Groundstates for Choquard type equations with weighted potentials and Hardy–Littlewood–Sobolev lower critical exponent

Shuai Zhou; Zhisu Liu; Jianjun Zhang

doi:10.1515/anona-2020-0186

Open Access Published by De Gruyter June 30, 2021

Groundstates for Choquard type equations with weighted potentials and Hardy–Littlewood–Sobolev lower critical exponent

Shuai Zhou , Zhisu Liu and Jianjun Zhang

From the journal Advances in Nonlinear Analysis

https://doi.org/10.1515/anona-2020-0186

Abstract

We are concerned with a class of Choquard type equations with weighted potentials and Hardy–Littlewood–Sobolev lower critical exponent

−Δu+V(x)u=Iα∗[Q(x)|u|N+αN]Q(x)|u|αN−1u,x∈RN.

By using variational approaches, we investigate the existence of groundstates relying on the asymptotic behaviour of weighted potentials at infinity. Moreover, non-existence of non-trivial solutions is also considered. In particular, we give a partial answer to some open questions raised in [D.~Cassani, J. Van Schaftingen and J. J. Zhang, Groundstates for Choquard type equations with Hardy-Littlewood-Sobolev lower critical exponent, Proceedings of the Royal Society of Edinburgh, Section A Mathematics, 150(2020), 1377–1400].

Keywords: Ground states; Choquard equation; Hardy–Littlewood–Sobolev inequality; critical growth

MSC 2010: 35B33; 35J61

1 Introduction and main results

In this paper, we are concerned with the following class of nonlocal equations

−Δu+V(x)u=Iα∗F(x,u)f(x,u),x∈RN, (1.1)

where N ≥ 3, V ∈ C(ℝ^N, ℝ) and I_α is the Riesz potential given for each x ∈ ℝ^N ∖ {0} by

Iα(x):=Aα|x|N−α,whereAα=Γ((N−α)/2)Γ(α/2)πN/22αand α∈(0,N).

Here Γ is the Euler gamma function and F is the primitive function of f ∈ C(ℝ^N × ℝ, ℝ) with respect to u and satisfies F(0) = 0. In the literature, problem (1.1) is known as Choquard’s type equation. Set F(x, u) = |u|^p/p and V ≡ a, problem (1.1) becomes

−Δu+au=(Iα∗|u|p)|u|p−2u,x∈RN. (1.2)

For N = 3, α = 2 and p = 2, (1.2) reduces to

−Δu+au=(I2∗u2)u,x∈R3. (1.3)

It seems that such equations appear first in the seminal work of S.I. Pekar ’54 [23], modeling the quantum Polaron and later were introduced by Choquard to study steady states of the one component plasma approximation in the context of Hartree-Fock theory [11]. Problem (1.1) has a variational structure, in the sense that H¹-solutions to (1.1) turn out to be critical points of the energy functional

E(u)=12∫RN(|∇u|2+V(x)u2)dx−∫RN[Iα∗F(x,u)]F(x,u)dx.

Due to the presence of convolution, problem (1.1) is nonlocal. In contrast with local problems, Choquard type equations carry some extra difficulty due to the nonlocal nature. By using a rearrangement approach, E. Lieb in [12] proved existence and uniqueness of positive solutions to (1.3). Subsequently, multiplicity results for (1.3) were obtained by P.L. Lions [14, 15] via the variational methods. Initiated by the papers of E. Lieb [12] and P.L. Lions [14, 15], Choquard equations have attracted a considerable attention in the past decades. We refer to [21] for a survey.

