A mass supercritical problem revisited

Louis Jeanjean (1) , Sheng-Sen Lu (2)

1 LMB - Laboratoire de Mathématiques de Besançon (UMR 6623)
2 Center for Applied mathematics, Tianjin University

In any dimension N ≥ 1 and for given mass m > 0, we revisit the nonlinear scalar field equation with an L 2 constraint:        −∆u = f (u) − µu in R N , u 2 L 2 (R N) = m, u ∈ H 1 (R N), (Pm) where µ ∈ R will arise as a Lagrange multiplier. Assuming only that the nonlinearity f is continuous and satisfies weak mass supercritical conditions, we show the existence of ground states to (Pm) and reveal the basic behavior of the ground state energy Em as m > 0 varies. In particular, to overcome the compactness issue when looking for ground states, we develop robust arguments which we believe will allow treating other L 2 constrained problems in general mass supercritical settings. Under the same assumptions, we also obtain infinitely many radial solutions for any N ≥ 2 and establish the existence and multiplicity of nonradial signchanging solutions when N ≥ 4. Finally we propose two open problems.

Mots clés: 35J60; 35Q55 Nonlinear scalar field equations; Mass supercritical cases; Ground states; Radial and nonradial solutions; Sign-changing solutions

Domaines: Equations aux dérivées partielles [math.AP]

Louis Jeanjean, Sheng-Sen Lu. A mass supercritical problem revisited. Calculus of Variations and Partial Differential Equations, 2020, 59 (5), ⟨10.1007/s00526-020-01828-z⟩. ⟨hal-03336029⟩