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.