In [19], V. Moroz and J. Van Schaftingen established existence of ground state solutions to (1.2). Thanks to a Pohozăev identity, they show that (1.2) admits a nontrivial solution in H¹(ℝ^N) if and only if

N+αN<p<N+αN−2. (1.4)

The endpoints 2α,∗:=N+αN and 2α∗:=N+αN−2 are sometimes called lower and upper Hardy-Littlewood-Sobolev critical exponents respectively in the sense of the Hardy-Littlewood-Sobolev inequality (see Lemma 2.1 below). Later, V. Moroz and J. Van Schaftingen [22] considered the autonomous form of Choquard equation (1.1)

−Δu+u=Iα∗F(u)f(u),x∈RN (1.5)

and investigated existence, symmetry and regularity of groundstates to problem (1.5) under almost necessary conditions on the nonlinearity F in the spirit of Berestycki and Lions. For the upper critical case, F. Gao and M. Yang [9] establish existence and nonexistence of solutions to the following Brezis-Nirenberg type problem of Choquard equation in bounded domains Ω ⊂ ℝ^N(N ≥ 3)

−Δu=∫Ω|u(y)|2α∗|x−y|N−αdy|u|2α∗−2u+λu,u∈H01(Ω).

By using the penalization argument introduced by J. Byeon and L. Jeanjean [4], D. Cassani and J. Zhang [6] investigated singularly perturbed problems related to equation (1.5) involving upper critical exponent

−ε2Δu+V(x)u=ε−αIα∗F(u)f(u),x∈RN

and obtained existence of single peak solutions around local minimal points of the potential V. With the help of the concentration compactness principle in the Choquard-type setting, S. Liang, P. Pucci and B. Zhang [16] established multiplicity results for Choquard-Kirchhoff type equations with Hardy-Littlewood-Sobolev critical exponents

−a+b∫RN|∇u|2dxΔu=αk(x)|u|q−2u+β∫RN|u(y)|2α∗|x−y|N−αdy|u|2α∗−2u,x∈RN

Compared with the upper critical case, the lower critical case has been less considered. Combining variational arguments with the concentration-compactness principle [13], V. Moroz and J. Van Schaftingen [20] considered Choquard equations with a purely lower critical nonlinearity

−Δu+V(x)u=Iα∗|u|N+αN|u|αN−1u,x∈RN (1.6)

and established a sufficient condition on existence of groundstates to problem (1.6). Subsequently, D. Cassani, J. Van Schaftingen and J. Zhang [7] investigated existence and nonexistence of groundstates to problem (1.6) and give a partial answer to some open questions raised in [7]. By variational methods, J. Van Schaftingen and J. Xia [25] proved existence of ground state solutions to Choquard equations with lower critical exponent and a subcritical perturbation. In [24], J. Seok considered problem (1.5) with both upper and lower critical exponents and obtained existence of nontrivial solutions in the higher dimensional case. For the related results on the Choquard equations with upper critical growth in the fractional setting and for the planar Choquard equations, we refer to [1, 2, 5, 8, 17, 26] and references therein.

We point out that due to the presence of the lower critical exponent 2_α,∗, the problem has a lack of compactness. Similarly to Sobolev critical problems, a Brezis-Nirenberg argument can be adopted to recover compactness. Actually, by imposing some suitable conditions on N, α and nonlinearities, one can get a candidate minimax value below a threshold, under which the compactness condition holds. In [7], compared with the high dimensional case N ≥ 3, dimension N = 3 becomes more tough. Precisely, in [7] to recover compactness in the three dimensional case, one sufficient condition is established on α as follows:

32<α<3 (1.7)

Moreover, a natural question is whether such restriction is necessary or not for the existence of groundstates.

In this paper, we consider the following class of equations

−Δu+V(x)u=Iα∗[Q(x)|u|N+αN]Q(x)|u|αN−1u,x∈RN (1.8)

and show that condition (1.7) can be replaced by demanding in the presence of weights a suitable asymptotic behavior at infinity. In the following, we perform the variational method to study existence and nonexistence of ground states to (1.8).

The associated functional with the Choquard equation (1.8) is given for any function u : ℝ^N → ℝ by

I(u)=12∫RN(|∇u|2+V(x)|u|2)dx−N2(N+α)∫RNIα∗[Q(x)|u|N+αN]Q(x)|u|N+αNdx

We assume that V and Q satisfy

infx∈RNV(x)>0,lim|x|→∞V(x)=1;
There exists μ ∈ ℝ such that

lim|x|→∞(1−V(x))|x|2=μ;
infx∈RNQ(x)>0,lim|x|→∞Q(x)=1;
There exist α ≥ 0 and ν_α ∈ ℝ such that

lim|x|→∞(Q(x)−1)|x|β=νβ.

