Existence of positive solutions for nonlinear boundary-value problems in unbounded domains of $mathbb{R}^{n}$
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, $$displaylines{ Delta u=f(.,u),quad u>0quad hbox{in }D,cr uig|_{partial D}=alpha varphi ,cr lim_{|x|o +...
| Main Authors: | , |
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| Format: | Article |
| Language: | English |
| Published: |
Texas State University
2005-12-01
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| Series: | Electronic Journal of Differential Equations |
| Subjects: | |
| Online Access: | http://ejde.math.txstate.edu/Volumes/2005/143/abstr.html |
| _version_ | 1818962282404642816 |
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| author | Faten Toumi Noureddine Zeddini |
| author_facet | Faten Toumi Noureddine Zeddini |
| author_sort | Faten Toumi |
| collection | DOAJ |
| description | Let $D$ be an unbounded domain in $mathbb{R}^{n}$ ($ngeq 2$) with a nonempty compact boundary $partial D$. We consider the following nonlinear elliptic problem, in the sense of distributions, $$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 , }$$ where $alpha ,eta,lambda $ are nonnegative constants with $alpha +eta >0$ and $varphi $ is a nontrivial nonnegative continuous function on $partial D$. The function $f$ is nonnegative and satisfies some appropriate conditions related to a Kato class of functions, and $h$ is a fixed harmonic function in $D$, continuous on $overline{D}$. Our aim is to prove the existence of positive continuous solutions bounded below by a harmonic function. For this aim we use the Schauder fixed-point argument and a potential theory approach. |
| first_indexed | 2024-12-20T12:26:51Z |
| format | Article |
| id | doaj.art-55415508c4cf48168bf5e18b43928d67 |
| institution | Directory Open Access Journal |
| issn | 1072-6691 |
| language | English |
| last_indexed | 2024-12-20T12:26:51Z |
| publishDate | 2005-12-01 |
| publisher | Texas State University |
| record_format | Article |
| series | Electronic Journal of Differential Equations |
| spelling | doaj.art-55415508c4cf48168bf5e18b43928d672022-12-21T19:40:50ZengTexas State UniversityElectronic Journal of Differential Equations1072-66912005-12-012005143114Existence of positive solutions for nonlinear boundary-value problems in unbounded domains of $mathbb{R}^{n}$Faten ToumiNoureddine ZeddiniLet $D$ be an unbounded domain in $mathbb{R}^{n}$ ($ngeq 2$) with a nonempty compact boundary $partial D$. We consider the following nonlinear elliptic problem, in the sense of distributions, $$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.http://ejde.math.txstate.edu/Volumes/2005/143/abstr.htmlGreen functionnonlinear elliptic equationpositive solutionSchauder fixed point theorem. |
| spellingShingle | Faten Toumi Noureddine Zeddini Existence of positive solutions for nonlinear boundary-value problems in unbounded domains of $mathbb{R}^{n}$ Electronic Journal of Differential Equations Green function nonlinear elliptic equation positive solution Schauder fixed point theorem. |
| title | Existence of positive solutions for nonlinear boundary-value problems in unbounded domains of $mathbb{R}^{n}$ |
| title_full | Existence of positive solutions for nonlinear boundary-value problems in unbounded domains of $mathbb{R}^{n}$ |
| title_fullStr | Existence of positive solutions for nonlinear boundary-value problems in unbounded domains of $mathbb{R}^{n}$ |
| title_full_unstemmed | Existence of positive solutions for nonlinear boundary-value problems in unbounded domains of $mathbb{R}^{n}$ |
| title_short | Existence of positive solutions for nonlinear boundary-value problems in unbounded domains of $mathbb{R}^{n}$ |
| title_sort | existence of positive solutions for nonlinear boundary value problems in unbounded domains of mathbb r n |
| topic | Green function nonlinear elliptic equation positive solution Schauder fixed point theorem. |
| url | http://ejde.math.txstate.edu/Volumes/2005/143/abstr.html |
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