Ground state for Choquard equation with doubly critical growth nonlinearity

Ground state for Choquard equation with doubly critical growth nonlinearity

Fuyi Li, Lei Long, Yongyan Huang and Zhanping Liang

School of Mathematical Sciences, Shanxi University, Taiyuan 030006, Shanxi, P.R. China

Received 9 November 2018, appeared 1 May 2019 Communicated by Gabriele Bonanno

Abstract. In this paper we consider nonlinear Choquard equation

−_∆u+V(x)u= (Iα∗F(u))f(u) inR^N,

where V ∈ C(R^N), Iα denotes the Riesz potential, f(t) = |t|^p−2t+|t|^q−2t for allt ∈ R, N > 5 and α ∈ (0,N−4). Under suitable conditions on V, we obtain that the Choquard equation with doubly critical growth nonlinearity, i.e., p = (N+α)/N,q= (N+α)/(N−2), has a nonnegative ground state solution by variational methods.

Keywords: Choquard equation, doubly critical nonlinearity, ground state solution.

2010 Mathematics Subject Classification: 35D30, 35J20.

1 Introduction and main results

In this paper we consider nonlinear Choquard equation

−_∆u+V(x)u= (I_α∗F(u))f(u) inR^N, (1.1) where N>^5,α∈(0,N−4),Iα is the Riesz potential given by

Iα(x) = ^Γ((N−α)/2)

2^απ^N/2Γ(α/2)|x|^{N−α, x}∈_R^N\ {0},

Γ denotes the Gamma function, F(t) = |t|^p/p+|t|^q/q,f(t) = |t|^p−2t+|t|^q−2t for all t ∈ _R, and the potential functionV∈ C(_R^N)and satisfies

(V) there existV₀,V_∞ >0 such thatV₀6^V(x)6^V∞for allx ∈_R^N, and lim_|x|→∞V(x) =V_∞. In the caseF(t) =|t|^p, f(t) =|t|^p−2t for allt∈_{R, and}V=1, the Choquard equation (1.1) reduces to the general Choquard equation

−_∆u+u= (Iα∗ |u|^p)|u|^p−2u inR^N. (1.2)

BCorresponding author. Email: lzp@sxu.edu.cn

When N =3,α= 2, and p = 2, the equation (1.2) has appeared in many interesting physical models and is known as the well-known Choquard–Pekar equation [6,15], the Schrödinger–

Newton equation [2,3,10,18], and the stationary Hartree equation. In this case, the existence of ground states of equation (1.2) was obtained in [6,8,9] by variational methods.

In view of the Hardy–Littlewood–Sobolev inequality, see Lemma2.1below, it can be shown that the energy functional corresponding to (1.2), for every α ∈ (0,N), is well defined on H¹(_R^N)and belongs toC¹if

N+α

N 6 ^p6 ^N+α N−₂^,

where (N+α)/N is called the lower critical exponent and (N+α)/(N−2) is called the upper critical exponent. V. Moroz and J. Van Schaftingen established the existence of ground state solutions to the Choquard equation (1.2) in [11] if p is in the subcritical range, namely p∈((N+α)/N,(N+α)/(N−2)), and some qualitative properties. By the Pohožaev identity [4,5,12], the Choquard equation (1.2) has no nontrivial ground state solution when p 6 (N+α)/N or p > (N+α)/(N−2). For the more content of the equation (1.2), we refer the interested reader to the guide [14].

WhenV is a positive constant,F∈ C¹and satisfies (F₁) there exists a positive constantCsuch that

|tF⁰(t)|6^C(|t|^(N+α)/N+|t|^{(N+α)/(N−2)}), t ∈_R, (F₂) lim_t→_∞F(t)/|t|^{(N+α)/(N−2)} =0 and lim_t→0F(t)/|t|^(N+α)/N =0, (F3) there exists a constantt0∈_R\ {0}such thatF(t0)6=0,

Moroz and Van Schaftingen [13] proved the existence of ground state to the equation (1.1).

