Global Solution Curves for Semilinear Elliptic Equations

The following sections are included:

• Preface

• Contents

The following sections are included:

• Continuation of solutions

• Symmetric domains in R²

• Turning points and the Morse index

• Convex domains in R²

• Pohozaev’s identity and non-existence of solutions for elliptic systems
  • Non-existence of solutions in the presence of supercritical and lower order terms
  • Non-existence of solutions for a class of systems
  • Pohozhaev’s identity for a version of p-Laplace equation

• Problems at resonance

The following sections are included:

• Preliminary results

• Positivity of solution to the linearized problem

• Uniqueness of the solution curve

• Direction of a turn and exact multiplicity

• On a class of concave-convex equations

• Monotone separation of graphs

• The case of polynomial f(u) in two dimensions

• The case when f(0) < 0

• Symmetry breaking

• Special equations

• Oscillations of the solution curve
  • Asymptotics of some oscillatory integrals
  • Reduction to the oscillatory integrals

• Uniqueness for non-autonomous problems
  • Radial symmetry for the linearized equation

• Exact multiplicity for non-autonomous problems

• Numerical computation of solutions
  • Using power series approximation
  • Application to singular solutions

• Radial solutions of Neumann problem
  • A computer assisted study of ground state solutions

• Global solution curves for a class of elliptic systems
  • Preliminary results
  • Global solution curves for Hamiltonian systems
  • A class of special systems

• The case of a “thin” annulus

• A class of p-Laplace problems

The following sections are included:

• Positive solutions of autonomous problems

• Direction of the turn

• Stability and instability of solutions
  • S-shaped curves of combustion theory
  • An extension of the stability condition

• S-shaped solution curves

• Computing the location and the direction of bifurcation
  • Sign changing solutions

• A class of symmetric nonlinearities

• General nonlinearities

• Infinitely many curves with pitchfork bifurcation

• An oscillatory bifurcation from zero: A model example

• Exact multiplicity for Hamiltonian systems

• Clamped elastic beam equation
  • Preliminary results
  • Exact multiplicity of solutions

• Steady states of convective equations

• Quasilinear boundary value problems
  • Numerical computations for the prescribed mean curvature equation

• The time map for quasilinear equations

• Uniqueness for a p-Laplace case

The following sections are included:

• Bibliography