2002-Fez conference on Partial Differential Equations, Electronic Journal of Differential Equations, Conference 09, 2002, pp 65–76. http://ejde.math.swt.edu or http://ejde.math.unt.edu ftp ejde.math.swt.edu (login: ftp)

Stationary Solutions for a Schrödinger-Poisson System in ℝ^{3} ^{∗}

Khalid Benmlih

Abstract Under appropriate, almost optimal, assumptions on the data we prove existence of standing wave solutions for a nonlinear Schrödinger equation in the entire space ℝ^{3} when the real electric potential satisfies a linear Poisson equation.

1 Introduction

Consider the time-dependent system which couples the Schrödinger equation

i∂_{t}u = −\frac{1}{2}Δu + (V + Ṽ)u  (1.1)

with initial value u(x, 0) = u(x), and the Poisson equation

−ΔV = |u|^{2} − n^{∗}.  (1.2)

The dopant-density n^{∗} and the effective potential Ṽ are given time-independent reals functions. There are many papers dealing with the physical problem modelled by this system from which we mention Markowich, Ringhofer & Schmeiser [8]; Illner, Kavian & Lange [3]; Nier [9]; Illner, Lange, Toomire & Zweifel [4], and references therein.

In this work we are mainly concerned with the proof of standing waves (actually ground states) of (1.1)–(1.2) in the entire space ℝ^{3}, i.e. solutions of the form

u(x, t) = e^{iωt}u(x)

with real number ω (frequency) and real wave function u. Hence we are interested in the stationary system

−\frac{1}{2}Δu + (V + Ṽ)u + ωu = 0 in ℝ^{3}  (1.3)
−ΔV = |u|^{2} − n^{∗} in ℝ^{3}  (1.4)

∗Mathematics Subject Classifications: 35J50, 35Q40.

Key words: Schrödinger equation, Poisson equation, standing wave solutions,

variational methods.

\begin{matrix}c \\ ◯ \end{matrix}2002 Southwest Texas State University.

Published December 28, 2002.

under appropriate, almost optimal, assumptions on Ṽ and n^{∗}. We suppose first that Ṽ ∈ L_{loc}^{1}(ℝ^{3}) and n^{∗} ∈ L^{6/5}(ℝ^{3}).

Let us remark that if V_{0} is such that −ΔV_{0} = −n^{∗} then (0, V_{0}) is a solution of the system (1.3)-(1.4). But here, we deal with solutions (u, V) in H^{1}(ℝ^{3}) × D^{1,2}(ℝ^{3}) such that u ≢ 0.

F. Nier [9] has studied the system (1.3)-(1.4). He has showed the existence of a solution for small data i.e. when ∥Ṽ∥_{L^{2}} and ∥n^{∗}∥_{L^{2}} are small enough. Conversely to our approach here, he has began by solving (1.3) for a fixed V and investigate the Poisson equation then obtained.

In this paper we solve first explicitly the Poisson equation (1.4) for a fixed u in H^{1}(ℝ^{3}). Next we substitute this solution V = V (u) in the Schrödinger equation (1.3) and look into the solvability of

−\frac{1}{2}Δu + (V (u) + Ṽ)u + ωu = 0 in ℝ^{3}.  (1.5)

Using the explicit formula of V (u), this equation appears as a Hartree equation studied by P.L. Lions [6] in the case where n^{∗} ≡ 0 and Ṽ(x) := −2/|x|. The fact that Ṽ in [6] converges to zero at infinity plays a crucial role to prove existence of solutions. However, in this paper we show that a slight modification of the arguments used in that paper allows us to prove existence of a ground state in the case Ṽ satisfying (1.7), (1.9) and n^{∗} not necessarily zero (but satisfying (1.8) and (1.9) as below).

Before giving our hypotheses on Ṽ and n^{∗} let us define a decomposition which will be useful in the sequel.