J. Seok [17] acts against the subcriticality condition (F₂), and consider thatFis doubly critical, i.e.,

F(t) = ¹

p|t|^p+¹

q|t|^q, p= ^N+α

N , q= ^N+α N−2. The functionalR

R^N(_Iα∗F(_u))_F(_u)contains two termsR

R^N(_Iα∗ |u|^p)|u|^pandR

R^N(_Iα∗ |u|^q))|u|^q. For the related critical problems involving only a single critical exponent, we refer to [1,12,16].

However, few work concerns the case that F is doubly critical. J. Seok cleverly estimated the energy, overcome the lack of compactness, and proved that the equation (1.1) admits a nontrivial solution under appropriate assumptions onαandN in radial space H_r¹(_R^N). Two natural questions arise. Does the solution has the least energy among nontrivial solutions of equation (1.1) in H¹(_R^N)? Furthermore, does the equation (1.1) has ground state solution in H¹(_R^N) if V is not a constant? To the best of our knowledge, there are no results on these questions. The present paper is devoted to these aspects and answers these questions. Our main result is as follows.

Theorem 1.1. Let N > ^5,α ∈ (0,N−4), the potential V satisfy the condition (V), and f(t) =

|t|^p−2t+|t|^q−2t for all t∈_{R, where p}= (N+α)/N and q= (N+α)/(N−2). Then the equation (1.1)has a nonnegative ground state solution provided V(_x)<_{V∞for all x} ∈_R^N.

We say that a functionu∈ H¹(_R^N)is a solution to (1.1) if J⁰(u) =0, for the definition of J, see (2.2) below. The solutionuobtained in Theorem1.1is a ground state solution in the sense that it minimizes the corresponding energy functional J among all nontrivial solutions.

Since the appearance of the potentialV breaks down the invariance under translations in R^N, we cannot use the translation invariant argument directly. To overcome this challenge, we need use the comparison arguments between the minimax level of the energy functional corresponding to (1.1) and that to the limit equation

−_∆u+V_∞u= (I_α∗F(u))f(u) inR^N. (1.3) Thus, we first need to study the existence of ground state solution to the equation (1.3). The result is stated as follows.

Theorem 1.2. Let N > ^5,α ∈ (_0,N−4), f(t) = |t|^p−2t+|t|^q−2t for all t ∈ _R, where p = (N+α)/N and q = (N+α)/(N−2). Then the equation (1.3) has a nonnegative ground state solution.

The proof of Theorem1.2relies on two ingredients: the nontrivial nature of solution to the equation (1.3) up to translation under the strict inequality

c<min 1

2

1− ¹ p

(pV_∞^pS₁^p)^1/(p−1),1 2

1− ¹

q

(qS^q₂)^1/(q−1)

obtained by a concentration-compactness argument (Lemma 3.2) and the proof of the latter strict inequality (Lemma3.1).

The rest of this paper is organized as follows. We give some preliminaries in Section 2.

Theorems1.2and1.1are proved in Sections 3 and 4, respectively.

Throughout this paper we always use the following notations. The letters C_i,i = 1, 2, . . . andCare positive constants which may change from line to line. R+ = [_0,∞)_. B_R(y)_denotes the open ball centered at y with radius R in R^N. For each s ∈ [1,∞), L^s(_R^N) denotes the Lebesgue space with the norm|u|_s= R

R^N|u|^s1/s,u∈ L^s(_R^N).

2 Preliminaries

In this section, we give some preliminaries. When Vsatisfies the condition (V), the following lemmas are all set up.

LetH¹(_R^N)be the usual Sobolev space. According to the conditions of the functionV, we can define an equivalent norm on H¹(_R^N),

(u,v) =

Z

R^N(∇u· ∇v+Vuv), kuk= (u,u)^1/2, u,v∈ H¹(_R^N).