Our main results are the following:

Theorem 1.1

Assume (V₁), (V₂), (Q₁), (Q₂) hold, then (1.8) admits a positive ground state solution, provided one of the following conditions holds

β=0,μ>N2(N−2)4(N+1),infx∈RNQ(x)≥1;
0 < α < 2, ν_α > 0;
β>2,μ>N2(N−2)4(N+1).

Combining a Pohozăev identity(see Proposition 2.1 below) with Hardy’s inequality, we have the following non-existence result for problem (1.8).

Theorem 1.2

Assume V, Q ∈ C¹(ℝ^N) ∩ L^∞(ℝ^N) and

supx∈RN|x|2〈x,∇V(x)〉<(N−2)22,infx∈RN〈x,∇Q(x)〉≥0,

then (1.8) admits in H¹(ℝ^N) ∩ Wloc2,2 (ℝ^N) only trivial solution.

Remark 1.1

The restriction infx∈RN V(x) > 0 is used to guarantee c_∗(see below) is finite. It can be replaced by a weaker condition infσ(−Δ + V) > 0.

Remark 1.2

In the case α = 2, Theorem 1.1 is still valid if μ > μ_∗, where

μ∗=N2(N−2)4(N+1)−2Nνβ(N+α)c∞∫RN|U(x)|2|x|2dx−1∫RNIα∗|U|N+αN|x|−2|U|N+αNdx.

Here c_∞ and U are given in Section 2.

As a special case, for the external Schrödinger potential V_μ,ν : ℝ^N → ℝ

Vμ,ν(x)=1−μν2+|x|2,for μ∈R,ν>0and x∈RN

and the weighted potential Q_α : ℝ^N → ℝ

Qβ(x)=1+νβ1+|x|β,for νβ∈R,β≥0and x∈RN,

problem (1.8) reduces to the following

−Δu+Vμ,ν(x)u=Iα∗[Qβ(x)|u|N+αN]Qβ(x)|u|αN−1u,x∈RN. (1.9)

Denote by μ^ν the best constant of the embedding H¹(ℝ^N) ↪ L²(ℝ^N, (ν² + |x|²)⁻¹ d x), that is,

μν:=infu∈H1(RN)∖{0}∫RN|∇u|2+|u|2∫RN|u(x)|2ν2+|x|2dx.

It has been shown in [7] that μν>(N−2)24+ν2.

Corollary 1.1

Problem (1.9) admits a positive ground state solution, provided one of the following conditions holds

α = 0, N2(N−2)4(N+1) < μ < μ^ν, ν_α = 0;
0 < α < 2, μ < μ^ν, ν_α > 0;
α > 2, N2(N−2)4(N+1) < μ < μ^ν, ν_α > − 1

and has no non-trivial solutions if μ < (N−2)24 and ν_α ≤ 0.

In the case μ > μ^ν, the operator − Δ + V_μ,ν is not positively definite and the problem becomes more complicated. By the linking theorem, we have the following result.

Theorem 1.3

Assume that either N ≥ 4 or N = 3, ker (− Δ + V_μ,ν) = {0}. If μ > max{μ^ν, N2(N−2)4(N+1) }, then (1.8) admits a ground state solution (necessarily sign changing) provided when N = 3, one of the followings holds

α = 0, ν_α = 0, 32 < α < 3;
0 < α < min{ 96−α , 2}, ν_α > 0
α > 2, ν_α > − 1, 32 < α < 3;

and when N ≥ 4, one of the followings holds
α = 0, ν_α = 0;
0 < α < 2, ν_α > 0;
α > 2, ν_α > − 1.

Notations.

∥u∥_p := (∫_ℝ^N|u|^p d x)^1/p for u ∈ L^p(ℝ^N), p ∈ [1, ∞).
∥u∥=∥∇u∥22+∥u∥2212,u∈H1(RN).

2 Proof of Theorem 1.1-1.2

Before proving Theorem 1.1, let us introduce some preliminary results. First, the following Hardy–Littlewood–Sobolev inequality will be frequently used in the sequel.

