On the Schrödinger–Maxwell equations under the effect of a general nonlinear term

www.elsevier.com/locate/anihpc

On the Schrödinger–Maxwell equations under the effect of a general nonlinear term

A. Azzollini

, P. d’Avenia

, A. Pomponio

aDipartimento di Matematica ed Informatica, Università degli Studi della Basilicata, Via dell’Ateneo Lucano 10, I-85100 Potenza, Italy bDipartimento di Matematica, Politecnico di Bari, Via E. Orabona 4, I-70125 Bari, Italy

Received 14 May 2009; received in revised form 9 November 2009; accepted 9 November 2009 Available online 3 December 2009

Abstract

In this paper we prove the existence of a nontrivial solution to the nonlinear Schrödinger–Maxwell equations inR³, assuming on the nonlinearity the general hypotheses introduced by Berestycki and Lions.

©2009 Elsevier Masson SAS. All rights reserved.

Résumé

Dans cet article on démontre l’existence d’une solution non-banale et positive pour les équations non-linéaires de Schrödinger–

Maxwell dansR³en supposant que le terme non-linéaire satisfait les hypothèses introduites par Berestycki et Lions.

©2009 Elsevier Masson SAS. All rights reserved.

1. Introduction

In the recent years, the following electrostatic nonlinear Schrödinger–Maxwell equations, also known as nonlinear Schrödinger–Poisson system,

−u+qφu=g(u) inR³,

−φ=qu² inR³, (SM)

have been object of interest for many authors. Indeed a similar system arises in many mathematical physics contexts, such as in quantum electrodynamics, to describe the interaction between a charge particle interacting with the electro- magnetic field, and also in semiconductor theory, in nonlinear optics and in plasma physics. We refer to [4] for more details in the physics aspects.

The greatest part of the literature focuses on the study of the previous system for the very special nonlinearity g(u)= −u+ |u|^p−1u, and existence, nonexistence and multiplicity results are provided in many papers for this particular problem (see [1,2,10,12–14,19–21,24,28]). In [9,11,27], also the linear and the asymptotic linear cases have

✩ The authors are supported by M.I.U.R.-P.R.I.N. “Metodi variazionali e topologici nello studio di fenomeni non lineari”.

* Corresponding author.

E-mail addresses:antonio.azzollini@unibas.it (A. Azzollini), p.davenia@poliba.it (P. d’Avenia), a.pomponio@poliba.it (A. Pomponio).

0294-1449/$ – see front matter ©2009 Elsevier Masson SAS. All rights reserved.

doi:10.1016/j.anihpc.2009.11.012

been studied, whereas in [22] the problem has been dealt with in a bounded domain, with Neumann conditions on the boundary.

The aim of this paper is to study the Schrödinger–Maxwell system assuming the same very general hypotheses introduced by Berestycki & Lions, in their celebrated paper [7]. Actually, we assume that the following hold forg:

(g1) g∈C(R,R);

(g2) −∞<lim infs→0+g(s)/slim sup_s→0+g(s)/s= −m <0;

(g3) −∞lim sup_s→+∞g(s)/s⁵0;

(g4) there existsζ >0 such thatG(ζ ):=ζ

0 g(s) ds >0.

Using similar assumptions on the nonlinearity g, [3,18] and [23] studied, respectively, a nonlinear Schrödinger equation in presence of an external potential and a system of weakly coupled nonlinear Schrödinger equations. We mention also [5] where the Klein–Gordon and in Klein–Gordon–Maxwell equations are considered.

The main result of the paper is

Theorem 1.1.Ifgsatisfies(g1)–(g4), then there existsq0>0such that, for any0< q < q0, problem(SM)admits a nontrivial positive radial solution(u, φ)∈H¹(R³)×D^1,2(R³).

Some remarks on this result are in order:

• the assumptions are trivially satisfied by nonlinearities likeg(u)= −u+ |u|^p−1u, for anyp∈ ]1,5[;

• hypotheses ongarealmost necessaryin the sense specified in [7, Subsection 2.2];

• the fact that the result is obtained for smallqis not surprising for at least two reasons: first, because smallqmakes, in some sense, less strong the influence of the termφu, which constitutes, in the first equation, a perturbation with respect to the classical nonlinear Schrödinger equation treated in [7]; second, there is a nonexistence result for largeq andg(u)= −u+ |u|^p−1uwithp∈ ]1,2](see [24]).