H¹(_R^N)is embedded continuously intoL^s(_R^N)for eachs ∈ [2, 2^∗]. Thus, for eachs∈ [2, 2^∗], there exists a positive constant C_s such that

|u|_s6^Cskuk, u∈ H¹(_R^N). (2.1) The energy functional J associated to the equation (1.1) is defined by

J(u) = ¹ 2

Z

R^N(|∇u|²+Vu²)− ¹ 2

Z

R^N(I_α∗F(u))F(u), u∈ H¹(_R^N). (2.2) By the Hardy–Littlewood–Sobolev inequality, see Lemma 2.1 below, we know that J is well defined on H¹(_R^N)and belongs toC¹, and its derivative is given by

hJ⁰(u),vi=

Z

R^N(∇u· ∇v+Vuv)−

Z

R^N(Iα∗F(u))f(u)v, u,v∈ H¹(_R^N).

Therefore, a weak solution of the equation (1.1) corresponds to a critical point of the energy functionalJ.

We consider the following constraint minimization problem c:=inf

N J, (2.3)

whereN denotes the Nehari manifold

N ={u∈ H¹(_R^N)\ {0}:I(u):=hJ⁰(u),ui=0}, I(u) =kuk²−

Z

R^N

I_α∗

1

p|u|^p+¹ q|u|^q

(|u|^p+|u|^q), u ∈H¹(_R^N), (2.4) andN isC¹.

To study the constraint minimization problem related with (1.1), we need to recall the following well-known Hardy–Littlewood–Sobolev inequality, see [7].

Lemma 2.1(Hardy–Littlewood–Sobolev inequality). Let r,s >1andµ∈ (0,N)with 1

r + ^µ N +¹

s =2.

Then there exists a sharp constsnt C(N,µ,r)>₀such that for all u∈ L^r(_R^N)and v∈ L^s(_R^N),

Z

R^N

Z

R^N

u(x)v(y)

|x−y|^{µ dxdy}

6^C(N,µ,r)|u|_r|v|_s. (2.5) The sharp constant satisfies that

C(N,µ,r)6 ^N

N−µ(|_S^N−1|/N)^{µ/N 1} rs

µ/N 1−1/r

µ/N

+

µ/N 1−1/s

µ/N! . Ifr =s=2N/(2N−µ), then

C(N,µ,r) =C(N,µ) =π^µ/2Γ(N/2−µ/2) Γ(N−µ/2)

Γ(N/2) Γ(N)

−1+µ/N

, and there is equality in (2.5) if and only ifv=Cuand

u(x) =A(γ²+|x−a|²)^{−(2N−µ)/2} for someA∈_{C, 0}6=γ∈ _Randa ∈_R^N.

Notice that, when µ= N−α, by the Hardy–Littlewood–Sobolev inequality, for each u ∈ H¹(_R^N), the integral

Z

R^N

Z

R^N

|u(x)|^β|u(y)|^θ

|x−y|^{N−α dxdy} is well defined if

β,θ ∈

N+α

N ,N+α N−2

. Let

Φ(u) =

Z

R^N(Iα∗ |u|^β)|u|^θ, u∈ H¹(_R^N),

where α ∈ (0,N), and β,θ ∈ [(N+α)/N,(N+α)/(N−2)]. By the Hardy–Littlewood–

Sobolev and the Hölder inequalities, a standard analysis shows that the following properties hold.

Lemma 2.2. If{u_n} ⊂ H¹(_R^N)is a sequence converging weakly to u in H¹(_R^N)as n → _{∞, then} we have

Φ(u)6^{lim inf}

n→_∞ Φ(u_n). (2.6)

h_Φ⁰(un),vi → h_Φ⁰(u),vi, v∈ H¹(_R^N). (2.7) Proof. Assume that {v_n}is an arbitrary subsequence of {u_n}. Since v_n → u in L_loc^s (_R^N) for s∈[1, 2^∗), there exists a subsequence{w_n}of {v_n}such thatw_n →ua.e. onR^N.

By Fatou’s lemma, we haveΦ(u)6^{lim inf}n→_∞Φ(wn). Thus, (2.6) holds.

