Electronic Journal of Differential Equations,
Vol. 2005(2005), No. 143, pp. 1-14.

Title: Existence of positive solutions for 
       nonlinear boundary-value problems in 
       unbounded domains of $\mathbb{R}^{n}$
       
Authors: Faten Toumi (Faculte des Sciences, Tunis, Tunisia)
         Noureddine Zeddini (Faculte des Sciences, Tunis, Tunisia)

Abstract: 
 Let $D$ be an unbounded domain in $\mathbb{R}^{n}$ ($n\geq 2$)
 with a nonempty compact boundary $\partial D$. We consider the  
 following nonlinear elliptic problem, in the sense of distributions,
 $$\displaylines{
 \Delta u=f(.,u),\quad u>0\quad \hbox{in }D,\cr
 u\big|_{\partial D}=\alpha \varphi ,\cr
 \lim_{|x|\to +\infty }\frac{u(x)}{h(x)}=\beta \lambda ,
 }$$
 where $\alpha ,\beta,\lambda $ are nonnegative constants with
 $\alpha +\beta >0$ and $\varphi $ is a nontrivial nonnegative
 continuous function on $\partial D$. The function
 $f$ is nonnegative and satisfies some appropriate conditions related 
 to a Kato class of functions, and $h$ is a fixed harmonic
 function in $D$, continuous on $\overline{D}$. Our aim is to prove
 the existence of positive continuous solutions bounded below by a
 harmonic function. For this aim we use the Schauder fixed-point
 argument and a potential theory approach.
 
Submitted September 30, 2005. Published December 8, 2005.
Math Subject Classifications: 34B15, 34B27.
Key Words: Green function; nonlinear elliptic equation; 
           positive solution; Schauder fixed point theorem.