Teklehaimanot, Ataklti Araya (2017) On Entire Solutions of Quasilinear Elliptic Equations. PhD thesis, Addis Ababa University.
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Abstract
In this thesis, we investigate entire solutions of the quasilinear equation (y) ∆φu = h(x; u) where ∆φu := div(φ(jruj)ru): Under suitable assumptions on the right-hand side we will show the existence of infinitely many positive solutions that are bounded and bounded away from zero in RN: All these solutions converge to a positive constant at infinity. The analysis that leads to these results is based on a fixed-point theorem attributed to Shcauder-Tychonoff. Under appropriate assumptions on h(x; t), we will also study ground state solutions of (y) whose asymptotic behavior at infinity is the same as a fundamental solution of the φ-Laplacian operator ∆φ: Ground state solutions are positive solutions that decay to zero at infinity. An investigation of positive solutions of (y) that converge to prescribed positive constants at infinity will be considered when the right-hand side in (y) assumes the form h(x; t) = a(x)f(t): After establishing a general result on the construction of positive solutions that converge to positive constants, we will present simple sufficient conditions that apply to a wide class of continuous functions f : R ! R so that the equation ∆φu = a(x)f(u) admits positive solutions that converge to prescribed positive constants at infinity. We will also study Cauchy-Liuoville type problems associated with the equation ∆φu = f(u) in RN: More specifically, we will study sufficient conditions on f : R ! R in order that the equation ∆φu = f(u) admits only constant positive solution provided that f has at least one real root. Our result in this direction can best be illustrated by taking φ(t) = ptp−2 + qtq−2 for some 1 < p < q which leads to the so called (p; q)-Laplacian, ∆(p;q)u := ∆pu + ∆qu:
| Item Type: | Thesis (PhD) |
|---|---|
| Subjects: | Q Science > Q Science (General) Q Science > QA Mathematics |
| Divisions: | Africana |
| Depositing User: | Selom Ghislain |
| Date Deposited: | 06 Sep 2018 13:49 |
| Last Modified: | 06 Sep 2018 13:49 |
| URI: | http://thesisbank.jhia.ac.ke/id/eprint/4967 |
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