Next, we will prove (2.7). Using the Hardy–Littlewood–Sobolev inequality and the sym- metry property of convolution, we deduce that, for β,θ ∈[(N+α)/N,(N+α)/(N−2)],

Z

R^N|(I_α∗ |w_n|^β)|w_n|^θ−2w_nv−(I_α∗ |u|^β)|u|^θ−2uv| 6

Z

R^N|(I_α∗ |w_n|^β)(|w_n|^θ−2w_n− |u|^θ−2u)v|+

Z

R^N|I_α∗(|w_n|^β− |u|^β)|u|^θ−2uv| 6^C|wn|^β_2βN/(N+

α)|(|wn|^θ−2wn− |u|^θ−2u)v|_2N/(N+α)+

Z

R^N|(Iα∗ |u|^θ−2uv)(|wn|^β− |u|^β)|. Set 2N/(N+α) = r. Since |wn|^θ−2wn− |u|^θ−2u

r is bounded in L^θ/(θ−1)(_R^N) and

|w_n|^θ−2w_n → |u|^θ−2ua.e. onR^N, it follows from|v|^r ∈ L^θ(_R^N)that|(|w_n|^θ−2w_n− |u|^θ−2u)v|_r

→0. Further, since{w_n}is bounded inL^βr(_R^N), we see that C|w_n|^β

2βN/(N+α)|(|w_n|^θ−2w_n− |u|^θ−2u)v|_2N/(N+α) →0. (2.8) Since Iα∗ |u|^θ−2uv∈ L^r/(r−1)(_R^N),{|wn|^β− |u|^β}is bounded inL^r(_R^N)and|wn|^β → |u|^β a.e.

on R^N, we see that

Z

R^N|(Iα∗ |u|^θ−2uv)(|w_n|^β− |u|^β)| →0. (2.9) It follows from (2.8) and (2.9) thath_Φ⁰(w_n),vi → h_Φ⁰(u),vi. Thus, (2.7) is true.

Lemma 2.3. For each u∈ H¹(_R^N)\ {0}, there exists a unique tu>0such that tuu∈ N. Moreover, J(t_uu) =max_t∈_R+ J(tu).

Proof. For each u ∈ H¹(_R^N)\ {₀}, the function g(t) _:= J(tu) takes the formC₁t²−C₂t^2p− C₃t^p+q−C₄t^2q for all t ∈ _R+. By Remark 2.4 below, we see that g has a unique positive critical pointt_u corresponding to its maximum, i.e., g⁰(t_u) = 0 and g(t_u) =max_R+g. Hence, I(t_uu) =t_ug⁰(t_u) =_{0 and}J(t_uu) =_maxt∈R+ J(tu)_.

Remark 2.4. Let a,b,cbe positive constants. By elementary calculation one obtains that the function

g(t) =t²−at^2p−bt^p+q−ct^2q, t ∈_R+,

has a unique positive critical point t₀ with g⁰(t) > _{0 for all}t ∈ (_0,t₀)_{, and} g⁰(t) < _{0 for all} t∈(t₀,∞). Thus, gtakes the maximum att =t₀.

Lemma 2.5. There exist positive constants δ andρ such that kuk > ^{δ and}hI⁰(u),ui 6 −ρ for all u∈ N.

Proof. Because of the definition ofN, by (2.4), the Hardy–Littlewood–Sobolev inequality and (2.1), we can derive that

kuk²6 ¹ p

Z

R^N(Iα∗(|u|^p+|u|^q))(|u|^p+|u|^q)

6^C1kuk^2p+C2kuk^p+q+C3kuk^2q, u∈ N. (2.10) Sincep,q>1, there exists a positive constant δsuch thatkuk>^{δfor allu}∈ N.

Furthermore, by (2.4) and (2.10), we have that

−hI⁰(u),ui= ²(p−1) p

Z

R^N(Iα∗ |u|^p)|u|^p+²(q−1) q

Z

R^N(Iα∗ |u|^q)|u|^q + (p+q−2)(p+q)

pq

Z

R^N(Iα∗ |u|^p)|u|^q

> ²(p−1) p

_Z

R^N(I_α∗ |u|^p)|u|^p+

Z

R^N(I_α∗ |u|^q)|u|^q+2 Z

R^N(I_α∗ |u|^p)|u|^q

>²(p−1)kuk²>²(p−1)δ². Setρ=2(p−1)δ². The proof is completed.

