. Since we are concerned with positive radial solutions, we consider the initial value problem

$$\left\{ \begin{array}{@{\,}ll} \displaystyle{u_{rr}+\frac{n... ...ty), \vspace{5pt}\\ u(0) = \alpha > 0.\\ \end{array}\right.$$ (1.3)

Here we assume that $p > 1$, $n>2$, and $K(r)$ satisfies

$$\mbox { (K) } \ \left \{ \ \begin{array}{@{\,}ll} K(r) \in... ...r^{n-1-(n-2)p} K(r) \in L^{1}(1, \infty). \end{array} \right.$$

We note that $K(r)$ may be unbounded at $r=0$. Under the first and second conditions, it is shown in [7,9] that the initial value problem (1.3) is uniquely solvable if and only if $rK(r) \in L^1(0, 1)$. We denote the unique solution by $u(r;\alpha)$. It is known [11] that the solution of (1.3) is classified as

1. a crossing solution: $u(r;\alpha)$ has a zero in $(0,\infty)$,
2. a slowly decaying solution: $u(r;\alpha) > 0$ on $[0,\infty)$ and $r^{n-2} u(r;\alpha) \to\infty$ as $r\to\infty$,
3. a rapidly decaying solution: $u(r;\alpha) > 0$ on $[0,\infty)$ and $\lim_{r \to \infty} r^{n-2}u(r;\alpha)$ exists and is positive.

Finally, it is known [1,6] that if $r^{n-1-(n-2)p} K(r) \not\in L^{1}(1, \infty)$, then any solution of (1.3) (whether or not it satisfies the initial condition) cannot be positive near $\infty$ so that $u(r;\alpha)$ is a crossing solution for any $\alpha\in(0,\infty)$.

It is shown in [11] that the Pohozaev identities

$$\begin{array}{l} \displaystyle\frac {d}{dr} \left \{ \displ... ..._r(r;K) u^{p+1} \equiv H_r(r;K) \{r^{n-2}u\}^{p+1}, \end{array}$$ (1.4)

play a crucial role for the structure of solution of (1.3), where $G(r;K)$ and $H(r;K)$ are functions on $(0,\infty)$ defined by

$$G(r;K) := \displaystyle{\frac{1}{p+1}} r^n K(r) - \displaystyle{\frac{n-2}{2}} \int_0^r r^{n-1} K(r)\,dr,$$ (1.5)

$$H(r;K) := \displaystyle{\frac{1}{p+1}} r^{2-(n-2)p} K(r) - \displaystyle{\frac{n-2}{2}} \int_r^\infty r^{1-(n-2) p} K(r)\,dr,$$ (1.6)

respectively. We define

$$\begin{array}{l} r_{G} := \inf \, \{ r \in (0,\infty) ; ~G(... ...H} := \sup \, \{ r \in (0,\infty) ; ~H(r;K) < 0 \}. \end{array}$$

Here we put $r_G = \infty$ if $G(r;K) \ge 0$ on $(0,\infty)$, and $r_H = 0$ if $H(r;K) \ge 0$ on $(0,\infty)$.

The following result was obtained in [11].

Theorem A Suppose that (K) holds. Then the structure of solutions for (1.3) is as follows:

(i)

If $G(r;K)\equiv 0$ on $(0,\infty)$, then the structure is of Type R: $u(r;\alpha)$ is a rapidly decaying solution for every $\alpha > 0$.

(ii)

Suppose that $G(r;K)\not\equiv 0$ on $(0,\infty)$.

(a)

If $r_G = \infty$, then the structure is of Type C: $u(r;\alpha)$ is a crossing solution for every $\alpha > 0$.

(b)

If $0 = r_H \le r_G < \infty$, then the structure is of Type S: $u(r;\alpha)$ is a slowly decaying solution for every $\alpha > 0$.

(c)

If $0 < r_H \le r_G < \infty$, then the structure is of Type M: There exists $\alpha^*\in(0,\infty)$ such that
$u(r;\alpha)$ is a $crossing ~solution$ for $\alpha \in (\alpha^*, \infty)$,
$u(r;\alpha^*)$ is a rapidly decaying solution, and
$u(r;\alpha)$ is a slowly decaying solution for $\alpha \in (0, \alpha^*)$.

(iii)

Let $a$ and $b$ be any given numbers with $0 \le a < b \le \infty$. Then there exists $K(r)$ satisfying (K), $r_G = a$ and $r_H= b$ such that the structure is of none of Types R, C, S, and M.

Thus, the structure of solutions for (1.3) can be determined completely in the case $r_H\leq r_G$, but cannot be determined from only $r_G$ and $r_H$ in the case $r_G < r_H$. We note that $G(r;K)\equiv 0$ on $(0,\infty)$ if and only if

$$K(r) = c \cdot r^{\frac{(n-2)p-(n+2)}{2}}$$ (1.7)

for some constant $c > 0$. In this case, the identity $H(r;K) \equiv 0$ holds on $(0,\infty)$ and the solution of (1.3) is explicitly written as

$$u(r;\alpha) = \alpha \left \{ 1 + {\displaystyle{\frac{2 \,c... ... (n-2)^2}}}\, {r^{(n-2)(p-1)/2 }} \right \}^{-\frac{2}{p-1}}.$$ (1.8)

The proof of Theorem A in [11] is based on the effective use of the Pohozaev identities (1.4). Theorem A is a quite powerful tool to derive precise information on the structure of solutions, but it was pointed out by Kwong and Zhang [5] that the method is not applicable directly to boundary value problems on bounded domains, mainly because the above Pohozaev identities are not related with the boundary conditions. Nonetheless, we will show by modifying the method that similar results can be obtained for the boundary value problems (see [4] about general results).