From the technical point of view, dealing with (SM) under the effect of a general nonlinear term presents several difficulties. Indeed the lack of the following global Ambrosetti–Rabinowitz growth hypothesis ong:

there exists μ >2 such that 0< μG(s)g(s)s, for alls∈R,

brings on two obstacles to the standard Mountain Pass arguments both in checking the geometrical assumptions in the functional and in proving the boundedness of its Palais–Smale sequences. To overcome these difficulties, we will use a combined technique consisting in a truncation argument (see [17,21]) and a monotonicity trickà laJeanjean [15]

(see also Struwe [26]).

It is natural to ask about multiplicity of solutions of (SM). However, our approach does not seem to work in this direction.

The paper is organized as follows. In Section 2 we introduce the functional framework for solving our problem by a variational approach. In Section 3 we define a sequence of modified functionals on which we can easily apply the Mountain Pass Theorem. Then we study the convergence of the sequence of critical points obtained. Finally Appendix A is devoted to prove a Pohozaev type identity which we use, in Section 3, as a fundamental tool in our arguments.

Notation.

• For any 1s+∞, we denote by · s the usual norm of the Lebesgue spaceL^s(R³);

• H¹(R³)is the usual Sobolev space endowed with the norm u²:=

R³

|∇u|²+u²;

• D^1,2(R³)is completion ofC₀^∞(R³)(the compactly supported functions inC^∞(R³)) with respect to the norm u²_D1,2(R³):=

R³

|∇u|²;

• for brevity, we denoteα=12/5.

2. Functional setting

We first recall the following well-known facts (see, for instance [4,6,12,24]).

Lemma 2.1.For everyu∈H¹(R³), there exists a uniqueφ_u∈D^1,2(R³)solution of

−φ=qu², inR³. Moreover,

(i) φ_u²_D1,2(R³)=q

R³φ_uu²; (ii) φu0;

(iii) for anyθ >0:φu_θ(x)=θ²φu(x/θ ), whereuθ(x)=u(x/θ );

(iv) there existC, C>0independent ofu∈H¹(R³)such that φ_uD^1,2(R³)Cqu²_α,

and

R³

φ_uu²Cqu⁴_α; (1)

(v) ifuis a radial function thenφ_uis radial, too.

Following [7], defines0:=min{s∈ [ζ,+∞[ |g(s)=0}(s0= +∞ifg(s) =0 for anysζ). We setg˜:R→R the function such that

˜ g(s)=

⎧⎨

⎩

g(s) on[0, s0];

0 onR₊\ [0, s0];

(g(−s)−ms)⁺−g(−s) onR₋.

(2)

By the strong maximum principle and by (ii) of Lemma 2.1, ifuis a nontrivial solution of (SM) withg˜in the place ofg, then 0< u < s₀and so it is a positive solution of (SM). Therefore we can suppose thatgis defined as in (2), so that (g1), (g2), (g4) and then the following limit

s→±∞lim g(s)

s⁵ =0 (3)

hold.

We set g₁(s):=

(g(s)+ms)⁺, ifs0, 0, ifs <0, g₂(s):=g₁(s)−g(s), fors∈R. Since

slim→0

g₁(s) s =0,

s→±∞lim g₁(s)

s⁵ =0, (4)

and

g₂(s)ms, ∀s0, (5)

by some computations, we have that for anyε >0 there existsCε>0 such that

g₁(s)C_εs⁵+εg₂(s), ∀s0. (6)

If we set G_i(t ):=

t 0

g_i(s) ds, i=1,2, then, by (5) and (6), we have

G₂(s)m

2s², ∀s∈R (7)

and for anyε >0 there existsCε>0 such that G1(s)C_ε

6 s⁶+εG2(s), ∀s∈R. (8)

The solutions(u, φ)∈H¹(R³)×D^1,2(R³)of (SM) are the critical points of the action functionalE:H¹(R³)× D^1,2(R³)→R, defined as

Eq(u, φ):=1 2

R³

|∇u|²−1 4

R³

|∇φ|²+q 2

R³

φu²−

R³

G(u).