To obtain a (PS)csequence of the energy functional J, we show that the functional has the mountain pass geometry.

Lemma 2.6. The functional J satisfies the mountain pass geometry, that is,

(i) there exist r,η > 0 such that J(u) > ^{η for all u} ∈ ∂Br = {u ∈ H¹(_R^N) : kuk = r}, and J(u)>0for all u with0<kuk6^r;

(ii) there exists u₀∈ H¹(_R^N)such thatku₀k>r and J(u₀)<0.

Proof. (i) By the Hardy–Littlewood–Sobolev inequality and (2.1), we derive that J(u)> ¹

2kuk²−C₁kuk^2p−C₂kuk^p+q−C₃kuk^2q. Then (i) follows ifr >0 is small enough.

(ii) For any givenu∈ H¹(_R^N)\ {0}, the functiong(t):= J(tu)take the formC₁t²−C₂t^2p− C₃t^p+q−C₄t^2q for allt∈ _R+. Sinceg(0) = 0 and lim_t→_∞g(t) = −_∞, there exists t₀ > 0 large enough such that (ii) holds foru0= t0u.

We define

c₁= inf

γ∈_Γmax

t∈[0,1]J(γ(t)),

whereΓ = {γ∈ C([0, 1],H¹(_R^N)) :γ(0) =0,J(γ(1))< 0}. Then it follows from Lemma2.6 (i) thatc₁>0. Furthermore, we can show that the minimax valuec₁also can be characterized byc=c₁= c₂, wherecis defined in (2.3), and

c₂= inf

H¹(_R^N)\{0}max

t∈_R+

J(tu).

According to Lemma2.6and [19, Theorem 2.8, p.41], there is a (PS)csequence{un} ⊂H¹(_R^N), that is,

J⁰(u_n)→_0, J(u_n)→c.

Lemma 2.7. Let{u_n} ⊂ H¹(_R^N)be a (PS)_csequence of J. Then{u_n}is bounded in H¹(_R^N). Proof. Since{u_n} ⊂H¹(_R^N)is a (PS)_c sequence, we have that

c+o(1) +o(1)ku_nk= J(u_n)− ¹

2pI(u_n)> ¹ 2

1− ¹

p

ku_nk². Because of p>1, the above inequality induce that{u_n}is bounded.

Lemma 2.8. Let{un} ⊂ H¹(_R^N)be a (PS)c sequence of J and un * u in H¹(_R^N). If u 6= 0, then the equation(1.1)has a nonnegative ground state solution of the equation(1.1).

Proof. Sinceu_n * uin H¹(_R^N), it follows from (2.7) that J⁰(u) = 0. Becauseu 6= 0, we know that u∈ N is a nonzero critical point of J. Using (2.6), we obtain

c6 ^J(u)−¹ 2I(u)

= ^p−1 2p²

Z

R^N(Iα∗ |u|^p)|u|^p+ ^q−1 2q²

Z

R^N(Iα∗ |u|^q)|u|^q+ ^p+q−2 2pq

Z

R^N(Iα∗ |u|^p)|u|^q 6 ^{lim inf}

n→_∞

p−1 2p²

Z

R^N(I_α∗ |u_n|^p)|u_n|^p+ ^q−1 2q²

Z

R^N(I_α∗ |u_n|^q)|u_n|^q + ^p+q−2

2pq Z

R^N(Iα∗ |un|^p)|un|^q

= lim inf

n→_∞

J(u_n)− ¹ 2I(u_n)

=c, which implied that J(u) =c.

Consider w = |u|. An easy computation shows that w ∈ N _and J(w) = J(u) = c. It follows from the Lagrange multiplier theorem that J⁰(w) = λI⁰(w)for some λ ∈ _{R. Hence,} λhI⁰(w),wi = hJ⁰(w),wi = 0. By Lemma2.5, we know that hI⁰(w),wi 6 −ρ. Thus, λ = 0, which implies that w is a nonnegative solution of the equation (1.1). Since J(w) = c, it is a nonnegative ground state solution to the equation (1.1).