Definition 1.1 We say that g satisfies the decomposition (1.6) if:

(i) g ∈ L_{loc}^{1}(ℝ^{3}),

(ii) g ≥ 0, and

(iii) There exists q_{0} ∈ [3/2, ∞] : ∀λ > 0 ∃g_{1λ} ∈ L^{q_{0}}(ℝ^{3}), q_{λ} ∈]3/2, ∞[ and g_{2λ} ∈ L^{q_{λ}}(ℝ^{3}) such that

g = g_{1λ} + g_{2λ} and lim_{λ→0} ∥g_{1λ}∥_{L^{q_{0}}} = 0.  (1.6)

For convenience, we use throughout this paper the following notations:

• ∥.∥ denotes the norm ∥.∥_{L^{2}} on L^{2}(ℝ^{3}),

• 𝕀_{A} denotes the characteristic function of the set A ⊂ ℝ^{3},

• [F ≤ λ] denotes the set {x; F (x) ≤ λ} for a function F and λ ∈ ℝ.

Let us give now two examples of functions satisfying the conditions in Definition 1.1.

Example 1.2 The following two functions satisfy the decomposition (1.6):

(i) g(x) := 1/|x|^{α} for some 0 < α < 2.

(ii) |g| where g is a function in L^{r}(ℝ^{3}) for some r > 3/2.

Proof. To prove (i) we write, for λ > 0,

\frac{1}{|x|^{α}} := \underbrace{\frac{1}{|x|^{α}} 𝕀_{[|x|>1/λ]}}_{g_{1λ}} + \underbrace{\frac{1}{|x|^{α}} 𝕀_{[|x|≤1/λ]}}_{g_{2λ}} .

Elementary calculations give

∥g_{1λ}∥_{L^{q_{0}}}^{q_{0}} = \frac{4π}{αq_{0} − 3}(λ)^{αq_{0}−3} and ∥g_{2λ}∥_{L^{q}}^{q} = \frac{4π}{3 − αq} (\frac{1}{λ})^{3−αq}.

Hence it suffices to choose any finite numbers q_{0} , q such that 3/2 < q < 3/α < q_{0}. To show (ii) write, as above,

|g| := \underbrace{|g|𝕀_{[|g|≤λ]}}_{g_{1λ}} + \underbrace{|g|𝕀_{[|g|>λ]}}_{g_{2λ}} .

It is clear that ∥g_{1λ}∥_{L^{∞}} ≤ λ (q_{0} = ∞) and ∥g_{2λ}∥_{L^{r}} ≤ ∥g∥_{L^{r}} (q_{λ} = r). □

Hypotheses. In what follows we assume that

Ṽ^{+} ∈ L_{loc}^{1}(ℝ^{3}) and Ṽ^{−} satisfies the decomposition (1.6) ,  (1.7)

where Ṽ^{+}(x) := max(Ṽ(x), 0) and Ṽ^{−}(x) := max(−Ṽ(x), 0). We suppose also that

n^{∗} ∈ L^{1} ∩ L^{6/5}(ℝ^{3})  (1.8)

and finally if we denote by

ϱ(x) := 2Ṽ(x) − \frac{1}{2π} ∫_{ℝ^{3}} \frac{n^{∗}(y)}{|x − y|} dy

we assume that

inf { ∫_{ℝ^{3}} (|∇φ|^{2} + ϱ(x)φ^{2})dx, ∫_{ℝ^{3}} |φ|^{2} = 1} < 0.  (1.9)

Remark that in the case of [6] (where n^{∗} ≡ 0 and Ṽ(x) := −2/|x|), all the three hypotheses above are satisfied. Indeed, (1.7) and (1.8) follow from (i) of Example 1.2. Moreover, if we consider Φ(x) := e^{−2|x|} then it verifies

−ΔΦ − 4\frac{Φ}{|x|} = −4Φ,

and consequently

inf \{ ∫_{ℝ^{3}} |∇φ|^{2} − 4 ∫_{ℝ^{3}} \frac{φ^{2}}{|x|}dx, ∫_{ℝ^{3}} |φ|^{2} = 1\} < 0

i.e.(1.9) is satisfied also.

