Infinitely many positive solutions of a Gross-Pitaevskii equation in the presence of a harmonic potential and combined nonlinearities

Abstract

The main goal of this paper is to address an important conjecture in the field of differential equations in the presence of a harmonic potential. While in the subcritical case, the uniqueness of positive solution has been addressed by Hirose and Ohta in [12] in 2007, the problem has remained open for years in the supercritical case. In [9], the author obtained interesting numerical computations suggesting that for some bifurcating parameter λ, the equation has many positive solutions that vanish at infinity. In this paper, we provide a proof to this claim by constructing an accountable number of solutions that bifurcate from the unique singular solutions with λ close to the first eigenvalue λ 1 of the harmonic operator −∆ + |x| 2 . Our method hinges on a matching argument, and applies to the supercritical case, and to the supercritical case in the presence of a subcritical, critical or supercritical perturbation.