Lemma 2.1

(Hardy–Littlewood–Sobolev inequality [10, Theorem 4.3]). Let s, r > 1 and 0 < α < N with 1/s + 1/r = 1 + α/N, f ∈ L^s(ℝ^N) and g ∈ L^r(ℝ^N), then there exists a positive constant C(s, N, α) (independent of f, g) such that

∫RN∫RNf(x)|x−y|α−Ng(y)dxdy≤C(s,N,α)∥f∥s∥g∥r.

In particular, if s = r = 2N/(N + α), the sharp constant is given by

Cα:=πN−α2Γ(α/2)Γ((N+α)/2)Γ(N/2)Γ(N)−α/N.

Due to the presence of the lower critical exponent N+αN , the compactness fails in general. In fact, the convolution term enjoys the invariance of dilation, that is, for any u ∈ L²(ℝ^N) and t > 0, one has

G∞(ut)=G∞(u),ut(⋅)=tN2u(t⋅),

where

G∞(u)=∫RNIα∗|u|N+αN|u|N+αNdx.

To recover the compactness, the following Brezis-Lieb type lemma plays a crucial role in the decomposition of the maximization sequence for c_∗ given below.

For any u ∈ H¹(ℝ^N), let

G(u)=∫RNIα∗[Q(x)|u|N+αN]Q(x)|u|N+αNdx,

T(u)=∫RN(|∇u|2+V(x)|u|2)dx.

Lemma 2.2

(Brezis-Lieb type Lemma). Assume that (V₁) and (Q₁) hold and let {un}n=1∞ be a bounded sequence in L²(ℝ^N) and for some u ∈ L²(ℝ^N), u_n → u strongly in Lloc2 (ℝ^N) as n → ∞, then, up to a subsequence, there holds

limn→∞[G(un)−G(u)−G∞(un−u)]=0.

Proof

Without loss of generality, we may assume that u_n → u almost everywhere on ℝ^N as n → ∞. Let v_n = Q u_n, v = Q u, it follows from [19, Lemma 2.4] that

limn→∞[G(un)−G(u)−G(un−u)]=0.

In the following, we show that

limn→∞[G(un−u)−G∞(un−u)]=0.

Let w_n = u_n − u,

G(un−u)=∫RNIα∗[Q(x)|wn|N+αN]Q(x)|wn|N+αNdx=∫RNIα∗[Q(x)|wn|N+αN](Q(x)−1)|wn|N+αNdx+∫RNIα∗[(Q(x)−1)|wn|N+αN]|wn|N+αNdx+G∞(wn):=J1,n+J2,n+G∞(wn).

By virtue of Lemma 2.1, we have

J1,n≤Cαmaxx∈RNQ(x)∫RN|wn|2dxN+α2N∫RN|Q(x)−1|2NN+α|wn|2dxN+α2N.

Since Q(x) → 1 as |x| → ∞ and w_n → 0 strongly in Lloc2 (ℝ^N) as n → ∞, one can get that

limn→∞∫RN|Q(x)−1|2NN+α|wn|2dx=0,

which implies that J_1,n → 0 as n → ∞. Similarly, we have J_2,n → 0 as n → ∞. The proof is complete. □

Set

c∞:=supG∞(u):∫RN|u|2dx=1,u∈L2(RN)

and

c∗:=supG(u):T(u)=1,u∈H1(RN).

Then by Lemma 2.1, 0 < c_∗, c_∞ < ∞ and it follows from [10, Theorem 4.3] that c_∞ can be achieved by the family of functions

U(x)=CλN2(λ2+|x|2)−N2,

for some fixed C > 0 and λ ∈ ℝ⁺ as parameters.

Lemma 2.3

(Compactness). If c_∗ > c_∞, then c_∗ can be achieved.