Our main result is the following. We prove that the Schrödinger–Poisson system (1.3)-(1.4) has a ground state, minimizing the energy functional corresponding to (1.5), given by (see Lemma 2.2):

E(φ) := \frac{1}{4} ∫_{ℝ^{3}} |∇φ|^{2}dx+ \frac{1}{4} ∫_{ℝ^{3}} |∇V (φ)|^{2}dx+ \frac{1}{2} ∫_{ℝ^{3}} Ṽφ^{2}dx+ \frac{ω}{2} ∫_{ℝ^{3}} φ^{2}dx (1.10)

Theorem 1.3 Under the assumptions (1.7), (1.8), and (1.9) there exists ω_{∗} > 0 such that for all 0 < ω < ω_{∗} the equation (1.5) has a nonnegative solution u ≢ 0 which minimizes the functional E:

E(u) = min_{φ∈H^{1}(ℝ^{3})} E(φ).

The remainder of this paper is organized as follows: In section 2 we present some preliminary lemmas which will be useful in the sequel. In section 3, we conclude by proving our main result.

2 Preliminary results

In this section we present a few preliminary lemmas which shall be required in several proofs. Recall (cf. [7, Theorem I.1] or [10, p.151]) that D^{1,2}(ℝ^{3}) is the completion of C_{0}^{∞}(ℝ^{3}) for the norm

∥φ∥_{D^{1,2}} = ( ∫_{ℝ^{3}} |∇φ|^{2} dx)^{1/2}.

By a Sobolev inequality, D^{1,2}(ℝ^{3}) is continuously embedded in L^{6}(ℝ^{3}), an equivalent characterization is

D^{1,2}(ℝ^{3}) := {φ ∈ L^{6}(ℝ^{3}); |∇φ| ∈ L^{2}(ℝ^{3})} .

For the solvability of the Poisson equation (1.3) we state the following lemma.

Lemma 2.1 For all f ∈ L^{6/5}(ℝ^{3}), the equation

−ΔW = f in ℝ^{3}  (2.1)

has a unique solution W ∈ D^{1,2}(ℝ^{3}) given by

W (f)(x) = \frac{1}{4π} ∫_{ℝ^{3}} \frac{f(y)}{|x − y|} dy .  (2.2)

Proof. The existence and the uniqueness of the solution of (2.1) follow from corollary 3.1.4 of reference [5], by minimizing on D^{1,2}(ℝ^{3}) the functional

J(v) = \frac{1}{2} ∫_{ℝ^{3}} |∇v|^{2}dx − ∫_{ℝ^{3}} fvdx.

For this, using Hölder’s and Sobolev’s inequalities we check easily that J is coercive (that is J(v_{n}) → +∞ as ∥v_{n}∥_{D^{1,2}} → ∞), strictly convex, lower semicontinuous and C^{1} on D^{1,2}(ℝ^{3}). Hence J attains its minimum at W ∈ D^{1,2}(ℝ^{3}) which is the unique solution of (2.1).

By uniqueness, W is the Newtonian potential of f and has (cf. [1, p.235]) an explicit formula given by (2.2). Furthermore, multiplying (2.1) by W and integrating we obtain

∥∇W ∥^{2} = ∫_{ℝ^{3}} f(x)W (x)dx.

After using Hölder and Sobolev inequalities we get

∥∇W ∥ ≤ S_{∗}^{1/2}∥f∥_{L^{6/5}}  (2.3)

where S_{∗} is the best Sobolev constant in

∥v∥_{L^{6}(ℝ^{3})}^{2} ≤ S_{∗}∥∇v∥_{L^{2}(ℝ^{3})}^{2}.  (2.4)

Hence the linear mapping f ↦ W is continuous from L^{6/5}(ℝ^{3}) into D^{1,2}(ℝ^{3}).

□

Now in order to find a solution of equation (1.5), we are going to show that the operator

v ↦ −\frac{1}{2}Δv + (W (|v|^{2} − n^{∗}) + Ṽ)v + ωv

is the derivative of a functional I : H^{1}(ℝ^{3}) → ℝ and hence equation (1.5) has a variational structure. To this end, we have the following lemma (see also [3])

Lemma 2.2 Let n^{∗} ∈ L^{6/5}(ℝ^{3}). For φ ∈ H^{1}(ℝ^{3}) we denote by V (φ) := W (|φ|^{2} − n^{∗}) the unique solution of (2.1) when f := |φ|^{2} − n^{∗} . Define

I(φ) := \frac{1}{4} ∫_{ℝ^{3}} |∇V (φ)|^{2}dx.