3 Proof of Theorem 1.2

Before giving the proof of Theorem1.1, we need give the proof of Theorem1.2. In this section, V =V_∞.

The following two inequalities are special cases of the Hardy–Littlewood–Sobolev inequal- ity. The first one is

S_{1 Z}

R^N(I_α∗ |u|^(N+α)/N)|u|^(N+α)/N

N/(N+α)

6

Z

R^Nu², u∈ H¹(_R^N). (3.1) The second one is

S₂ Z

R^N(I_α∗ |v|^{(N+α)/(N−2)})|v|^{(N+α)/(N−2)}

(N−2)/(N+α)

6

Z

R^N|∇v|²_, v∈ H¹(_R^N)_{. (3.2)} By Lemma 2.1 and [1, Lemma 2.2], we see that the best constants S₁ and S₂ are achieved if and only if u_λ(x) = C₁λ^N/2/(λ²+|x|²)^N/2 for all x ∈ _{R, and} v_λ(x) = C₂λ^(N−2)/2/ (λ²+|x|²)^(N−2)/2 for all x ∈ R, respectively. The next lemma comes from [17], for reader’s convenience, we give a detailed proof.

Lemma 3.1. There exists u₀∈ H¹(_R^N)\ {0}such that sup

t∈_R+

J(tu₀)<min 1

2

1− ¹ p

(pV_∞^pS^p₁)^1/(p−1),1 2

1− ¹

q

(qS^q₂)^1/(q−1)

.

Proof. Let us define two functions u_λ(x):=C₁ λ^N/2

(λ²+|x|²)^{N/2, vλ}(x):=C₂ λ^(N−2)/2

(λ²+|x|²)^{(N−2)/2, x}∈_R^N, λ>0, which are the extremal functions of inequalities (3.1) and (3.2), respectively. Since N > ^5, u_λ,v_λ ∈ H¹(_R^N). By computing, we have that for eachλ>0

u_λ(x) =λ^−N/2u₁(x/λ), v_λ(x) =λ^−(N−2)/2v₁(x/λ), x ∈_R^N,

|u_λ|₂=|u₁|₂, |v_λ|₂=λ|v₁|₂, (3.3)

|∇u_λ|₂=λ⁻¹|∇u₁|₂, |∇v_λ|₂ =|∇v₁|₂, (3.4) Z

R^N(I_α∗ |u_λ|^p)|u_λ|^p =

Z

R^N(I_α∗ |u₁|^p)|u₁|^p, (3.5) Z

R^N(I_α∗ |u_λ|^p)|u_λ|^q=λ^−q Z

R^N(I_α∗ |u₁|^p)|u₁|^q, (3.6) Z

R^N(I_α∗ |u_λ|^q)|u_λ|^q=λ^−2q Z

R^N(I_α∗ |u₁|^q)|u₁|^q, (3.7) Z

R^N(Iα∗ |v_λ|^p)|v_λ|^p =λ^2p Z

R^N(Iα∗ |v₁|^p)|v₁|^p, Z

R^N(Iα∗ |v_λ|^p)|v_λ|^q=λ^p Z

R^N(Iα∗ |v₁|^p)|v₁|^q, Z

R^N(Iα∗ |vλ|^q)|vλ|^q=

Z

R^N(Iα∗ |v₁|^q)|v₁|^q. The constantsC₁ andC₂ are chosen to satisfy

Z

R^Nu²₁=

Z

R^N(Iα∗ |u₁|^p)|u₁|^p, Z

R^N|∇v₁|²=

Z

R^N(Iα∗ |v₁|^q)|v₁|^q. (3.8) Lets_λ >0 andt_λ >0 satisfy

J(s_λu_λ) =max

s∈_R+

J(su_λ), J(t_λv_λ) =max

t∈_R+

J(tv_λ). (3.9)

Then there exist ¯s_λ >s_λ and ¯t_λ >t_λ such thatJ(s¯_λu_λ)<0 and J(t^¯_λv_λ)<0. Thus, by defining γ₁(t) =ts¯_λu_λ andγ₂(t) =t¯t_λv_λ for all t∈[_{0, 1}], we see that

c6^min

tmax∈[0,1]J(γ₁(t)), max

t∈[0,1]J(γ2(t))

=min{J(s_λu_λ),J(t_λv_λ)}.