The action functionalEqexhibits a strong indefiniteness, namely it is unbounded both from below and from above on infinite dimensional subspaces. This indefiniteness can be removed using the reduction method described in [4,6], by which we are led to study a one variable functional that does not present such a strongly indefinite nature. Hence, it can be proved that(u, φ)∈H¹(R³)×D^1,2(R³)is a solution of (SM) (critical point of functionalEq) if and only if u∈H¹(R³)is a critical point of the functionalI_q:H¹(R³)→Rdefined as

I_q(u)=1 2

R³

|∇u|²+q 4

R³

φ_uu²−

R³

G(u),

andφ=φ_u.

We will look for critical points ofI_qonH_r¹(R³):= {u∈H¹(R³)|uis radial}, which is a natural constraint.

3. The perturbed functional

Kikuchi, in [21], considered (SM), whereg(u)= −u+ |u|^p−1u, with 1< p <5. To overcome the difficulty in finding bounded Palais–Smale sequences for the associated functionalIq, following [17], he introduced the cut-off functionχ∈C^∞(R+,R)satisfying

⎧⎪

⎨

⎪⎩

χ (s)=1, fors∈ [0,1], 0χ (s)1, fors∈ ]1,2[, χ (s)=0, fors∈ [2,+∞[, χ_∞2,

(9)

and studied the following modified functionalI_q^T :H¹(R³)→R I_q^T(u)=1

2

R³

|∇u|²+q 4k˜_T(u)

R³

φ_uu²−

R³

G(u),

where, for everyT >0, k˜T(u)=χ

u² T²

.

With this penalization, forT sufficiently large and forqsufficiently small, he is able to find a critical pointu¯such that ¯uT and so he concludes thatu¯is a critical point ofI_q.

Let us observe that ifg(u)=f (u)−uwithf satisfying the Ambrosetti–Rabinowitz growth condition, the argu- ments of Kikuchi still hold with slide modifications.

On the other hand, in presence of nonlinearities satisfying Berestycki–Lions assumptions, further difficulties arise about the geometry of our functional and compactness. First of all, as in [21], we introduce a similar truncated func- tionalI_q^T :H_r¹(R³)→R

I_q^T(u)=1 2

R³

|∇u|²+q 4k_T(u)

R³

φ_uu²−

R³

G(u),

where, now, kT(u)=χ

u^α_α T^α

.

TheC¹-functionalI_q^T satisfies the geometrical assumptions of the Mountain–Pass Theorem but, under our general as- sumptions on the nonlinearity, we are not able to obtain the boundedness of the Palais–Smale sequences. Therefore we use an indirect approach developed by Jeanjean. We apply the following slight modified version of [15, Theorem 1.1]

(see [16]).

Theorem 3.1.Let(X, · )be a Banach space andJ⊂R+an interval. Consider the family ofC¹functionals onX I_λ(u)=A(u)−λB(u), ∀λ∈J,

withBnonnegative and eitherA(u)→ +∞orB(u)→ +∞asu → ∞and such thatIλ(0)=0.

For anyλ∈J we set Γ_λ:=

γ∈C

[0,1], X γ (0)=0, Iλ

γ (1)

<0 . If for everyλ∈J the setΓλis nonempty and

c_λ:= inf

γ∈Γλ

tmax∈[0,1]I_λ γ (t )

>0, (10)

then for almost everyλ∈J there is a sequence(v_n)_n⊂Xsuch that (i) (v_n)_nis bounded;

(ii) I_λ(v_n)→c_λ;

(iii) (I_λ)(v_n)→0in the dualX⁻¹ofX.

In our case,X=H_r¹(R³), A(u):=1

2

R³

|∇u|²+q 4kT(u)

R³

φuu²+

R³

G2(u),

B(u):=

R³

G1(u),

so that the perturbed functional we study is I_q,λ^T (u)=1

2

R³

|∇u|²+q 4k_T(u)

R³

φ_uu²+

R³

G₂(u)−λ

R³

G₁(u).