Then I is C^{1} on H^{1}(ℝ^{3}) and its derivative is given by

⟨I′(φ), ψ⟩ = ∫_{ℝ^{3}} V (φ)φψdx ∀ψ ∈ H^{1}(ℝ^{3}).  (2.5)

Proof. Note that if φ ∈ H^{1}(ℝ^{3}) then, by interpolation, |φ|^{2} ∈ L^{6/5}(ℝ^{3}). So taking f = |φ|^{2}−n^{∗} and multiplying the equation (2.1) by V (φ) := W (|φ|^{2}−n^{∗}) we deduce that ∥∇V (φ)∥^{2} = ∫ f(x)V (φ)(x)dx, and hence in view of (2.2) we get

I(φ) = \frac{1}{16π} ∫ ∫ \frac{(|φ|^{2} − n^{∗})(x)(|φ|^{2} − n^{∗})(y)}{|x − y|} dx dy.  (2.6)

Using this expression, we show easily that (2.5) holds for the Gâteaux differential of I i.e. for all φ, ψ ∈ H^{1}(ℝ^{3})

lim_{t→0^{+}} \frac{I(φ + tψ) − I(φ)}{t} = ∫_{ℝ^{3}} V (φ)φψ dx,

and that the mapping φ ↦ φV (φ) is continuous on H^{1}(ℝ^{3}). Thus I is Frechet differentiable and C^{1} on H^{1}(ℝ^{3}) and its derivative satisfies (2.5). □

At certain steps of our proof of Theorem 1.3, we need some estimates for which we will use the next inequalities.

Lemma 2.3 (i) If θ ∈ L^{r}(ℝ^{3}) for some r ≥ 3/2 then ∀δ > 0, ∃C_{δ} > 0 such that

∫_{ℝ^{3}} θ(x)|φ(x)|^{2}dx ≤ δ∥∇φ∥^{2} + C_{δ}∥φ∥^{2} ∀φ ∈ H^{1}(ℝ^{3})  (2.7)

(ii) For all φ ∈ D^{1,2}(ℝ^{3}) and y ∈ ℝ^{3} one has

∫_{ℝ^{3}} \frac{|φ(x)|^{2}}{|x − y|^{2}} dx ≤ 4∥∇φ∥^{2}  (2.8)

(iii) For any δ > 0 and all y ∈ ℝ^{3}

∫_{ℝ^{3}} \frac{|φ(x)|^{2}}{|x − y|} dx ≤ δ∥∇φ∥^{2} + \frac{4}{δ}∥φ∥^{2} ∀φ ∈ H^{1}(ℝ^{3})  (2.9)

Proof. In order to prove (i) we show first that (2.7) holds for any θ ∈ L^{∞}+L^{3/2} and conclude since L^{r}(ℝ^{3}) ⊂ L^{∞}(ℝ^{3})+L^{3/2}(ℝ^{3}) for all r ≥ 3/2. Let θ = θ_{1} +θ_{2} with θ_{1} ∈ L^{∞} and θ_{2} ∈ L^{3/2}. Then for each λ > 0 we have

∫_{ℝ^{3}} θ(x)|φ(x)|^{2}dx ≤∥θ_{1}∥_{L^{∞}}∥φ∥^{2} + λ ∫_{[|θ_{2}|≤λ]} |φ|^{2}dx + ∫_{[|θ_{2}|>λ]} |θ_{2}||φ|^{2}dx
≤ (∥θ_{1}∥_{L^{∞}} + λ) ∥φ∥^{2} + ∥θ_{2}∥_{L^{3/2}([|θ_{2}|>λ])}∥φ∥_{L^{6}}^{2}
≤ (∥θ_{1}∥_{L^{∞}} + λ) ∥φ∥^{2} + S_{∗}∥θ_{2}^{λ}∥_{L^{3/2}}∥∇φ∥^{2}

