Boundary Value Problems for a Super-Sublinear Asymmetric Oscillator: The Exact Number of Solutions

Abstract

Properties of asymmetric oscillator described by the equation 𝑥 󸀠󸀠 = −𝜆(𝑥 + )𝑝 + 𝜇(𝑥 − ) 𝑞 (i), where 𝑝 ≥ 1 and 0 < 𝑞 ≤ 1, are studied.A set of (𝜆, 𝜇) such that the problem (i), 𝑥(0) = 0 = 𝑥(1) (ii), and |𝑥 󸀠 (0)| = 𝛼 (iii) have a nontrivial solution, is called 𝛼-spectrum.We give full description of 𝛼-spectra in terms of solution sets and solution surfaces. The exact number of nontrivial solutions of the two-parameter Dirichlet boundary value problem (i), and (ii) is given.