Boundary value problems for a super-sublinear asymmetric oscillator: the exact number of solutions (Q1950018)

Summary: Properties of an asymmetric oscillator described by the equation \[ x'' = -\lambda(x^+)^p + \mu(x^-)^q\tag{1} \] with \(p \geq 1\) and \(0 < q \leq 1\) are studied. The set of parameter tupels \(\mu,\lambda\) such that the problem (1) has a nontrivial solution satisfying \[ x(0) = 0 = x(1)\tag{2} \] and \(|x'(0)| = \alpha\) is the called \(\alpha\)-spectrum. We give a full description of \(\alpha\)-spectra. The exact number of nontrivial solutions of the two-parameter Dirichlet boundary value problem (1) and (2) is given.