Abstract
We consider the semilinear elliptic PDE Δu + f(λ, u) = 0 with the zero-Dirichlet boundary condition on a family of regions, namely stadions. Linear problems on such regions have been widely studied in the past. We seek to observe the corresponding phenomena in our nonlinear setting. Using the Gradient Newton Galerkin Algorithm (GNGA) of Neuberger and Swift, we document bifurcation, nodal structure, and symmetry of solutions. This paper provides the first published instance where the GNGA is applied to general regions. Our investigation involves both the dimension of the stadions and the value λ as parameters. We find that the so-called crossings and avoided crossings of eigenvalues as the dimension of the stadions vary influences the symmetry and variational structure of nonlinear solutions in a natural way.
Original language | English (US) |
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Pages (from-to) | 447-464 |
Number of pages | 18 |
Journal | Communications in Nonlinear Science and Numerical Simulation |
Volume | 12 |
Issue number | 4 |
DOIs | |
State | Published - Jul 2007 |
Keywords
- Bifurcation
- GNGA
- Newton's method
- Semilinear elliptic BVP
- Stadia
- Stadium
- Variational method
ASJC Scopus subject areas
- Numerical Analysis
- Modeling and Simulation
- Applied Mathematics