Proof

For any minimization sequence {un}n=1∞ ⊂ H¹(ℝ^N) of c_∗, i. e, G(u_n) → c_∗, n → ∞ with

∫RN(|∇un|2+V(x)|un|2)dx=1,

without loss of generality, we assume that u_n is non-negative for all n and for some u₀ ∈ H¹(ℝ^N), u_n → u₀ ≥ 0 weakly in H¹(ℝ^N), strongly in Lloc2 (ℝ^N) and a. e. on ℝ^N as n → ∞. Thanks to Lemma 2.2,

c∗=G(u0)+G∞(un−u0)+on(1), (2.1)

where o_n(1) → 0 as n → ∞. Moreover, set w_n = u_n − u₀, we have

1=∫RN(|∇u0|2+V(x)|u0|2)dx+∫RN(|∇wn|2+|wn|2)dx+on(1). (2.2)

On the other hand, by the definitions of c_∗ and c_∞, it is easy to know that

G(u)≤c∗∫RN(|∇u|2+V(x)|u|2)dxN+αN,for anyu∈H1(RN)

and

G∞(u)≤c∞∫RN|u|2dxN+αN,for anyu∈L2(RN).

Then by (2.1) and (2.2),

c∗≤c∗∫RN(|∇u0|2+V(x)|u0|2)dxN+αN+c∞∫RN|wn|2dxN+αN+on(1)≤c∗∫RN(|∇u0|2+V(x)|u0|2)dxN+αN+c∞∫RN(|∇wn|2+|wn|2)dxN+αN+on(1)≤c∗∫RN(|∇u0|2+V(x)|u0|2)dx+c∞∫RN(|∇wn|2+|wn|2)dx+on(1).

If c_∗ > c_∞, we claim that w_n → 0 strongly in H¹(ℝ^N) as n → ∞. Otherwise, we have

c∗<c∗∫RN(|∇u0|2+V(x)|u0|2)dx+c∗∫RN(|∇wn|2+|wn|2)dx+on(1)=c∗+on(1),

which is a contradiction. So w_n → 0 strongly in H¹(ℝ^N) as n → ∞ and then u_n → u₀ strongly in H¹(ℝ^N) as n → ∞. This implies that u₀ is a non-negative maximizer of c_∗. The proof is complete. □

In the following, we give a lower bound estimate for c_∗. For any ε > 0, set

uε(x)=εN2U(εx),

where U(x) is a maximizer of c_∞ and given above with λ = 1. Following [20], we have

Lemma 2.4

(Energy estimate). Assume (V₁), (V₂), (Q₁), (Q₂) hold, then c_∗ > c_∞ if one of the following conditions holds

α = 0, μ > N2(N−2)4(N+1) , infx∈RN Q(x) ≥ 1;
0 < α < 2, ν_α > 0;
α > 2, μ > N2(N−2)4(N+1) .

Proof

Observe that for any ε > 0,

∫RN(Iα∗|uε|N+αN)|uε|N+αN=∫RN(Iα∗|U|N+αN)|U|N+αN=c∞,

∫RN|uε|2=∫RN|U|2=1,∫RN|∇uε|2=ε2∫RN|∇U|2<+∞.

Let

mε:=∫RN|∇uε|2+V(x)|uε|2,

then m_ε = 1 + ε² 𝓘_μ(ε), where

Iμ(ε)=ε−2∫RN[|∇uε(x)|2+(V(x)−1)|uε(x)|2]dx.

Since

∫RN∇uε2=N2(N−2)4(N+1)∫RNuε(x)2|x|2 dx,

Iμ(ε)=ε−2∫RN[|∇uε(x)|2+(V(x)−1)|uε(x)|2]dx.

By Lebesgue’s monotone convergence theorem, we obtain

Iμ(ε)=∫RN[N2(N−2)4(N+1)+ε−2(V(ε−1x)−1)|x|2]|U(x)|2|x|2dx→[N2(N−2)4(N+1)−μ]∫RN|U(x)|2|x|2dx,as ε→0.

Let

aμ:=[N2(N−2)4(N+1)−μ]∫RN|U(x)|2|x|2dx,

then we get that

mε=1+aμε2+o(ε2),asε→0

and

aμ>0,ifμ<N2(N−2)4(N+1);=0,ifμ=N2(N−2)4(N+1);<0,ifμ>N2(N−2)4(N+1).

