REMARKS ON NORMALIZED GROUND STATES OF SCHR ¨ODINGER EQUATION WITH AT LEAST MASS CRITICAL NONLINEARITY

We are concerned with the nonlinear Schr¨odinger equation − ∆ u + λu = g ( u ) in R N , λ ∈ R , with prescribed L 2 -norm (cid:82) R N u 2 dx = ρ 2 . Under general assumptions about the nonlinearity which allows at least mass critical growth, we prove the existence of a ground state solution to the problem via a clear constrained minimization method.


Introduction
In this paper, we consider the following nonlinear Schrödinger equation where ρ > 0 is a prescribed constant, N ≥ 3 and λ ∈ R will appear as Lagrange multiplier.
Such problems are motivated in particular by searching for solitary waves or stationary states in nonlinear equations of the Schrödinger or Klein-Gordon type.For physical reasons, it is natural to study the existence of solutions with prescribed L 2 -norm. Let where H 1 (R N ) is endowed with the usual norm u = (|∇u| 2 2 + |u| 2 2 ) 1/2 and | • | q stands for the L q -norm.Under suitable assumptions provided below, solutions to (1.1) are critical points of J : where G(u) = u 0 g(s)ds, on the constraint S with λ ∈ R being the Lagrange multiplier.Set where It is known that, thanks to the Pohozaev identity in [4], any solution to (1.1) stays in M. For In this paper, we make the following assumptions.
Let C N,p be the optimal constant of the Gagliardo-Nirenberg inequality where δ = N ( 1 2 − 1 p ).Since G can have L 2 -critical growth at the origin by (G 1 ), we need the assumption which implies that ρ or η is small.In recent years, the existence of normalized solutions for nonlinear Schrödinger equations has been studied widely under variant assumptions about g for instance in [1, 2, 5, 7-9, 11-13, 15, 17, 18] and the references therein.Let 2 N = 2 + 4 N .In the L 2 -subcritical case, i.e.G(u) ∼ |u| p with 2 < p < 2 N , one can obtain the existence of a global minimizer of J directly on S, see [16].In the L 2 -supercritical and Sobolev subcritical (2 N < p < 2 * ) case, the energy functional J is unbounded from above and from below and minimization does not work.For this case, using a mountain-pass argument developed on S, Jeanjean [12] showed the existence of one normalized solution.A different mini-max approach based on the σ-homotopy stable family of compact subsets of M has been applied in [2,3].Note that in [2,3,12] the nonlinearity was assumed to satisfy the following condition of Ambrosetti-Rabinowitz type: there exist 2 Recently, Bieganowski and Mederski in [5] considered general growth conditions on G in the spirit of Berestycki and Lions [4] and obtained ground states by a direct minimization method.The delicate approach in [5] consists of minimizing J on the constraint D ρ ∩ M.Among other results, they obtained a normalized ground state solution for (1.1) under assumptions (G 0 ) − (G 5 ), (P 0 ) and In this paper, we investigate the existence of ground state solutions for (1.1) and deal with general nonlinearities in a version slightly different from the ones in [5].We weaken some assumptions about g and refine some key ingredients of the arguments in [5], such as the determination of the sign of the corresponding Lagrange multipliers and the nonexistence of nontrivial solutions for the associated elliptic equation.Precisely, we prove the following theorem.
) is an abstract assumption.As shown in [5], (G 6 ) holds under the assumptions (G 0 ) − (G 5) and ( G 6 ) for u ∈ R. In fact, if u ∈ D ρ \{0} is a weak solution to −∆ u = g( u), then by regularity, u is continuous.Moreover, Then it follows that there is an open interval I ⊂ R such that 0 ∈ I and g(u)u − 2 * G(u) = 0 for u ∈ I. Then we deduce that G(u) = C|u| 2 * for some C > 0 and u ∈ I, contradicting the definition of .As an immediate corollary of this observation, we obtain a normalized ground state solution of (1.1) under assumptions (G 0 ) − (G 5 ), (P 0 ) and ( G 6 ).Therefore, Theorem 1.1 can be regarded as a generalization of the existence result in [5, Theorem 1.1].Moreover, inspired by [10], we know in advance the sign of the corresponding Lagrange multiplier λ > 0 by Clark's Theorem [6,10].
Under (G 0 ) − (G 5 ) and (P 0 ), we can prove (G 6 ) if N ∈ {3, 4} and g is odd.Therefore, the following corollary is a generalization of the result in [5, Theorem 1.1(b)] where additional assumption g(u)u ≤ 2 * G(u) for u ∈ R was assumed.
Next, we replace (G 6 ) by the following assumption which is simpler to check.
We introduce the following additional assumption about ρ.
As a corollary of Theorem 1.
Then g satisfies (G 7 ) with C 0 = 1, (G 1 ) with η = 0 and (G 2 ) − (G 5 ).Therefore, Here and in the sequel, C denotes a generic positive constant which may vary from one equation to another.In the next section, we give the proof of the main results.

Proof of main results
First, we recall the property on the notion Proof.By (G 0 ), (G 1 ) and (G 3 ), for any ε > 0 there exists R ε > 0 such that By (2.1), we deduce that for some p ∈ (2 + 4/N, 2 * ), there exists c ε > 0 such that This implies that for u ∈ D ρ ∩ M, Then the Gagliardo-Nirenberg inequality and the Sobolev embedding inequality imply that where δp := N (p − 2)/2 > 2 and C is a positive constant.Therefore, we see that |∇u| Note that (G 1 ), (G 3 ) and (G 5 ) yield that for any ε > 0 there is c ε > 0 such that