A NUMERICAL METHOD FOR FINDING SIGN-CHANGING SOLUTIONS OF
SUPERLINEAR DIRICHLET PROBLEMS
(to appear in Nonlinear World)
John M. Neuberger
In a recent result
(See Castro-Cossio-Neuberger, to appear in Rocky Mountain J. of Math.),
it was shown via a variational argument that a class of
superlinear elliptic boundary value problems has
at least three nontrivial solutions,
a pair of one sign and one which sign changes exactly once.
These three and all other nontrivial solutions are saddle points
of an action functional,
and are characterized as local minima of that functional
restricted to a codimension one submanifold of the Hilbert space
H-0-1-2, or an appropriate higher codimension subset of that manifold.
In this paper we present a numerical Sobolev steepest descent algorithm
for finding these three solutions.
Of primary interest is the method of projecting
iterates of elements in our Hilbert space onto the submanifold and
When applied to the ordinary differential equation,
the algorithm is extended to find additional solutions
possessing a greater number of internal zeroes,
and in that case the solutions are compared to independent numerical
calculations obtained by Euler's method.
We further test the algorithm on partial differential equations
on the unit square.
With or without a symmetric nonlinearity,
numerical computations for PDEs on the square yield
four exactly-once sign-changing solutions of Morse Index 2
and supply evidence that suggests that there may exist four more of MI 3.
In modifying the code to run on arbitrary regions such as the disk,
annulus, and dumbell, we are obtaining numerical evidence complementary to
our current analysis.
In particular, as we show in a current paper (in submittion), the minimal
action function valued sign-changing solution on the disk is nonradial.
This also holds on the annulus.
This algorithm can be used to investigate the nodal structure of a wide
range of superlinear elliptic PDEs and should be modifiable to include
Neumann boundary conditions. Accuracy and convergence are quite
satisfactory though improvements and refinements are clearly possible.
Particularly gratifying are the ways we can use the algorithm to
visualize the infinite dimensional submanifold's structure and its relation
to the eigenfunctions of the negative Laplacian.