| Summary: | Let $D$ be an unbounded domain in $mathbb{R}^{n}$ ($ngeq 2$) with a nonempty compact boundary $partial D$. We consider the following nonlinear elliptic problem, in the sense of distributions, $$displaylines{ Delta u=f(.,u),quad u>0quad hbox{in }D,cr uig|_{partial D}=alpha varphi ,cr lim_{|x|o +infty }frac{u(x)}{h(x)}=eta lambda , }$$ where $alpha ,eta,lambda $ are nonnegative constants with $alpha +eta >0$ 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.
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