This book provides an introduction to the bifurcation theory approach to global solution curves and studies the exact multiplicity of solutions for semilinear Dirichlet problems, aiming to obtain a complete understanding of the solution set. This understanding opens the way to efficient computation of all solutions. Detailed results are obtained in case of circular domains, and some results for general domains are also presented. The author is one of the original contributors to the field of exact multiplicity results.

Description-Table Of Contents

1. Curves of solutions on general domains. 1.1. Continuation of solutions. 1.2. Symmetric domains in R[symbol]. 1.3. Turning points and the Morse index. 1.4. Convex domains in R[symbol]. 1.5. Pohozaev's identity and non-existence of solutions for elliptic systems. 1.6. Problems at resonance -- 2. Curves of solutions on balls. 2.1. Preliminary results. 2.2. Positivity of solution to the linearized problem. 2.3. Uniqueness of the solution curve. 2.4. Direction of a turn and exact multiplicity. 2.5. On a class of concave-convex equations. 2.6. Monotone separation of graphs. 2.7. The case of polynomial f(u) in two dimensions. 2.8. The case when f(0) < 0. 2.9. Symmetry breaking. 2.10. Special equations. 2.11. Oscillations of the solution curve. 2.12. Uniqueness for non-autonomous problems. 2.13. Exact multiplicity for non-autonomous problems. 2.14. Numerical computation of solutions. 2.15. Radial solutions of Neumann problem. 2.16. Global solution curves for a class of elliptic systems. 2.17. The case of a """"thin"""" annulus. 2.18. A class of p-Laplace problems -- 3. Two point boundary value problems. 3.1. Positive solutions of autonomous problems. 3.2. Direction of the turn. 3.3. Stability and instability of solutions. 3.4. S-shaped solution curves. 3.5. Computing the location and the direction of bifurcation. 3.6. A class of symmetric nonlinearities. 3.7. General nonlinearities. 3.8. Infinitely many curves with pitchfork bifurcation. 3.9. An oscillatory bifurcation from zero: a model example. 3.10. Exact multiplicity for Hamiltonian systems. 3.11. Clamped elastic beam equation. 3.12. Steady states of convective equations. 3.13. Quasilinear boundary value problems. 3.14. The time map for quasilinear equations. 3.15. Uniqueness for a p-Laplace case.