where S_{∗} is the best Sobolev constant in (2.4) and θ_{2}^{λ} denotes θ_{2}^{λ} := θ_{2}𝕀_{[|θ_{2}|>λ]}. It is clear that |θ_{2}^{λ}| ≤ |θ_{2}| for all λ > 0 and that θ_{2}^{λ} → 0 pointwise a.e. when λ → +∞. Since θ_{2} ∈ L^{3/2} then by Lebesgue convergence theorem we infer that ∥θ_{2}^{λ}∥_{L^{3/2}} converges to zero. Hence for any δ > 0 there exists K_{δ} > 0 such that if λ ≥ K_{δ} one has S_{∗}∥θ_{2}^{λ}∥_{L^{3/2}} ≤ δ. Choosing C_{δ} := ∥θ_{1}∥_{L^{∞}} + K_{δ} we deduce that (2.7) holds for all θ ∈ L^{∞}(ℝ^{3}) + L^{3/2}(ℝ^{3}).

Regarding (ii), (2.8) is the classical Hardy inequality (see [2]). Finally, to show (iii) for all δ > 0 and any y ∈ ℝ, we write

∫_{ℝ^{3}} \frac{|φ(x)|^{2}}{|x − y|} dx = ∫_{|x−y|< \frac{δ}{4}} \frac{|φ(x)|^{2}}{|x − y|^{2}} |x − y|dx + ∫_{|x−y|≥ \frac{δ}{4}} \frac{|φ(x)|^{2}}{|x − y|} dx

≤\frac{δ}{4} ∫_{ℝ^{3}} \frac{|φ(x)|^{2}}{|x − y|^{2}} dx + \frac{4}{δ} ∫_{ℝ^{3}} |φ(x)|^{2}dx

and (2.9) holds by using Hardy inequality (2.8). □

Remark 2.4 Note that Ṽ^{−} satisfies the inequality (2.7) i.e. ∀δ > 0∃C_{δ} > 0 such that

∫_{ℝ^{3}} Ṽ^{−}(x)|φ(x)|^{2}dx ≤ δ∥∇φ∥^{2} + C_{δ}∥φ∥^{2} ∀φ ∈ H^{1}(ℝ^{3}).  (2.10)

Indeed, by (1.7) Ṽ^{−} satisfies the decomposition (1.6). Then for a fixed λ > 0 we have

Ṽ^{−} = Ṽ_{1λ}^{−} + Ṽ_{2λ}^{−}

where for i = 1, 2, Ṽ_{iλ}^{−} ∈ L^{s}(ℝ^{3}) for some s ∈ [3/2, ∞] (s = q_{0} or s = q_{λ}). Hence by Lemma 2.3 each Ṽ_{iλ}^{−} satisfies the inequality (2.7) and consequently Ṽ^{−} also.

To finish this section we state the following convergence Lemma.

Lemma 2.5 Let ψ ∈ L^{r}(ℝ^{3}) for some r > 3/2. If v_{n} ⇀ 0 weakly in H^{1}(ℝ^{3}) then

∫_{ℝ^{3}} ψ(x)v_{n}^{2}(x)dx → 0 as n → +∞

Proof. Consider the subset of ℝ^{3}, A_{λ} := [|ψ| > λ] and a compact subset K of A_{λ} suitably chosen later. We write

∫_{ℝ^{3}} |ψ|(x)v_{n}^{2}(x)dx = ∫_{ℝ^{3}−A_{λ}} |ψ|v_{n}^{2}dx + ∫_{A_{λ}−K} |ψ|v_{n}^{2}dx + ∫_{K} |ψ|v_{n}^{2} dx
≤ λ∥v_{n}∥^{2} + ∥ψ∥_{L^{r}(A_{λ}−K)}∥v_{n}∥_{L^{2r′} (ℝ^{3})}^{2} + ∥ψ∥_{L^{r}(ℝ^{3})}∥v_{n}∥_{L^{2r′} (K)}^{2}
≤ λC_{0} + C_{1}∥ψ∥_{L^{r}(A_{λ}−K)} + ∥ψ∥_{L^{r}(K)}∥v_{n}∥_{L^{2r′} (K)}^{2}

where \frac{1}{r′} + \frac{1}{r} = 1. In the last inequality we used that (v_{n})_{n} is bounded in H^{1}(ℝ^{3}) (note that 2 < 2r′ < 6). For a given arbitrary δ > 0, we fix first λ such

that λC_{0} ≤ \frac{δ}{3}. Next we choose a compact subset K ⊂ A_{λ} such that

C_{1}∥ψ∥_{L^{r}(A_{λ}−K)} ≤ \frac{δ}{3}

and finally since v_{n} ⇀ 0 in H^{1}(ℝ^{3}) and 2 < 2r′ < 6 then up a subsequence ∥v_{n}∥_{L^{2r′} (K)}^{2} converges to 0 and therefore there exists N_{δ} ∈ ℕ such that for all n ≥ N_{δ} we get

