Date of Award
Master of Science in Applied Mathematics
Department of Mathematics and Statistics
Aihua W. Wood, PhD
We examine two problems concerning semilinear elliptic equations. We consider single equations of the form Δu = p(x)uα + q(x)uβ for 0 <α ≤ β ≤1 and systems Δu = p(| x |) f (v), Δv = q(| x |)g(u) , both in Euclidean n -space, n ≥ 3 . These types of problems arise in steady state diffusion, the electric potential of some bodies, subsonic motion of gases, and control theory. For the single equation case, we present sufficient conditions on p and q to guarantee existence of nonnegative bounded solutions on the entire space. We also give alternative conditions that ensure existence of nonnegative radial solutions blowing up at infinity. Similarly, for systems, we provide conditions on p,q, f , and g that guarantee existence of nonnegative solutions on the entire space. The main requirement for f and g will be closely related to a growth requirement known as the Keller-Osserman condition. Further, we demonstrate the existence of solutions blowing up at infinity and describe a set of initial conditions that would generate such solutions. Lastly, we examine several specific examples numerically to graphically demonstrate the results of our analysis.
DTIC Accession Number
Peterson, Jesse D., "Entire Blow-Up Solutions of Semilinear Elliptic Equations and Systems" (2008). Theses and Dissertations. 2658.