Introduction

In this paper we are concerned with the Dirichlet problem (hereafter called (DP)) for the quasilinear degenerate elliptic equation :

$$-g(\vert x\vert,u) \Delta u + f(\vert x\vert,u) = 0 \quad {\rm in} \quad B_R \leqno(1.1)$$

$$u=\beta \quad {\rm on} \quad \partial B_R, \leqno(1.2)$$

where $B_R=\{x \in {\bf R}^N ; \vert x\vert<R\}, N \geq 2, g:[0,R]\times {\bf R} \to {\bf R}^+=[0,\infty)$ is a given continuous function, $\Delta$ is the Laplacian, and $\beta$ is a real number such that $f(R,\beta)=0$.

This investigation is a sequel of our previous work [3] where we studied the existence, uniqueness, nonuniqueness and radial property of viscosity solutions of the Dirichlet problem for the semilinear degenerate elliptic equation

$$-g(\vert x\vert) \Delta u + f(\vert x\vert,u) = 0 \quad {\rm in} \quad B_R, \leqno(1.3)$$

where $g$ is a nonnegative continuous function. We refer the reader to the Monograph by Crandall, Ishii and Lions [1] for definitions, details and references of viscosity solutions.

The main purpose of the present paper is to prove existence of viscosity solutions, and to give a sufficient condition assuring the uniqueness and the radial symmetry of viscosity solutions of (DP). In what follows we consider the problem (DP) in case $N=2$, since we can treat it in case $N\geq 3$ by the same arguments.

Throughout this paper we make the following assumptions:

(H1) $f(t,y) \in C([0,R] \times {\bf R})$ is strictly increasing in $y$ for each fixed $t \in [0,R]$.

(H2) There exists an implicit function $\varphi(t)$ of $f(t,y)=0$ satisfying

$$\sup_{0\leq s\leq R,s\not=t}{\displaystyle{ \mid{{\varphi(s)-\varphi(t)}\over {s-t}} \mid}}=\Psi(t) \in L^1(0,R).$$

It is clear that $\varphi(t)$ is continuous on $[0,R]$ by (H1) and (H2).

We state our existence theorem.

Theorem 1   Under the assumptions (H1) and (H2) there exists a radial viscosity solution of (DP).

In order to establish the uniqueness of viscosity solutions for (DP) we introduce additional assumptions and a notion of standard viscosity solution.

(H3) $\Psi(t) \in L^\infty(0,R),$ here $\Psi$ is the function defined in (H2).

(H4) The function $g$ satisfies the condition : if $g(t_1,y_1)=0$ then

$$\left\{ \begin{array}{rcl} g(s,y_1) &\leq& {\rm Const.}\mid ... ...or} \quad \forall (s,y) \in N(t_1,y_1), \\ \end{array}\right.$$

where $N(t_1)$ and $N(t_1,y_1)$ are small neighborhoods of $t_1$ and $(t_1,y_1)$, respectively.

(H5) For $f$ and $g$, we impose the following structure condition : if $0\leq t \leq R$, $\ y_1<y_2$ and $g(t,y_1)+g(t,y_2)>0$, then

$$g(t,y_1)f(t,y_2)-g(t,y_2)f(t,y_1) > 0.$$

Example Let $a \in C^2([0,R]), b \in C^2([0,R];[0,1])$. Define $g \in C^2([0,R]\times {\bf R})$ by

$$g(t,y)=1-\cos [h(y-a(t);b(t))]$$

for $(t,y) \in [0,R]\times {\bf R}$, where $h(y;b),0\leq b \leq 1$, is a $C^2({\bf R})$-function such that

(i) for every $0<b<1$,

(i-1) $h(y;b)=0$ for all $y \leq b-\tilde b$ and all $y \geq 2-b+\tilde b$,

(i-2) $h(y;b)=\tilde b$ for all $b \leq y \leq 2-b$,

(i-3) $${dh\over dy}(y;b) \geq 0$$ for all $b-\tilde b \leq y \leq b$,

(i-4) $${dh\over dy}(y;b) \leq 0$$ for all $2-b \leq y \leq 2-b+\tilde b$;

(ii) for $b=0,1, h(y;b)=0$ for all $y \in {\bf R}$, i.e., $h(y;0)=h(y;1)\equiv 0$ for $\forall y \in {\bf R}$.

Here, for every $b \in (0,1), \tilde b:=(1/2)\min \{b,1-b\}$.     Suppose $f(t,y)=y-\varphi (t)$, and that $\varphi (t) \in C^1([0,R])$ is a given function satisfying $a(t)+b(t) \leq \varphi (t) \leq a(t)+2-b(t)$ for every $t \in [0,R]$ such that $0<b(t)<1$. Then $f(t,y)$ and $g(t,y)$ satisfy the assumptions (H1)-(H5).

Definition. A function $u$ is called a standard viscosity solution of (DP) if $u$ is a viscosity solution and $u(x)=\varphi(x)$ for all $x \in B_R \setminus \{0\}$ satisfying $g(x,u(x))=0$ .

By making use of a notion of standard viscosity solutions, we shall prove the uniqueness for (DP) :

Theorem 2   Under the assumptions (H1)-(H5) there exists a unique viscosity solution $u$ of (DP). Moreover, every viscosity solution of (DP) is standard and radially symmetric.

The authors would like to express their hearty gratitude to the referee for kind and helpful advice.