American Institute of Mathematical Sciences

March  2007, 6(1): 237-246. doi: 10.3934/cpaa.2007.6.237

Monotone solutions to a class of elliptic and diffusion equations

 1 Department of Mathematical Sciences, Tsinghua University, Beijing 100084, China, China

Received  March 2006 Revised  August 2006 Published  December 2006

In this paper, we study the existence and properties of monotone solutions to the following elliptic equation in $\mathbf R^n$

$-\Delta u= F'(u),$ in $\mathbf R^n,$

$\partial_{x_n}u>0,$

and the diffusion equation

$u_t-\Delta u= F'(u),$ in $\mathbf R^n\times$ {$t>0$},

$\partial_{x_n}u>0, u|_{t=0}=u_0,$

where $\Delta$ is the standard Laplacian operator in $\mathbf R^n$, and $u_0$ is a given smooth function in $\mathbf R^n$ with some monotonicity condition. We show that under a natural condition on the nonlinear term $F'$, there exists a global solution to the diffusion problem above, and as time goes to infinity, the solution converges in $C_{l o c}^2(\mathbf R^n)$ to a solution to the corresponding elliptic problem. In particular, we show that for any two solutions $u_1(x')<$ $u_2(x')$ to the elliptic equation in $\mathbf R^{n-1}$:

$-\Delta u=F'(u),$ in $\mathbf R^{n-1},$

either for every $c\in (u_1(0),u_2(0))$, there exists an $(n-1)$ dimensional solution $u_c$ with $u_c(0)=c$, or there exists an $x_n$-monotone solution $u(x',x_n)$ to the elliptic equation in $\mathbf R^n$:

$-\Delta u=F'(u),$ in $\mathbf R^n,$

$\partial_{x_n}u>0,$ in $\mathbf R^n$

such that

$\lim_{x_n\to-\infty}u(x',x_n)=v_1(x')\leq u_1(x')$

and

$\lim_{x_n\to+\infty}u(x',x_n)=v_2(x')\leq u_2(x').$

A typical example is when $F'(u)=u-|u|^{p-1}u$ with $p>1$. Some of our results are similar to results for minimizers obtained by Jerison and Monneau [13] by variational arguments. The novelty of our paper is that we only assume the condition for $F$ used by Keller and Osserman for boundary blow up solutions.

Citation: Li Ma, Chong Li, Lin Zhao. Monotone solutions to a class of elliptic and diffusion equations. Communications on Pure & Applied Analysis, 2007, 6 (1) : 237-246. doi: 10.3934/cpaa.2007.6.237
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