Actually, this functional is the restriction to the radial functions of a C¹-functional defined on the whole space H¹(R³)and for everyu, v∈H¹(R³)

I_q^T (u), v

=

R³

(∇u| ∇v)+qkT(u)

R³

φuuv

+ qα 4T^αχ

u^αα

T^α

R³

φ_uu²

R³

|u|^α−2uv+

R³

g₂(u)v−λ

R³

g₁(u)v.

In order to apply Theorem 3.1, we have just to define a suitable intervalJ such thatΓ_λ = ∅, for anyλ∈J, and (10) holds.

Observe that, according to [7], as a consequence of (g4), there exists a functionz∈H_r¹(R³)such that

R³

G₁(z)−

R³

G₂(z)=

R³

G(z) >0. (11)

Then there exists 0<δ <¯ 1 such that

¯ δ

R³

G₁(z)−

R³

G₂(z) >0. (12)

We defineJ as the interval[¯δ,1]. Lemma 3.2.Γ_λ = ∅, for anyλ∈J. Proof. Letλ∈J. Setθ >¯ 0 andz¯=z(·/θ ).¯

Defineγ: [0,1] →H_r¹(R³)in the following way γ (t )=

0, ift=0,

¯

z(·/t ), if 0< t1.

It is easy to see thatγ is a continuous path from 0 toz. Moreover, we have that¯ I_q,λ^T

γ (1) θ¯

2

R³

|∇z|²+q 4θ¯⁵χ

θ¯³z^α_α T^α

R³

φzz²

+ ¯θ³

R³

G2(z)− ¯δ

R³

G1(z)

and then, ifθ¯is sufficiently large, by (12) and (9) we getI_q,λ^T (γ (1)) <0. 2 Lemma 3.3.There exists a constantc >˜ 0such thatc_λc >˜ 0for allλ∈J.

Proof. Observe that for anyu∈H_r¹(R³)andλ∈J, using (7) and (8) forε <1, we have I_q,λ^T (u)1

2

R³

|∇u|²+q 4kT(u)

R³

φuu²+

R³

G2(u)−

R³

G1(u)

1

2

R³

|∇u|²+(1−ε)m 2

R³

u²−C_ε 6

R³

|u|⁶

and then, by Sobolev embeddings, we conclude that there existsρ >0 such that, for anyλ∈Jandu∈H_r¹(R³)with u =0 anduρ,it resultsI_q,λ^T (u) >0. In particular, for anyu =ρ,we haveI_q,λ^T (u)c >˜ 0.Now fixλ∈J andγ ∈Γ_λ.Sinceγ (0)=0 and I_q,λ^T (γ (1)) <0, certainlyγ (1)> ρ. By continuity, we deduce that there exists tγ∈ ]0,1[such thatγ (tγ) =ρ.Therefore, for anyλ∈J,

cλ inf

γ∈Γ_λI_q,λ^T γ (tγ)

c >˜ 0. 2

We present a variant of the Strauss’ compactness result [25] (see also [7, Theorem A.1]). It will be a fundamental tool in our arguments:

Theorem 3.4.LetP andQ:R→Rbe two continuous functions satisfying

slim→∞

P (s) Q(s)=0,

(vn)n, vandwbe measurable functions fromR^NtoR, withzbounded, such that sup

n

R^N

Q vn(x)

wdx <+∞, P

vn(x)

→v(x) a.e. inR^N.

Then(P (v_n)−v)w_L¹_(B)→0, for any bounded Borel setB.

Moreover, if we have also

slim→0

P (s) Q(s)=0,

xlim→∞sup

n

v_n(x)=0,

then(P (vn)−v)wL¹(R^N)→0.

In analogy with the well-known compactness result in [8], we state the following result

Lemma 3.5.For anyλ∈J, each bounded Palais–Smale sequence for the functionalI_q,λ^T admits a convergent subse- quence.