∥ψ∥_{L^{r}(K)}∥v_{n}∥_{L^{2r′} (K)}^{2} ≤ \frac{δ}{3}

which completes the proof. □

3 Proof of Theorem 1.3

Now we are in position to prove our main result. To this end, we shall minimize the energy functional

E(φ) := \frac{1}{4} ∫ |∇φ|^{2}dx + I(φ) + \frac{1}{2} ∫ Ṽφ^{2}dx + \frac{ω}{2} ∫ φ^{2}dx

whose critical points correspond, on account of Lemma 2.2, to solutions of (1.5). Using (2.6), we may decompose E(φ) as

E(φ) = E_{1}(φ) − E_{2}(φ) + E_{3}(φ) + E(0)  (3.1)

where

E_{1}(φ) :=\frac{1}{4} ∫ |∇φ|^{2} dx + \frac{1}{2} ∫ Ṽ^{+}φ^{2}dx + \frac{ω}{2} ∫ φ^{2} dx
E_{2}(φ) :=\frac{1}{2} ∫ Ṽ^{−}φ^{2} dx + \frac{1}{8π} ∫ ∫ \frac{n^{∗}(y)}{|x − y|}φ^{2}(x) dx dy
E_{3}(φ) :=\frac{1}{16π} ∫ ∫ \frac{φ^{2}(x)φ^{2}(y)}{|x − y|} dx dy
E(0) :=\frac{1}{16π} ∫ ∫ \frac{n^{∗}(x)n^{∗}(y)}{|x − y|} dx dy.

The proof of Theorem 1.3 is divided into the four following Lemmas:

Lemma 3.1 Let ω > 0 and c ∈ ℝ. If the set [E ≤ c] is bounded in L^{2}(ℝ^{3}) then it is also bounded in H^{1}(ℝ^{3}).

Proof. By the expression (3.1), E(φ) ≤ c implies in particular

\frac{1}{4}∥∇φ∥^{2} − E_{2}(φ) ≤ c_{0}  (3.2)

where c_{0} := c − E(0) and since the other terms are nonnegative. To estimate E_{2}(φ) we use (2.9) which gives for any δ > 0,

∫ ∫_{ℝ^{3}×ℝ^{3}} \frac{n^{∗}(y)}{|x − y|}φ^{2}(x)dxdy ≤ (δ∥∇φ∥^{2} + \frac{4}{δ}∥φ∥^{2}) ∥n^{∗}∥_{L^{1}}.

Using this inequality, Remark 2.4 and choosing δ such that δ(\frac{1}{2} + \frac{∥n^{∗}∥_{L^{1}}}{8π} ) < \frac{1}{8} we obtain

E_{2}(φ) ≤ \frac{1}{8}∥∇φ∥^{2} + K_{0}∥φ∥^{2}  (3.3)

where K_{0} is a positive constant. In Consequence (3.2) gives

\frac{1}{8}∥∇φ∥^{2} ≤ K_{0}∥φ∥^{2} + c_{0}.

Lemma 3.2 For all ω > 0 and c ∈ ℝ the set [E ≤ c] is bounded in L^{2}(ℝ^{3}).

