We derive a non-linear differential equation that must be satisfied by the quantum potential, in the context of the Madelung equations, in one dimension for a particular class of wave functions. In this case, we exhibit explicit conditions leading to the blow-up of the quantum potential of a free particle at the boundary of the compact support of the probability density.

Abstract We show the variational structure of a multiplicity result of positive solutions u is an element of H(1) (R(N)) to the equation -Delta u + a(x)u = u(p), where N >= 2, p > 1 with p < 2* - 1 = N+2/N-2 if N >= 3 and the potential a(x) is a positive function enjoying a planar symmetry. We require suitable decay assumptions which are widely implied by those in [6], in which Wei and Yan have obtained an analogous multiplicity result by using different techniques.

We show the existence of infinitely many positive solutions u ∈ H1(R2) to the equation −Delta u + a(x)u = u^p, with p > 1 , without asking, on the positive potential a(x), any symmetry assumption as inWei and Yan (Calc Var Partial Differ Equ 37, 423–439, 2010) or Devillanova and Solimini (Adv Nonlinear Studies 12, 173–186, 2012) or small oscillation assumption as in Cerami et al. (Commun Pure Appl Math, doi:10.1002/cpa.21410, 2012) 6 and in Weiwei and Wei (Infinitely many positive solutions for Nonlinear equations with non-symmetric Potential, 2012).