Let us return to our original problem (1.1). In order to study positive radial solutions for the Dirichlet problem (1.1) with (1.2), we consider the initial value problems

$$\left\{ \begin{array}{@{\,}ll} \displaystyle{u_{rr}+\frac{n... ...,1), \vspace{5pt}\\ u(0) = \alpha > 0,\\ \end{array}\right.$$ (1.9)

and

$$\left\{ \begin{array}{@{\,}ll} \displaystyle{u_{rr}+\frac{n... ...ce{5pt}\\ u(1) = 0, ~u_r(1) = -\beta < 0. \end{array}\right.$$ (1.10)

For $Q(r)$, we will assume

$$\mbox{ (Q) } \ \left \{ \ \begin{array}{@{\,}ll} Q(r) \in ... ...{5pt}\\ (1-r)^p Q(r) \in L^{1}(1/2, 1). \end{array} \right.$$

We note that $Q(r)$ may be unbounded at $r=0$ or $r=1$. As is noted above, under these conditions, (1.9) has a unique solution $u=u(r;\alpha)$ on $(0,1)$. The last condition in (Q) is a necessary and sufficient condition on the existence of a unique solution for (1.10) (see Lemma 2.7 below). We denote the unique solution by $u(r;\beta)$.

By Lemma 2.3 given in the next section, the solution of (1.9) is classified as

1. a crossing solution: $u(r;\alpha)$ has a zero in $(0,1)$,
2. a singular solution: $u(r;\alpha) > 0$ on $[0,1)$ and $\lim_{r \to 1}u(r;\alpha)/(1-r) = \infty$,
3. a regular solution: $u(r;\alpha) > 0$ on $[0,1)$ and $\lim_{r \to 1} u(r;\alpha)/(1-r)$ exists and is positive.

We will see that if the last condition in (Q) does not hold, then $u(r;\alpha)$ is a crossing solution for any $\alpha\in(0,\infty)$. We note that depending on $Q(r)$, the singular solution satisfies $u(1)>0$ or $u(1)=0$. In the latter case, the singular solution is not differentiable at $r=1$. Thus, if $u(r;\alpha)$ is a regular solution, then $u=u(\vert x\vert;\alpha)$ is a solution of the Dirichlet problem (1.1) with (1.2) in the class $C(\overline B)\cap C^1(\overline B-\{0\}) \cap C^2(B-\{0\})$.

Similarly, the solution of (1.10) is classified as

1. a crossing solution: $u(r;\beta)$ has a zero in $(0,1)$,
2. a singular solution: $u(r;\beta) >0$ on $(0,1)$ and $\lim_{r \to 0}u(r;\beta) = \infty$,
3. a regular solution: $u(r;\beta) >0$ on $(0,1)$ and $\lim_{r \to 0} u(r;\beta)$ exists and is positive.

We will see that if the third condition in (Q) does not hold, then $u(r;\beta)$ is a crossing solution for any $\beta\in(0,\infty)$. Also, if $u(r;\beta)$ is a regular solution, then $u=u(\vert x\vert;\beta)$ is a solution of the Dirichlet problem (1.1) with (1.2) in the class $C(\overline B)\cap C^1(\overline B-\{0\}) \cap C^2(B-\{0\})$.

To derive an analog of Theorem A, we need to modify the Pohozaev identities and the functions $G(r;K)$ and $H(r;K)$ by taking the boundary condition (1.2) into account. We will see that the following identities are suitable for our purpose:

$$\begin{array}{l} \displaystyle\frac {d}{dr} \left \{\displa... ...ystyle\frac {r^{n-2} u }{1-r^{n-2} } \right)^{p+1}, \end{array}$$ (1.11)

where $G^b(r;Q)$ and $H^b(r;Q)$ are functions on $(0,1)$ defined by

$$G^b(r;Q) := \displaystyle{\frac{1}{p+1}} (1-r^{n-2}) r^n Q(r) - \displaystyle{\frac{n-2}{2}} \int_0^r r^{n-1} Q(r)\,dr,$$ (1.12)

$$H^b(r;Q) := \displaystyle{\frac{1}{p+1}} (1-r^{n-2})^{p+2} r^{2-(n-2)p} Q(r)$$ (1.13)

$$\qquad -\displaystyle{\frac{n-2}{2}} \int_r^1 (1 - r^{n-2})^{p+1} r^{1-(n-2) p} Q(r)\,dr.$$

We also define

$$\begin{array}{l} r^b_G := \inf \, \{ r \in (0,1) ; ~G^b(r;Q... ...r^b_H := \sup \, \{ r \in (0,1) ; ~H^b(r;Q) < 0 \}. \end{array}$$

Here we put $r_G^b = 1$ if $G^b(r;Q) \ge 0$ on $(0,1)$, and $r_H^b = 0$ if $H^b(r;Q) \ge 0$ on $(0,1)$. Thus we have $0 \le r^b_G, r^b_H \le 1$ by definition.