Proof. Assume by contradiction that there exists a sequence (u_{j})_{j} ⊂ H^{1}(ℝ^{3}) such that E(u_{j}) ≤ c and ∥u_{j}∥ → +∞. Let v_{j} := u_{j}/∥u_{j}∥ then ∥v_{j}∥ = 1 and from E(u_{j}) ≤ c we get

\frac{1}{4} ∫ |∇v_{j}|^{2}dx − E_{2}(v_{j}) + E_{3}(v_{j})∥u_{j}∥^{2} + \frac{ω}{2} ≤ \frac{c_{0}}{∥u_{j}∥^{2}} .  (3.4)

By using the estimate (3.3) for φ := v_{j} we obtain

\frac{1}{8}∥∇v_{j}∥^{2} + E_{3}(v_{j})∥u_{j}∥^{2} + \frac{ω}{2} ≤ \frac{c_{0}}{∥u_{j}∥^{2}} + K_{0}.  (3.5)

Since ω and E_{3}(v_{j}) are nonnegative, this inequality implies that (v_{j})_{j} is bounded in H^{1}(ℝ^{3}) and that E_{3}(v_{j})∥u_{j}∥^{2} is also bounded; i.e.

( ∫ ∫_{ℝ^{3}×ℝ^{3}} \frac{v_{j}^{2}(x)v_{j}^{2}(y)}{|x − y|}dxdy)∥u_{j}∥^{2} ≤ c_{1}.

Let then v ∈ H^{1}(ℝ^{3}) be such that for a subsequence of v_{j}, noted again v_{j}, we have v_{j} ⇀ v weakly in H^{1}(ℝ^{3}), v_{j} → v pointwise almost everywhere and v_{j}^{2} converging to v^{2} strongly in L_{loc}^{p}(ℝ^{3}) for any 1 ≤ p < 3. By Fatou’s Lemma we deduce that

∫ ∫_{ℝ^{3}×ℝ^{3}} \frac{v^{2}(x)v^{2}(y)}{|x − y|}dxdy ≤ lim inf_{j→+∞} ∫ ∫_{ℝ^{3}×ℝ^{3}} \frac{v_{j}^{2}(x)v_{j}^{2}(y)}{|x − y|} dx dy
≤ lim inf_{j→+∞} \frac{c_{1}}{∥u_{j}∥^{2}} = 0

and therefore v ≡ 0. On the other hand, it follows from (3.4) that

\frac{ω}{2} − E_{2}(v_{j}) ≤ \frac{c_{0}}{∥u_{j}∥^{2}} .  (3.6)

Set

h(x) := Ṽ^{−}(x) + V^{∗}(x)  (3.7)

where V^{∗}(x) := \frac{1}{4π} ∫ \frac{n^{∗}(y)}{|x−y|}dy is the Newtonian potential of n^{∗} given by Lemma 2.1 . Then (3.6) is equivalent to

ω − ∫_{ℝ^{3}} h(x)v_{j}^{2}(x)dx ≤ \frac{2c_{0}}{∥u_{j}∥^{2}} .  (3.8)

Using successively the hypothesis (1.7) and Lemma 2.5 we may show that

∫_{ℝ^{3}} h(x)v_{j}^{2}(x)dx → 0 as j → +∞.  (3.9)

Passing to the limit in (3.8) we infer that ω ≤ 0 which is a contradiction. In conclusion, any (u_{j})_{j} ⊂ H^{1}(ℝ^{3}) such that E(u_{j}) ≤ c is bounded in L^{2}(ℝ^{3}). □

Lemma 3.3 For any ω > 0 the functional E is weakly lower semi-continuous on H^{1}(ℝ^{3}) and attains its minimum on H^{1}(ℝ^{3}) at u ≥ 0.

Proof. First, to show that the functional E is weakly lower semi-continuous, remark that in the expression (3.1) the term E_{1} and E_{3} are continuous and convex (therefore weakly lower semi-continuous). Then we just have to prove that u ↦ ∫_{ℝ^{3}} h(x)u^{2}(x)dx is weakly sequentially continuous on H^{1}(ℝ^{3}) where h is defined by (3.7). Consider u_{j} ⇀ u weakly in H^{1}(ℝ^{3}) and write

∫ h(x)u_{j}^{2}(x)dx = ∫ h(x)(u_{j} − u)^{2}dx + 2 ∫ h(x)u(u_{j} − u)dx + ∫ h(x)u^{2}dx.

Taking (u_{j} − u) instead of v_{j} in (3.9) we infer that

∫_{ℝ^{3}} h(x)(u_{j} − u)^{2}dx → 0 as j → ∞.

