Summary: Remarks on time map for quasilinear equations
Tomasz Adamowicz and Philip Korman
Department of Mathematical Sciences
University of Cincinnati
Cincinnati Ohio 45221-0025
We present two different generalizations of R. Schaaf's  time map formula to quasilinear
equations, including the case of p-Laplacian. We give conditions for monotonicity and for
convexity of the time map, which imply uniqueness or multiplicity results for the corresponding
Dirichlet boundary value problem. Our time map formulas can be also used for effective
computations of the global solution curves.
Key words: Multiplicity of solutions, time maps, quasilinear equations, p-Laplacian.
AMS subject classification: 34B15, 35J60, 35J70.
The p-Laplace operator plays the fundamental role in nonlinear analysis. It serves as a model
quasilinear equation both in pure mathematics and in various areas of the applied sciences (see
e.g. [1, 4, 5, 9, 11, 13]). One of the mainstreams of the p-harmonic theory, with plethora of
papers published in recent years, is the so-called nonlinear eigenvalue problem (see e.g. [2, 10, 16]
and references therein). In the simplest form it reads