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Introduction

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

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


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

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

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

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

\begin{displaymath}\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).\end{displaymath}

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

\begin{displaymath}
\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.
\end{displaymath}

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

\begin{displaymath}
g(t,y_1)f(t,y_2)-g(t,y_2)f(t,y_1) > 0.
\end{displaymath}

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

\begin{displaymath}
g(t,y)=1-\cos [h(y-a(t);b(t))]
\end{displaymath}

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

(i-4) $\displaystyle{dh\over dy}(y;b) \leq 0$ 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.


next up previous
Next: Bibliography Up: Remarks on viscosity solutions Previous: Remarks on viscosity solutions
Nobuki Takayama 2002-04-24