On the Dirichlet Problem of Some Semilinear Elliptic Equation in Finite Balls

• Main Authors: Liu, Yi-Xian, 劉益先
• Other Authors: Chern Jann-Long
• Format: Others
• Language: zh-TW
• Published: 1997
• Online Access: http://ndltd.ncl.edu.tw/handle/21059730916156360690

碩士 === 國立中央大學 === 數學系 === 85 === In recent years mathematicians have paid much attention to theexistence of positive solutions of the elliptic equations involvingcritical exponents. However, in this paper we consider the Dirichlet problem of thesemilinear elliptic equation.$$\left\{\eqalign{&\lap u+k(|x|)u+K(|x|)u^{n+2\over n-2 }=0 \quad\hbox{in~}B_R \cr &u>0 \quad\hbox{in~} B_R \cr &u|_{\partial B_R}=0 \cr }\right.\eqno (1.1) $$where $n\geq 3, \lap=\sum_{i=1}^n{\partial^2\over {\partial x_i^2}}$,and $B_R=\{x=(x_1,x_2,\cdots,x_n)~|~x_1^2+x_2 ^2+\cdots +x_n^2<R^2\}$ and $k$ and $K$ are two given smooth functions. We shall give someexistence result of solutions of Eq.$(1.1)$ for suitable radius $R$. Eq.$(1.1)$ arises from Riemannian geometry. Here we give a briefdescription. Let $(M, g)$ be a Riemann manifold of dimension $n,~n\geq 2$, and $K( \cdot)$ be a given function on $M$. The following question has been raised: can we find a new metric $g_1$ on $M$ such that $K$ is the scalar curvature of $g_1$ and $g_1$ isconformal to $g$ (that is, $g_1=\phi g$ for some positive function $\phi$ on $M$)? In the case $n\geq 3$, if we write $\phi=u^{4\over n-2}, ~u>0$, then this is equivalent to the problem of solving the elliptic equation:$${4(n-1)\over{(n-2)}}\lap_gu-ku+Ku^{{n+2\over n-2}}=0\eqno (1.2) $$ on $M$, where $\lap_g$ and $k$ are the Laplace-Betrami operator and thescalar curvature on $M$ in the $g$-metric respectively. Here we are interested in the radial solutions of the equation$(1.1)$. For this purpose, the equation $(1.1)$ is equivalent to$$\left\{\eqalign{&u''(r)+{n-1\over r}u'(r)+k(r)u(r)+K(r)u^{n+2\over n-2}(r)=0\quad\hbox{in~} [0, R]\cr &u(0)=\alpha>0, u'(0)=0, u(R)=0 \cr }\right.\eqno (1.3) $$ Firstly, we consider, the following equation.$$\left\{\eqalign{&v''(r)+{n-1\over r}v'(r)+K(r)u^{n+2 \over n-2}=0\quadr\in [0,R] \cr&v(0)=\alpha>0, v'(0)=0, v(R)=0, v>0\quad\hbox{on~} [0,R)\cr}\right.\eqno (1.4)$$where $K(r)$ is a smooth function on $[0,R]$. For Eq. (1.4), we have the following theorem \Thm 1.1 Suppose that $K(r)$ is a smooth function, and satisfies$K(r)=K(0)+Ar^\sigma+o(r^\sigma)$, and $K'(r)=\sigma Ar^{\sigma -1}+o(r^{\sigma -1})$ as$r\to 0$ for some constants $K(0)>0,~~A>0$, and $0<\sigma<n-2$. Then there exists an $\alpha_1>0$, such that for each $\alpha\geq \alpha_1$, the equation $(1.4)$ has a solution on $[0,R]$ where $0<R=R(\alpha)$ isdependent on the initial value $v(0)=\alpha$. Before stating our main result we also need the following comparison lemma, \Lem 1,2 Suppose that $u(r),~v(r)$ satisfy$$ \eqalignno{&\lap u+K_1(r)u\leq 0 \quad x\in B_R&(1.5) \cr &\lap v+K_2(r)v\geq 0 \quad x\in B_R&(1.6) \cr}$$\nd respectively, where $r=|x|$, and $K_1(r)\geq K_2(r),~~u(r)>0$ and$u(0)=v(0)$. Then $v(r)\geq u(r)>0$ in $B_R$. Now coming back to our problem Eq.(1.3), we obtain the following main result:\Thm 1.1 Suppose that \itemitem{(i)}$K(r)$ is a smooth function, and satisfies $K(r)=K(0)+Ar^\sigma+o(r^\sigma)$, and $K'(r)=\sigmaAr^{\sigma -1}+o(r^{\sigma -1})$ as $r \to 0$ for some constants $K(0)>0,~A>0,$ and $0<\sigma <n-2$.\itemitem{(ii) } $k(r)$ is a continuous function and $k(r)\geqK(r)\alpha_2^{{4 \over n-2}}$ where $\alpha_2 \geq \alpha_1$ is a constant, and $\alpha_1>0$ is the constant given in $Theorem ~1.1$.Then for $\alpha_1\leq\alpha\leq\alpha_2$, the equation $(1.3)$ has a solution on $[0,R]$ where $0<R=R(\alpha)$ is dependent on the initial value$u(0)=\alpha$.