It follows from (3.9), (3.3)–(3.7) that 0= ^d

dt[J(tu_λ)]

t=s_λ

=s_λ Z

R^N|∇u_λ|²+s_λ Z

R^NV_∞u²_λ−^s

2p−1 λ

p Z

R^N(Iα∗ |u_λ|^p)|u_λ|^p

− (p+q)s_λ^p+q−1 pq

Z

R^N(Iα∗ |u_λ|^p)|u_λ|^q− ^s

2q−1 λ

q Z

R^N(Iα∗ |u_λ|^q)|u_λ|^q (3.10)

= ^sλ λ²

Z

R^N|∇u₁|²+s_λ Z

R^NV_∞u²₁− ^s

2p−1 λ

p Z

R^N(Iα∗ |u₁|^p)|u₁|^p

− (p+q)s_λ^p+q−1 pqλ^q

Z

R^N(Iα∗ |u₁|^p)|u₁|^q− ^s

2q−1 λ

qλ^2q Z

R^N(Iα∗ |u₁|^q)|u₁|^q, which implies that

06 ¹ λ²

Z

R^N|∇u₁|²+

Z

R^NV_∞u²₁− ^s

2p−2 λ

p Z

R^N(Iα∗ |u₁|^p)|u₁|^p. (3.11) Let s_∞ = lim sup_λ→∞s_λ. Suppose thats_∞ = ∞. Then we get a contradiction by (3.11). Thus s_∞ <∞. Taking againλ→_∞in (3.10), we obtain

s_∞ Z

R^NV_∞u²₁= ^s

2p−1

∞

p Z

R^N(Iα∗ |u₁|^p)|u₁|^p,

which from (3.8) implies s_∞ = (pV_∞)^1/(2p−2). Furthermore, we can prove that lim_λ→∞s_λ = (pV_∞)^1/(2p−2). Hence,

J(s_λu_λ) = ^s

2 λ

2 Z

R^N|∇u_λ|²+ ^s

2 λ

2 Z

R^NV_∞u²_λ− ^s

2p λ

2p² Z

R^N(Iα∗ |u_λ|^p)|u_λ|^p

− ^s

p+q λ

pq Z

R^N(Iα∗ |u_λ|^p)|u_λ|^q− ^s

2q λ

2q² Z

R^N(Iα∗ |u_λ|^q)|u_λ|^q

= ^s

2 λ

2λ² Z

R^N|∇u₁|²+^s

2 λ

2 Z

R^NV_∞u²₁− ^s

2p λ

2p² Z

R^N(I_α∗ |u₁|^p)|u₁|^p

− ^s

p+q λ

pqλ^q Z

R^N(I_α∗ |u₁|^p)|u₁|^q− ^s

2q λ

2q²λ^2q Z

R^N(I_α∗ |u₁|^q)|u₁|^q 6 ¹

2 V_∞s²_λ− ^s

2p λ

p²

! Z

R^Nu²₁− ¹ λ^q

"

s_λ^p+q pq

Z

R^N(I_α∗ |u₁|^p)|u₁|^q− ^s

2 λ

2λ^2−q Z

R^N|∇u₁|²

# . Note that the function f(s) _:= V_∞p²s²−s^2p,s ∈ _R+, attains its maximum at s = s_∞. This shows that

1

2 V_∞s²_λ− ^s

2p λ

p²

! Z

R^Nu²₁6 ¹ 2

1− ¹

p

(pV_∞^pS₁^p)^1/(p−1). It follows from 4+α<N thatq<2 and

lim

λ→_∞

"

s_λ^p+q pq

Z

R^N(I_α∗ |u₁|^p)|u₁|^q− ^s

2 λ

2λ^2−q Z

R^N|∇u₁|²

#

= ^s

p+q

∞

pq Z

R^N(I_α∗ |u₁|^p)|u₁|^q>0.

