Normalized solutions for a fourth-order Schrödinger equation with a positive second-order dispersion coefficient

Abstract

In this paper, we study normalized solutions to a fourth-order Schrödinger equation with a positive second-order dispersion coefficient in the mass supercritical regime. Unlike the well-studied case where the second-order term is zero or negative, the geometrical structure of the corresponding energy functional changes dramatically and this makes the solution set richer. Under suitable control of the second-order dispersion coefficient and mass, we find at least two radial normalized solutions, a ground state and an excited state, together with some asymptotic properties. It is worth pointing out that in the considered repulsive case, the compactness analysis of the related Palais-Smale sequences becomes more challenging. This forces the implementation of refined estimates of the Lagrange multiplier and the energy level to obtain normalized solutions.