Entire Blow-Up Solutions of Semilinear Elliptic Equations and Systems
Abstract
We examine two problems concerning semilinear elliptic equations. We consider single equations of the form Delta upsilon rho(chi)upsilon(alpha) + sigma(chi)upsilon(Beta) for 0 < alpha is less than or equal to Beta is less than or equal to 1 and systems Delta upsilon = rho(absolute value chi absolute value)phi(nu), Delta nu = sigma (absolute value chi absolute value) delta(upsilon), both in Euclidean nu-space, nu is greater than or equal to 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 phi and sigma 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 rho, sigma, phi, and delta 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.
Document Details
- Document Type
- Technical Report
- Publication Date
- Mar 01, 2008
- Accession Number
- ADA482963
Entities
People
- Jesse D. Peterson
Organizations
- Air Force Institute of Technology