Moreover, similarly to the proof of (3.9) we show that

∫_{ℝ^{3}} h(x)u(u_{j} − u)dx → 0 as j → ∞,

and consequently

∫_{ℝ^{3}} h(x)u_{j}^{2}(x)dx → ∫_{ℝ^{3}} h(x)u^{2}(x)dx as j → ∞.

This means that u ↦ ∫_{ℝ^{3}} h(x)u^{2}(x)dx is weakly sequentially continuous on H^{1}(ℝ^{3}) and therefore E is weakly lower semi-continuous on H^{1}(ℝ^{3}).

Next, if we denote μ := inf {E(φ); φ ∈ H^{1}(ℝ^{3})} and (u_{n})_{n} ⊂ H^{1}(ℝ^{3}) a minimizing sequence then by Lemmas 3.1 and 3.2, (u_{n})_{n} is bounded in H^{1}(ℝ^{3}) and therefore there exists u ∈ H^{1}(ℝ^{3}) such that u_{n} ⇀ u weakly in H^{1}(ℝ^{3}). The functional E being weakly lower semi-continuous on H^{1}(ℝ^{3}) we have

E(u) ≤ lim inf_{n→+∞} E(u_{n}) = μ

and consequently E(u) = μ. Since E is C^{1} on H^{1}(ℝ^{3}) then E′(u) = 0 and in view of Lemma 2.2, u is a solution of the equation (1.5).

Let us remark finally that by a simple inspection we have E(|u|) ≤ E(u) and therefore we may assume that u ≥ 0 . □

Lemma 3.4 There exists ω_{∗} > 0 such that if 0 < ω < ω_{∗} then E(u) < E(0) and thus u ≢ 0.

Proof. Assuming (1.9), there exist μ_{1} < 0 and φ_{1} ∈ H^{1}(ℝ^{3}) such that ∫ |φ_{1}|^{2} = 1 and

∫_{ℝ^{3}} |∇φ_{1}|^{2}dx + ∫_{ℝ^{3}} ϱ(x)φ_{1}^{2}(x)dx < μ_{1}.

From (3.1) we observe that

∫_{ℝ^{3}} |∇φ|^{2}dx + ∫_{ℝ^{3}} ϱ(x)φ^{2}(x)dx = 4E_{1}(φ) − 4E_{2}(φ) − 2ω ∫_{ℝ^{3}} φ^{2}(x)dx.

Then the last inequality gives

E_{1}(φ) − E_{2}(φ) − \frac{ω}{2} < \frac{μ_{1}}{4} .

Now, for t > 0 and using again (3.1) we compute easily

E(tφ_{1}) − E(0) = t^{2}E_{1}(φ_{1}) − t^{2}E_{2}(φ_{1}) + t^{4}E_{3}(φ_{1})
<\frac{t^{2}}{4} [(μ_{1} + 2ω) + 4t^{2}E_{3}(φ_{1})] .

Hence, if (μ_{1} + 2ω) < 0 there exists t_{∗} > 0 small enough such that for all 0 < t ≤ t_{∗},

(μ_{1} + 2ω) + 4t^{2}E_{3}(φ_{1}) < 0 .

In other words, setting ω_{∗} := −μ_{1}/2 then if 0 < ω < ω_{∗} we have E(tφ_{1}) < E(0) for 0 < t ≤ t_{∗}. Since E(u) := inf{E(φ); φ ∈ H^{1}(ℝ^{3})}, this implies that E(u) < E(0) and consequently u ≢ 0. The proof of Theorem 1.3 is thus complete. □

Remark 3.5 If n^{∗} is nonnegative then we may replace the assumption (1.9) by the next one

inf {∫ |∇φ|^{2}dx + 2 ∫ Ṽ(x)φ^{2}dx; ∫ |φ|^{2} = 1} < 0

which does not depend on n^{∗} and implies obviously (1.9).

Acknowledgments. The author acknowledges the hospitality of Laboratoire de Mathématiques Appliquées de Versailles (France) where a part of this work was done. He is grateful to Otared Kavian for very valuable discussions and suggestions.

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Khalid Benmlih Department of Economic Sciences, University of Fez P.O. Box 42A, Fez, Morocco. E-mail: kbenmlih@hotmail.com