Multiplicity of normalized semi-classical states for a class of nonlinear Choquard equations

1 Introduction and main results

In this article, we study the existence of multiple normalized semi-classical states to the following Choquard equation:

where a , ε > 0 , N ≥ 3 , α ∈ ( 0 , N ) , λ ∈ R , F ( t ) = ∫ 0 t f ( s ) d s . The Ruiz potential I α : R N → R is the Riesz potential defined as follows:

When λ ∈ R is a fixed and assigned parameter or even with an additional external, the existence of nontrivial solutions of (1.1) has been studied during the last decade. For example, when λ = 1 , ε = 1 , α = 2 , and F ( u ) = u 2 , (1.1) comes back to the description of the quantum theory of a polaron at rest by Pekar [39] and the modeling of an electron trapped in its own hole (in the work of Choquard in 1976), in a certain approximation to the Hartree-Fock theory of one-component plasma [29]. The equation is also known as the Schrödinger-Newton equation, which was proposed by Penrose [40] in 1996 as a model of self-gravitating matter. Under this condition, the existence of nontrivial solutions was investigated by various variational methods by Lieb [29] and Menzala [30] and also by ordinary differential equations methods [14,26,31,44]. There are also many articles investigating the Choquard equation under the general pure nonlinearity condition. In 2013, Moroz and Van Schaftingen [32] considered the semilinear elliptic problem:

where N ≥ 3 , α ∈ ( 0 , N ) . They proved that there is a positive ground state solution to (1.2) when N + α N < p < N + α N − 2 . In 2015, Moroz and Van Schaftingen [33] also considered semi-classical states of nonlocal problems:

where N ≥ 1 , α ∈ ( 0 , N ) , and ε > 0 is a small parameter. By using variational methods and a new nonlocal penalization technique, they proved that equation (1.3) has a family of solutions centered on the local minimum of potential V for each small ε > 0 . Moroz and Schaftingen [36] proved problem (1.3) has a groundstate solution if ε = 1 , λ = 0 , V ( x ) ≡ 1 and the nonlinearity satisfies the following almost necessary conditions:

1. there exists C > 0 such that for every s ∈ R , ∣ s f ( s ) ∣ ≤ C ( ∣ s ∣ N + α N + ∣ s ∣ N + α N − 2 ) ;

2. lim ∣ s ∣ → 0 F ( s ) ∣ s ∣ − N + α N = 0 and lim ∣ s ∣ → ∞ F ( s ) ∣ s ∣ − N + α N − 2 = 0 ;

3. there exists s 0 ∈ R \ { 0 } such that F ( s 0 ) = 0 .

where g ( s ) is of the type g ( s ) = ∑ i = 1 m a i ∣ s ∣ σ 1 s with for a i > 0 , 0 < σ i < N N − 2 for N ≥ 3 , and σ i > 0 if N = 1 , 2 , and i = 1 , … , m with m ∈ N . Since then, the study of normalized solutions for Schrödinger equations is attracting much attention of more and more researchers in the past few years, and there have been many scholars devoted to the existence of multiple normalized solution, for instances, we refer to Bartsch et al. [7], Hirata and Tanaka [19], Ikoma and Tanaka [20], Bartsch and Soave [11], Bartsch and de Valeriola [10], Jeanjean and Lu [22], Cingolani and Jeanjean [12], Lan et al. [25], Noris et al. [38], Jeanjean and Le [23,24], Guo and Jeanjean [17], Soave [42,43], Wang and Sun [45], and the references therein.

We notice that some scholars have studied the normalized solutions for Schrödinger equations with nonconstant potentials in recent years. Especially, Bartsch et al. [9] studied the normalized solutions for mass supercritical Schrödinger equations with potential

They proved the existence of a solution ( u , λ ) ∈ H 1 ( R N ) × R + with prescribed L 2 -norm ‖ u ‖ 2 = ρ under various conditions on the potential V : R N → R , positive and vanishing at infinity, including potentials with singularities, by using a new min-max argument. Alves [1] focused on existence of multiple normalized solutions to the following Schrodinger equation with nonconstant potential

where a , ε > 0 , f has L 2 -subcritical growth, and h : R N → [ 0 , ∞ ) is a continuous function with the following assumptions:

1. h ∈ C ( R N ) and 0 < h 0 = inf x ∈ R N h ( x ) ≤ max x ∈ R N h ( x ) = h max ;

2. h ∞ = lim ∣ x ∣ → + ∞ h ( x ) < h max ;

3. h − 1 ( { h max } ) = { a 1 , a 2 , … , a l } with a 1 = 0 and a j ≠ a s if j ≠ s .

where V satisfies the Rabinowitz’s global condition [41] or del Pino and Felmer’s local condition [15], and the Lusternik-Schnirelmann theory is applied.

