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Calculus of Variations

A mass supercritical problem revisited Louis Jeanjean1 · Sheng-Sen Lu2,3 Received: 29 March 2020 / Accepted: 31 July 2020 © Springer-Verlag GmbH Germany, part of Springer Nature 2020

Abstract 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 , ⎪ ⎪ ⎨ u2L 2 (R N ) = m, (Pm ) ⎪ ⎪ ⎩ 1 N u ∈ H (R ), 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 E m 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 sign-changing solutions when N ≥ 4. Finally we propose two open problems. Mathematics Subject Classification 35J60 · 35Q55

Communicated by P. Rabinowitz.

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Louis Jeanjean [email protected] Sheng-Sen Lu [email protected]

1

Laboratoire de Mathématiques (CNRS UMR 6623), Université de Bourgogne Franche-Comté, 25030 Besançon, France

2

Center for Applied Mathematics, Tianjin University, Tianjin 300072, People’s Republic of China

3

Present Address: School of Mathematical Sciences and LMAM, Peking University, Beijing 100871, People’s Republic of China 0123456789().: V,-vol

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L. Jeanjean, S.-S. Lu

1 Introduction We are concerned with the nonlinear scalar field equation with an L 2 constraint: ⎧ −u = f (u) − μu in R N , ⎪ ⎪ ⎨ u2L 2 (R N ) = m, ⎪ ⎪ ⎩ u ∈ H 1 (R N ).

(Pm )

Here N ≥ 1, f ∈ C(R, R), m > 0 is a given constant and μ ∈ R will arise as a Lagrange multiplier. In particular μ ∈ R does depend on the solution u ∈ H 1 (R N ) and is not a priori given. The main feature of (Pm ) is that the desired solutions have an a priori prescribed L 2 norm. In the literature, solutions of this type are often referred to as normalized solutions. A strong motivation to study (Pm ) is that it arises naturally in the search of stationary waves of nonlinear Schrödinger equations of the following form iψt + ψ + g(|ψ|2 )ψ = 0,

ψ : R+ × R N → C.

(1.1)

Here by stationary waves we mean solutions of (1.1) of the special form ψ(t, x) = eiμt u(x) with a constant μ ∈ R and a time-independent real valued function u ∈ H 1 (R N ). The research of such type of equations started roughly forty years ago [14,15,31,32,42] and it now lies at the root of several models directly linked with current applications, such as nonlinear optics, the theory of water waves. For these equations, finding solutions with a prescribed L 2 -norm is particularly relevant since this quantity is preserved along the time evolution. Under mild conditions on f , one can introduce the C 1