Proof. Letλ∈J and(u_n)_nbe a bounded (PS) sequence forI_q,λ^T , namely I_q,λ^T (u_n)

n is bounded, I_q,λ^T

(u_n)→0 in H_r¹

R³

. (13)

Up to a subsequence, we can suppose that there existsu∈H_r¹(R³)such that u_n u weakly inH_r¹

R³

, (14)

un→u inL^p R³

, 2< p <6, (15)

u_n→u a.e. inR^N. (16)

If we apply Theorem 3.4 for P (s)=g_i(s), i=1,2, Q(s)= |s|⁵, (v_n)_n=(u_n)_n, v=g_i(u), i =1,2 and w∈ C₀^∞(R^N),by (3), (4) and (16) we deduce that

R³

g_i(u_n)w→

R³

g_i(u)w, i=1,2.

Moreover, by (15) and [24, Lemma 2.1], we have k_T(u_n)

R³

φ_unu_nw→k_T(u)

R³

φ_uuw,

χ u_n^αα

T^α

R³

φ_unu²_n

R³

|u_n|²⁵u_nw→χ u^αα

T^α

R³

φ_uu²

R³

|u|²⁵uw.

As a consequence, by (13) and (14) we deduce(I_q,λ^T )(u)=0 and hence

R³

|∇u|²+qkT(u)

R³

φuu²+ qα 4T^αχ

u^α_α T^α

u^αα

R³

φuu²+

R³

g2(u)u=λ

R³

g1(u)u. (17)

By weak lower semicontinuity we have:

R³

|∇u|²lim inf

n

R³

|∇un|². (18)

Again, by (15) we have kT(un)

R³

φu_nu²_n→kT(u)

R³

φuu², (19)

χ un^αα

T^α

u_n^α_α

R³

φ_unu²_n→χ u^αα

T^α

u^α_α

R³

φ_uu². (20)

If we apply Theorem 3.4 forP (s)=g1(s)s, Q(s)=s²+s⁶, (vn)n=(un)n, v=g1(u)u,andw=1,by (3), (4) and (16), we deduce that

R³

g₁(u_n)u_n→

R³

g₁(u)u. (21)

Moreover, by (16) and Fatou’s lemma

R³

g₂(u)ulim inf

n

R³

g₂(u_n)u_n. (22)

By (17), (19), (20), (21) and (22), and since(I_λ)(u_n), u_n →0, we have lim sup

n

R³

|∇u_n|²=lim sup

n

λ

R³

g₁(u_n)u_n−

R³

g₂(u_n)u_n

−qk_T(u_n)

R³

φ_unu²_n− qα 4T^αχ

u_n^α_α T^α

u_n^αα

R³

φ_unu²_n

λ

R³

g₁(u)u−

R³

g₂(u)u−qk_T(u)

R³

φ_uu²− qα 4T^αχ

u^αα

T^α

u^α_α

R³

φ_uu²

=

R³

|∇u|². (23)

By (18) and (23), we get limn

R³

|∇u_n|²=

R³

|∇u|², (24)

hence limn

R³

g₂(u_n)u_n=

R³

g₂(u)u. (25)

Sinceg₂(s)s=ms²+h(s), withha positive and continuous function, by Fatou’s lemma we have

R³

h(u)lim inf

n

R³

h(un),

R³

u²lim inf

n

R³

u²_n.

These last two inequalities and (25) imply that, up to a subsequence,

R³

u²=lim

n

R³

u²_n,

which, together with (24), shows thatun→ustrongly inH_r¹(R³). 2

Lemma 3.6.For almost everyλ∈J, there existsu^λ∈H_r¹(R³),u^λ =0, such that(I_q,λ^T )(u^λ)=0andI_q,λ^T (u^λ)=c_λ. Proof. By Theorem 3.1, for almost everyλ∈J, there exists a bounded sequence(u^λ_n)n⊂H_r¹(R³)such that

I_q,λ^T u^λ_n

→c_λ; (26)

I_q,λ^T u^λ_n

→0 in H_r¹

R³

. (27)

Up to a subsequence, by Lemma 3.5, we can suppose that there existsu^λ∈H_r¹(R³)such thatu^λ_n→u^λ inH_r¹(R³).

