Electron. J. Diff. Eqns., Vol. 2005(2005), No. 143, pp. 1-14.

Existence of positive solutions for nonlinear boundary-value problems in unbounded domains of $\mathbb{R}^{n}$

Faten Toumi, Noureddine Zeddini

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,
$$\displaylines{ \Delta u=f(.,u),\quad u greater than 0\quad \hbox{in }D,\cr u\big|_{\partial D}=\alpha \varphi ,\cr \lim_{|x|\to +\infty }\frac{u(x)}{h(x)}=\beta \lambda , }$$
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.

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Faten Toumi
Département de Mathématiques
Faculté des Sciences de Tunis
Campus Universitaire, 2092 Tunis, Tunisia
email: Faten.Toumi@fsb.rnu.tn
Noureddine Zeddini
Département de Mathématiques
Faculté des Sciences de Tunis
Campus Universitaire, 2092 Tunis, Tunisia
email: Noureddine.Zeddini@ipein.rnu.tn

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