| Author(s) |
Du, Yihong
Efendiev, Messoud
|
| Publication Date |
2017
|
| Abstract |
<p>We consider positive solutions of quasilinear elliptic problems of the form Δ<i>p<sup>u</sup></i> + ƒ(<i>u</i>) = 0 over the quarter-space <i>Q</i> = {<i>x</i> ∈ ℝ<i><sup>N</sup></i> : <i>x</i><sub>1</sub> > 0, <i>x</i><sub>2</sub> > 0}, with <i>u</i> = 0 on ∂<i>Q</i>. For a general class of nonlinearities ƒ ≥ 0 with finitely many positive zeros, we show that, for each <i>z</i> > 0 such that ƒ(<i>z</i>) = 0, there is a bounded positive solution satisfying </p> <p>lim u(<i>x</i>1, <i>x</i>2, ..., <i>x<sub>N</sub></i>) = <i>V</i> (x2), <br/> <i>x</i>1→∞ <br/>lim u(<i>x</i>1, <i>x</i>2, ..., <i>x<sub>N</sub></i>) = <i>V</i> (<i>x</i>1),<br/><i>x</i>2→∞</p> <p>where <i>V</i> is the unique solution of the one-dimensional problem</p> <p>ΔpV + ƒ(<i>V</i>) = 0 in [0,∞), <i>V</i> (0) = 0, <i>V</i> (<i>t</i>) > 0 for <i>t</i> > 0, <i>V</i> (∞) = <i>z</i>.</p> <p>When <i>p</i> = 2, we show further that such a solution is unique, and there are no other types of bounded positive solutions to the quarter-space problem. Thus in this case the number of bounded positive solutions to the quarter-space problem is exactly the number of positive zeros of ƒ.</p>
|
| Citation |
Patterns of Dynamics, p. 128-137
|
| ISBN |
9783319641737
9783319641720
9783319877419
|
| Link | |
| Publisher |
Springer
|
| Series |
Springer Proceedings in Mathematics & Statistics
|
| Edition |
1
|
| Title |
Existence and Exact Multiplicity for Quasilinear Elliptic Equations in Quarter-Spaces
|
| Type of document |
Book Chapter
|
| Entity Type |
Publication
|
| Name | Size | format | Description | Link |
|---|---|---|---|---|
| closedpublished/ExistenceDu2017BookChapter.pdf | 227.443 KB | application/pdf | Published version | View document |
| administrative/PatternsOfDynamics2017FrontMatter.pdf | 97.627 KB | application/pdf | Front matter | View document |