By Lemma 3.3, (26) and (27) we conclude. 2

Therefore there exist(λn)n⊂J and(un)n⊂H_r¹(R³)such that I_q,λ^T

n(u_n)=c_λn, I_q,λ^T

n

(u_n)=0. (28)

Lemma 3.7.Letu_nbe a critical point forI_q,λ^T

nat levelc_λn. Then, forT >0sufficiently large, there existsq₀=q₀(T ) such that for any0< q < q₀, up to a subsequence,u_nαT, for anyn1.

Proof. We will argue by contradiction.

First of all, since(I_q,λ^T

n)(u_n)=0,u_nsatisfies the following Pohozaev type identity 1

2

R³

|∇un|²+5q 4 kT(un)

R³

φu_nu²_n+ 3q T^αχ

u_n^α_α T^α

un^αα

R³

φu_nu²_n

=3λn

R³

G₁(u_n)−3

R³

G₂(u_n) (29)

(see Appendix A for the proof).

Moreover, combining (29) with the first of (28) and by (1), we get

R³

|∇un|²=3cλ_n+q 2kT(un)

R³

φu_nu²_n+ 3q T^αχ

u_n^α_α T^α

un^αα

R³

φu_nu²_n

3cλ_n+C1q²kT(un)un⁴α+C2χ u_n^α_α

T^α q²

T^αun⁴α^+α. (30)

We are going to estimate the right part of the previous inequality. By the min–max definition of the Mountain Pass level, we have

c_λnmax

θ I_q,λ^T

n

z(·/θ )

max

θ

θ 2

R³

|∇z|²+θ³

R³

G₂(z)− ¯δ

R³

G₁(z)

+max

θ

q 4θ⁵χ

θ³z^αα

T^α

R³

φ_zz²

=A1+A2(T )

wherezis the function such that (11) holds.

Now, ifθ³2T^α/z^ααthenA₂(T )=0,otherwise we compute A₂(T )q

4 2T^α

z^α_α ⁵₃

R³

φ_zz²=C₃q²T⁴.

We also have

C₁q²k_T(u_n)u_n⁴αC₄q²T⁴ C₂χ

un^αα

T^α q²

T^αu_n⁴_α^+αC₅q²T⁴. Then, from (30) we deduce that

R³

|∇un|²3A1+C6q²T⁴. (31)

On the other hand, since(I_q,λ^T

n)(un), (un) =0, by (6) we have that

R³

|∇u_n|²+qk_T(u_n)

R³

φ_unu²_n+ qα 4T^αχ

u_n^αα

T^α

u_n^α_α

R³

φ_unu²_n+

R³

g₂(u_n)u_n

=λ_n

R³

g₁(u_n)u_nC_ε

R³

|u_n|⁶+ε

R³

g₂(u_n)u_n. (32)

Now, by (5) and (32), we obtain m(1−ε)

R³

u²_n(1−ε)

R³

g2(un)un

C_ε

R³

|u_n|⁶− qα 4T^αχ

u_n^αα

T^α

u_n^αα

R³

φ_nu²_n

C

R³

|∇u_n|² 3

+ ¯Cq²T⁴

C

3A1+C₆q²T⁴3

+ ¯Cq²T⁴ (33)

where in the last inequality we have used (31).

We suppose by contradiction that there exists no subsequence of (un)n which is uniformly bounded byT in the α-norm. As a consequence, for a certainn¯it should result that

u_nα> T , ∀nn.¯ (34)

Without any loss of generality, we are supposing that (34) is true for anyu_n. Therefore, by (31) and (33), we conclude that

T²<u_n²_αCu_n²C₇+C₈q²T⁴+C₉q⁴T⁸+C₁₀q⁶T¹²

which is not true forT large andqsmall enough: indeed we can findT₀>0 such thatT₀²> C₇+1 andq₀=q₀(T₀) such thatC₈q²T₀⁴+C₉q⁴T₀⁸+C₁₀q⁶T₀¹²<1, for anyq < q₀, and we find a contradiction. 2

Proof of Theorem 1.1. LetT , q₀ be as in Lemma 3.7 and fix 0< q < q₀. Let u_n be a critical point for I_q,λ^T

n at levelc_λn. We prove that(u_n)_nis aH¹-bounded Palais–Smale sequence forI_q.