When V ( x ) ≡ 0 and ε = 1 , Li and Ye [28] considered the normalized solution to equation (1.1) as follows:

where F satisfies some superlinear growth, but Sobolev subcritical conditions, they proved the existence of a positive solution of (1.8) under prescribed L 2 -norm by using minimax procedure and concentration compactness. Yuan et al. [48] revisited problem (1.8) under the weaker conditions imposed on the nonlinearity f , by using a minimax procedure and some new analytical techniques, and showed that for any a > 0 , (1.8) possesses at least a couple of weak solution. Recently, Bartsch et al. [8] proved the existence of a least energy solution of (1.8) based on a linking argument, and obtained the existence of infinitely many normalized solutions by using a cohomological index theory when f is odd in u . When f ( u ) = ∣ u ∣ 2 α * − 2 u , and 2 α * = N + α N − 2 is the Hardy-Littlewood-Sobolev critical exponent, problem (1.8) reduces to the following type of problem with a subcritical perturbation μ ∣ u ∣ q − 2 u :

where N ≥ 3 , μ , a > 0 , q ∈ ( 2 , 2 + 4 N ) . Li [27] showed the existence of the ground state, and its radial symmetry, exponential decay and orbital stability, instability. Ye et al. [47] proved that (1.9) with μ ∣ u ∣ q − 2 u being replaced by a nonlocal perturbation μ ( I α ∗ ∣ u ∣ q ) ∣ u ∣ q − 2 u has normalized ground states and mountain-pass type solutions, under the L 2 -subcritical growth, L 2 -critical, or L 2 -supercritical growth.

Motivated by the aforementioned works, in this article, we consider the normalized semiclassical states to the following Choquard problem:

where a > 0 is a constant, ε > 0 is a parameter, N ≥ 3 , α ∈ ( 0 , N ) , λ ∈ R is unknown and appears as a Lagrange multiplier, and the potential V is a bounded continuous function and satisfies the following assumptions.

1. V ∈ C ( R N , R ) ∩ L ∞ ( R N ) , V ( x ) ≥ 0 for each x ∈ R N ;

2. There exists a bounded set Λ ⊂ R N such that

   min x ∈ Λ ¯ V ( x ) < min x ∈ ∂ Λ V ( x ) = V ∞ .

   Without loss of generality, we usually assume that 0 ∈ Λ and V ( 0 ) = min x ∈ Λ ¯ V ( x ) .

1. f is an odd function and lim s → 0 ∣ f ( s ) ∣ ∣ s ∣ q 0 − 1 = β > 0 , q 0 ∈ 1 + α N , 1 + 2 + α N ;

2. There are constants c 1 , c 2 > 0 and p 1 ∈ 1 + α N , 1 + 2 + α N satisfying

   ∣ f ( s ) ∣ ≤ c 1 + c 2 ∣ s ∣ p 1 − 1 , ∀ s ∈ R ;

3. There exists q ∈ q 0 , 1 + 2 + α N such that f ( s ) ⁄ s q − 1 is an increasing function for s ∈ ( 0 , + ∞ ) .

The main result of the article can be formulated as below.

After a simple change of variables, problem (1.10) can be transformed into the following equivalent form:

To study the existence of multiple normalized solutions of problem (1.11), we shall look for the critical points of the corresponding energy functional

constrained to the S ( a ) given below:

We would like to recall that if Y is a closed subset of a topological space X , the Lusternik- Schnirelmann category cat X ( Y ) is the least number of closed and contractible sets in X which cover Y . If X = Y , we use the notation cat ( X ) . For more details about this subject, we refer to [46].

The main structure of the article is as follows: In Section 2, we consider the autonomous case of the equation (1.11) and show some technical results. In Section 3, we focus on the nonautonomous case and study the penalized problem by using truncation techniques. In this section, we prove that the modified energy functional satisfies the Palais-Smale condition, and obtain some useful tools, which are useful for the proof of the following main conclusions. In Section 4, we prove the Theorem 1.1.

Throughout this article, we use the notation ‖ ⋅ ‖ p to denote the norm in L p ( R N ) and use C , C j , j = 1 , 2 , … , to denote generic constants whose values might be irrelevant. But the ones which we need to emphasize will be denoted by the special characteristics.

2 The autonomous case

In this section, we first consider the autonomous case and prove some relevant conclusions to facilitate the subsequent argument. We start by reviewing the following two inequalities, which will be used frequently used.

The Gagliardo-Nirenberg inequality: If N ≥ 3 , then for each l ∈ [ 2 , 2 N N − 2 ) , there exists C = C ( N , l ) > 0 such that

Next, we focus on the existence of a normalized solution to the following problem:

where a , μ > 0 , λ ∈ R is unknown and appears as a Lagrange multiplier and f satisfies ( f 1 ) – ( f 3 ) . Our goal is to find the critical point of the energy functional:

constrained to the sphere S ( a ) given in (1.12).

Our main conclusion in this section, is stated as follows.

To prove the Theorem 2.1, we first propose several lemmas.

From the previous lemma, we know that the real number

is well defined. Next, we discuss some properties of I μ .

Next we will introduce the compactness theorem on S ( a ) , which plays an important role in the analysis of both autonomous case and nonautonomous case.