Normalized solutions to mass supercritical Schrodinger equations with negative potential

Abstract

We study the existence of positive solutions with prescribed L 2 -norm for the Schrödinger equationWe treat two cases. Firstly, under an explicit smallness assumption on V and no condition on the mass, we prove the existence of a mountain pass solution at positive energy level, and we exclude the existence of solutions with negative energy. Secondly, requiring that the mass is smaller than some explicit bound, depending on V , and that V is not too small in a suitable sense, we find two solutions: a local minimizer with negative energy, and a mountain pass solution with positive energy. Moreover, a nonexistence result is proved.

Normalized Solutions to Kirchhoff Equation with Nonnegative Potential

“…Indeed, new difficulties arise due to the simultaneous occurrence of nonnegative potential and nonlocal term. As we mentioned before, the classical method in [15,23,35] does not work in our situation. Thus, we adopt the linking structure constructed by [2].…”

“…The case of negative potential V ≤ 0 is considered in [10] and [23], they both obtained the ground state solution. In fact, under some explicit smallness assumption on V (x), one has E V (c) < E 0 (c), thus the method in [3,15,28] is still valid and the trapping nature of the potential provides enough compactness.…”

Normalized solution to the Schödinger equation with potential and general nonlinear term: Mass super-critical case

“…By constructing a suitable linking structure, the authors of [2] can obtain the existence of solutions with high Morse index. The case of negative potential V (x) ≤ 0 and vanishing at infinity, is also considered in [17]. Under some explicit smallness assumption on V (x), the authors of [17] can obtain the existence of solutions with prescribed L 2 -norm.…”

“…The case of negative potential V (x) ≤ 0 and vanishing at infinity, is also considered in [17]. Under some explicit smallness assumption on V (x), the authors of [17] can obtain the existence of solutions with prescribed L 2 -norm. However, for the the mass super-critical case involves potential, all the literatures mentioned above, only consider the case of g(u) = |u| p−2 u.…”

Normalized Solutions to the mixed fractional Schrodinger equations with potential and general nonlinear term

“…By constructing a suitable linking structure, the authors proved the existence of normalized solutions with high Morse index. The case of negative potential is considered in [33] with the particular nonlinearity g(u) = |u| p . Very recently, in [18] Ding et al treated the case of negative potential and a more general nonlinearity.…”