Since by Lemma 3.7

u_nαT , (35)

the boundedness in theH¹-norm trivially follows from arguments such as those in (31) and (33). Finally, by (35), certainly we have that

I_q,λ^T

n(un)=1 2

R³

|∇u|²+q 4

R³

φuu²+

R³

G2(u)−λn

R³

G1(u),

and then, sinceλn1, we can prove that(un)nis a (PS) sequence forIqby similar argument as in [3, Theorem 1.1].

Now we conclude arguing as in Lemma 3.5. 2 Appendix A. A Pohozaev type identity

In this section we show that ifu, φ∈H_loc² (R³)solve

⎧⎪

⎪⎨

⎪⎪

⎩

−u+qk_T(u)φu+q α T^αχ

u^αα

T^α

|u|^2/5u

R³

φu²=g(u) inR³,

−φ=qu² inR³,

(36)

then the following Pohozaev type identity 1

2

R³

|∇u|²+5q 4 k_T(u)

R³

φu²+ 3q T^αχ

u^αα

T^α

u^αα

R³

φu²=3

R³

G(u) (37)

holds.

Indeed, by [13, Lemma 3.1], for everyR >0, we have

BR

−u(x· ∇u)= −1 2

BR

|∇u|²− 1 R

∂BR

|x· ∇u|²+R 2

∂BR

|∇u|², (38)

BR

φu(x· ∇u)= −1 2

BR

u²(x· ∇φ)−3 2

BR

φu²+R 2

∂BR

φu², (39)

BR

g(u)(x· ∇u)= −3

BR

G(u)+R

∂BR

G(u), (40)

B_R

|u|^2/5u(x· ∇u)= −3 α

B_R

|u|^α+R α

∂B_R

|u|^α, (41)

whereB_Ris the ball ofR³centered in the origin and with radiusR.

Multiplying the first equation of (36) byx· ∇uand the second equation byx· ∇φand integrating onB_R, by (38), (39), (40) and (41) we get

−1 2

B_R

|∇u|²− 1 R

∂B_R

|x· ∇u|²+R 2

∂B_R

|∇u|²

−q 2k_T(u)

BR

u²(x· ∇φ)−3q 2 k_T(u)

BR

φu²+Rq 2 k_T(u)

∂BR

φu²

− 3q T^αχ

u^αα

T^α

R³

φu²

BR

|u|^α+Rq T^αχ

u^αα

T^α

R³

φu²

∂BR

|u|^α

= −3

B_R

G(u)+R

∂B_R

G(u) (42)

and q

B_R

u²(x· ∇φ)= −1 2

B_R

|∇φ|²− 1 R

∂B_R

|x· ∇φ|²+R 2

∂B_R

|∇φ|². (43)

Substituting (43) into (42) we obtain

−1 2

B_R

|∇u|²−3q 2 kT(u)

B_R

φu²+1 4kT(u)

B_R

|∇φ|²

− 3q T^αχ

u^αα

T^α

R³

φu²

B_R

|u|^α+3

B_R

G(u)

= 1 R

∂BR

|x· ∇u|²−R 2

∂BR

|∇u|²− 1 2Rk_T(u)

∂BR

|x· ∇φ|²

+R 4k_T(u)

∂B_R

|∇φ|²−Rq 2 k_T(u)

∂B_R

φu²

−Rq T^αχ

u^α_α T^α

R³

φu²

∂B_R

|u|^α+R

∂B_R

G(u).

As in [13], the right-hand side goes to zero asR→ +∞and so we get

−1 2

R³

|∇u|²−3q 2 kT(u)

R³

φu²+1 4kT(u)

R³

|∇φ|²

− 3q T^αχ

u^α_α T^α

R³

φu²

R³

|u|^α+3

R³

G(u)=0.

If(u, φ_u)∈H¹(R³)×D^1,2(R³)is a solution of (36), by standard regularity results,u, φ_u∈H_loc² (R³)and, by (i) of Lemma 2.1, we get (37).

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