#### Date of Award

3-2006

#### Document Type

Thesis

#### Degree Name

Master of Science

#### Department

Department of Mathematics and Statistics

#### First Advisor

Aihua W. Wood, PhD

#### Abstract

We consider the semilinear elliptic equation Δu = *p(x)f(u)* on a domain Ω ⊆ **R**^{n}, *n* ≥ 3, where *f* is a nonnegative function which vanishes at the origin and satisfies *g _{1}* ≤

*f*≤

*g*where

_{2}*g*are nonnegative, nondecreasing functions which also vanish at the origin, and

_{1}; g_{2}*p*is a nonnegative continuous function with the property that any zero of

*p*is contained in a bounded domain in such that

*p*is positive on its boundary. For Ω bounded, we show that a nonnegative solution u satisfying u(x) → ∞ as x → ∂Ω exists provided the function 𝜓(s) ≡ ⌠

^{s}

_{0}

*f(t) dt*satisfies ⌠

^{∞}

_{1}[𝜓(s)]

^{-1/2}ds < ∞ For Ω unbounded (including Ω =

**R**

^{n}), we show that a similar result holds where u(x) → ∞ as |x| → ∞ within Ω and u(x) → ∞ as

*x*→ ∂Ω if

*p(x)*decays to zero rapidly as |x| → ∞.

#### AFIT Designator

AFIT-GAM-ENC-06-03

#### DTIC Accession Number

ADA446264

#### Recommended Citation

Proano, Zachary H., "Existence of Explosive Solutions to Non-Monotone Semilinear Elliptic Equations" (2006). *Theses and Dissertations*. 3348.

https://scholar.afit.edu/etd/3348