Abstract |
In this thesis, direct and inverse problems concerning nodal solutions associated with the one-dimensional p-Laplacian operators are studied. We first consider the eigenvalue problem on (0, 1), −(y0(p−1))0 + (p − 1)q(x)y(p−1) = (p − 1) λw(x)y(p−1) (0.1) Here f(p−1) := |f|p−2f = |f|p−1 sgn f. This problem, though nonlinear and degenerate, behaves very similar to the classical Sturm-Liouville problem, which is the special case p = 2. The spectrum {λk} of the problem coupled with linear separated boundary conditions are discrete and the eigenfunction yn corresponding toλn has exactly n−1 zeros in (0, 1). Using a Pr‥ufer-type substitution and properties of the generalized sine function, Sp(x), we solve the reconstruction and stablity issues of the inverse nodal problems for Dirichlet boundary conditions, as well as periodic/antiperiodic boundary conditions whenever w(x) λ 1. Corresponding Ambarzumyan problems are also solved. We also study an associated boundary value problem with a nonlinear nonhomogeneous term (p−1)w(x) f(y(x)) on the right hand side of (0.1), where w is continuously differentiable and positive, q is continuously differentiable and f is positive and Lipschitz continuous on R+, and odd on R such that f0 := lim y!0+ f(y) yp−1 , f1 := lim y!1 f(y) yp−1 . are not equal. We extend Kong’s results for p = 2 to general p > 1, which states that whenever an eigenvalue _n 2 (f0, f1) or (f1, f0), there exists a nodal solution un having exactly n − 1 zeros in (0, 1), for the above nonhomogeneous equation equipped with any linear separated boundary conditions. Although it is known that there are indeed some differences, Our results show that the one-dimensional p-Laplacian operator is still very similar to the Sturm-Liouville operator, in aspects involving Pr‥ufer substitution techniques. |