Now we state our main results on (1.9) and (1.10).

Theorem 1.1   Suppose that (Q) holds. Then the structure of solutions for (1.9) is as follows:

(i)

If $G^b(r;Q)\equiv 0$ on $(0,1)$, then the structure is of Type R: $u(r;\alpha)$ is a regular solution for every $\alpha > 0$.

(ii)

Suppose that $G^b(r;Q)\not\equiv 0$ on $(0,1)$.

(a)

If $0<r^b_H \leq r^b_G = 1$, then the structure is of Type C: $u(r;\alpha)$ is a crossing solution for every $\alpha > 0$.

(b)

If $0 = r^b_H \le r^b_G < 1$, then the structure is of Type S: $u(r;\alpha)$ is a singular solution for every $\alpha > 0$.

(c)

If $0 < r^b_H \le r^b_G < 1$, then the structure is of Type M: There exists $\alpha^b \in(0,\infty)$ such that
$u(r;\alpha)$ is a $crossing ~solution$ for $\alpha \in (\alpha^b, \infty)$,
$u(r;\alpha^b)$ is a regular solution, and
$u(r;\alpha)$ is a singular solution for $\alpha \in (0, \alpha^b)$.

(iii)

Let $a$ and $b$ be any given numbers with $0 \le a < b \le 1$. Then there exists $Q(r)$ satisfying (Q), $r^b_G = a$ and $r^b_H= b$ such that the structure is of none of Types R, C, S, and M.

Theorem 1.2   Suppose that (Q) holds. Then the structure of solutions for (1.10) is as follows:

(i)

If $G^b(r;Q)\equiv 0$ on $(0,1)$, then the structure is of Type R: $u(r;\beta)$ is a regular solution for every $\beta > 0$.

(ii)

Suppose that $G^b(r;Q)\not\equiv 0$ on $(0,1)$.

(a)

If $0=r^b_H \leq r^b_G< 1$, then the structure is of Type C: $u(r;\beta)$ is a crossing solution for every $\beta > 0$.

(b)

If $0 < r^b_H \le r^b_G = 1$, then the structure is of Type S: $u(r;\beta)$ is a singular solution for every $\beta > 0$.

(c)

If $0 < r^b_H \le r^b_G < 1$, then the structure is of Type M: There exists $\beta^b \in(0,\infty)$ such that
$u(r;\beta)$ is a $crossing ~solution$ for $\beta \in (\beta^b , \infty)$,
$u(r;\beta^b )$ is a regular solution, and
$u(r;\beta)$ is a singular solution for $\beta \in (0, \beta^b)$.

(iii)

Let $a$ and $b$ be any given numbers with $0 \le a < b \le 1$. Then there exists $Q(r)$ satisfying (Q), $r^b_G = a$ and $r^b_H= b$ such that the structure is of none of Types R, C, S, and M.

We note that $G^b(r;Q)\equiv 0$ if and only if

$$Q(r) = c \cdot r^{\frac{(n-2)p-(n+2)}{2}} (1-r^{n-2})^{-\frac{p+3}{2}}$$ (1.14)

for some $c > 0$. In this case, we have $H^b(r;Q) \equiv 0$ on $(0,1)$ and the solutions of (1.9) and (1.10) are explicitly written as

$$u(r;\alpha) = \alpha \left \{ 1 + {\displaystyle{\frac{2 \,c... ...r^{n-2}}}} \right )^{\frac{p-1}{2}} \right \}^{-\frac{2}{p-1}}$$ (1.15)

and

$$\qquad u(r;\beta) = \displaystyle\frac {\beta(1-r^{n-2})}{r... ...{r^{n-2}} \right )^{\frac{p-1}{2}} \right \}^{-\frac{2}{p-1}},$$ (1.16)

respectively.

There are two ways to prove the above theorems. One is to follow the proof in [11] for (1.3) by using the modified Pohozaev identities (1.11). The other is to transform (1.9) and (1.10) into the form of (1.3) by suitable changes of variables, and apply Theorem A to the transformed systems. Then we may inversely transform the results for (1.3) to those for (1.9) and (1.10). In this paper, we adopt the latter method.

This paper is organized as follows. In section 2, we give suitable changes of variables to transform (1.9) and (1.10) into the form of (1.3). Then the proofs of Theorems 1.1 and 1.2 are obtained easily from Theorem A. In section 3, as applications of Theorems 1.1 and 1.2, we give a few corollaries concerning the structure of solutions for the problems (1.9) and (1.10) with $Q(r) = r^\sigma/ (1-r^{n-2})^{\tau}$. We will also give an application to some exterior Dirichlet problem. Section 4 is devoted to proofs of the corollaries.