Analysis of classes of superlinear semipositone problems with nonlinear boundary conditions

UNCG Author/Contributor (non-UNCG co-authors, if there are any, appear on document)
Quinn A. Morris (Creator)
The University of North Carolina at Greensboro (UNCG )
Web Site:
Ratnasingham Shivaji

Abstract: We study positive radial solutions for classes of steady state reaction diffusion problems on the exterior of a ball with both Dirichlet and nonlinear boundary conditions. We consider p-Laplacian problems (p>1) with reaction terms which are superlinear at infinity and semipositone. In the case p=2, using variational methods, we establish the existence of a solution, and via detailed analysis of the Green's function, we prove the positivity of the solution. In the case p[unequal to]2, we again use variational methods to establish the existence of a solution, but the positivity of the solution is achieved via sophisticated a priori estimates. In the case p[unequal to]2, the Green's function analysis is no longer available. Our results significantly enhance the literature on superlinear semipositone problems. Finally, we provide algorithms for the numerical generation of exact bifurcation curves for one-dimensional problems. In the autonomous case, we extend and analyze a quadrature method, and using nonlinear solvers in Mathematica, generate bifurcation curves. In the nonautonomous case, we employ shooting methods in Mathematica to generate bifurcation curves.

Additional Information

Language: English
Date: 2017
Existence, P-superlinear, Quadrature method, Semipositone
Nonlinear boundary value problems $x Numerical solutions
Dirichlet problem $x Numerical solutions
Curves $x Rectification and quadrature
Diffusion $x Mathematical models
Laplacian operator

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