Thus,

J(s_λu_λ)< ¹ 2

1− ¹

p

(pV_∞^pS₁^p)^1/(p−1) for sufficiently largeλ>0.

Similarly we have 0= ^d

dt[J(tv_λ)]

t=t_λ

=t_λ Z

R^N|∇v_λ|²+t_λ Z

R^NV_∞v²_λ− ^t

2p−1 λ

p Z

R^N(Iα∗ |v_λ|^p)|v_λ|^p

− (p+q)t_λ^p+q−1 pq

Z

R^N(Iα∗ |v_λ|^p)|v_λ|^q− ^t

2q−1 λ

q Z

R^N(Iα∗ |v_λ|^q)|v_λ|^q (3.12)

=t_λ Z

R^N|∇v₁|²+t_λλ² Z

R^NV_∞v²₁− ^t

2p−1 λ λ^2p

p Z

R^N(Iα∗ |v₁|^p)|v₁|^p

− (p+q)t_λ^p+q−1λ^p pq

Z

R^N(Iα∗ |v₁|^p)|v₁|^q− ^t

2q−1 λ

q Z

R^N(Iα∗ |v₁|^q)|v₁|^q, which implies that

06

Z

R^N|∇v₁|²+λ² Z

R^NV_∞v²₁−^t

2q−2 λ

q Z

R^N(I_α∗ |v₁|^q)|v₁|^q. (3.13) Lett₀ := lim sup_λ→0t_λ. Then we can get that t₀ < _∞ by (3.13). Taking againλ→ 0 in (3.12), we get

t₀ Z

R^N|∇v₁|² = ^t

2q−1 0

q Z

R^N(Iα∗ |v₁|^q)|v₁|^q,

which impliest₀ =q^1/(2q−2). Furthermore, we can prove that lim_λ→0t_λ =q^1/(2q−2). Thus, J(t_λv_λ) = ^t

2 λ

2 Z

R^N|∇v_λ|²+ ^t

2 λ

2 Z

R^NV_∞v²_λ− ^t

2p λ

2p² Z

R^N(I_α∗ |v_λ|^p)|v_λ|^p

−^t

p+q λ

pq Z

R^N(Iα∗ |v_λ|^p)|v_λ|^q− ^t

2q λ

2q² Z

R^N(Iα∗ |v_λ|^q)|v_λ|^q

= ^t

2 λ

2 Z

R^N|∇v₁|²+ ^t

2 λλ²

2 Z

R^NV_∞v²₁− ^t

2p λ λ^2p

2p² Z

R^N(Iα∗ |v₁|^p)|v₁|^p

−^t

p+q λ λ^p

pq Z

R^N(Iα∗ |v₁|^p)|v₁|^q− ^t

2q λ

2q² Z

R^N(Iα∗ |v₁|^q)|v₁|^q 6 ¹

2 t²_λ−^t

2q λ

q²

! Z

R^N|∇v₁|²−λ^p

"

t_λ^p+q pq

Z

R^N(I_α∗ |v₁|^p)|v₁|^q− ^t

2 λλ^2−p

2 Z

R^NV_∞v²₁

# . Note that the functiong(t):=q²t²−t^2q,t∈_R+, attains its maximum att= t0. Hence,

1

2 t²_λ− ^t

2q λ

q²

! Z

R^N|∇v₁|²6 ¹ 2

1−¹

q

(qS^q₂)^1/(q−1). It follows fromp<2 that

lim

λ→0

"

t^p_λ^+q pq

Z

R^N(I_α∗ |v₁|^p)|v₁|^q− ^t

2 λλ^2−p

2 Z

R^NV_∞v²₁

#

= ^t

p+q 0

pq Z

R^N(I_α∗ |v₁|^p)|v₁|^q>0.