Let vε:=uεmε, then

∫RN|∇vε|2+V(x)|vε|2=1

and c_∗ ≥ G(v_ε). In the following, we show that G(v_ε) > c_∞ for ε > 0 small. In fact,

mεN+αNG(vε)=∫RNIα∗[Q(x)|uε|N+αN]Q(x)|uε|N+αNdx=∫RNIα∗[Q(ε−1x)|U|N+αN]Q(ε−1x)|U|N+αNdx=∫RNIα∗[Q(ε−1x)|U|N+αN][Q(ε−1x)−1]|U|N+αNdx+∫RNIα∗[(Q(ε−1x)−1)|U|N+αN]|U|N+αNdx+G∞(U).

By virtue of Lemma 2.1, it is easy to check that

∫RNIα∗[|x|−β|U|N+αN]|U|N+αNdx<∞,ifβ∈[0,N).

It follows from the Lebesgue’s monotone convergence theorem that, if α ∈ [0, N),

limε→0ε−β∫RNIα∗[(Q(ε−1x)−1)|U|N+αN]|U|N+αNdx=νβ∫RNIα∗[|x|−β|U|N+αN]|U|N+αNdx.

Similarly, if α ∈ [0, N),

limε→0ε−β∫RNIα∗[Q(ε−1x)|U|N+αN][Q(ε−1x)−1]|U|N+αNdx=νβ∫RNIα∗|U|N+αN[|x|−β|U|N+αN]dx.

Let

bβ,α:=2∫RNIα∗|U|N+αN[|x|−β|U|N+αN]dx>0,

then, for α ∈ [0, N), we have, as ε → 0,

G(vε)=mε−N+αN(c∞+bβ,ανβεβ+o(εβ))=(1+aμε2+o(ε2))−N+αN(c∞+bβ,ανβεβ+o(εβ))=1−N+αNaμε2+o(ε2)(c∞+bβ,ανβεβ+o(εβ))=c∞+bβ,ανβεβ−N+αNc∞aμε2+o(εβ)+o(ε2).

which implies that if 2 < α < N,

G(vε)=c∞−N+αNc∞aμε2+o(ε2),asε→0. (2.3)

Moveover, by (Q₁), if α = 0 and in addition infx∈RN Q(x) ≥ 1, we know ν_α = 0 and

G(vε)≥mε−N+αNc∞≥c∞−N+αNc∞aμε2+o(ε2),asε→0. (2.4)

It follows from (2.3) and (2.4) that G(v_ε) > c_∞ for ε > 0 small if

μ>N2(N−2)4(N+1),β=0or2<β<N.

For α ≥ N, by (Q₂) we have for some C > 0,

|x|2Q(x)−1≤C|x|β−2,x∈R∖{0}.

Noting that

∫RNIα∗[|x|−2|U|N+αN]|U|N+αNdx<∞,

due to the Lebesgue’s monotone convergence theorem, if α ≥ N,

limε→0ε−2∫RNIα∗[(Q(ε−1x)−1)|U|N+αN]|U|N+αNdx=0.

Similarly, if α ≥ N,

limε→0ε−2∫RNIα∗[Q(ε−1x)|U|N+αN][Q(ε−1x)−1]|U|N+αNdx=0.

This yields that, for α ≥ N, as ε → 0,

G(vε)=mε−N+αN(c∞+o(ε2))=(1+aμε2+o(ε2))−N+αN(c∞+o(ε2))=1−N+αNaμε2+o(ε2)(c∞+o(ε2))=c∞−N+αNc∞aμε2+o(ε2),

As a consequence, G(v_ε) > c_∞ for ε > 0 small if

μ>N2(N−2)4(N+1),β≥N.

Finally, if 0 < α < 2, then

G(vε)=c∞+bβ,ανβεβ+o(εβ),asε→0,

which implies that G(v_ε) > c_∞ for ε > 0 small and any μ ∈ ℝ, ν_α > 0. The proof is complete. □

Proof of Theorem 1.1 completed

Proof

As an immediate consequence of Lemma 2.3 and 2.4, there exists u_∗ ∈ H¹(ℝ^N) such that G(u_∗) = c_∗ and

∫RN(|∇u∗|2+V(x)|u∗|2)dx=1.

By the Lagrange multiplier theorem, there holds that for some κ ∈ ℝ such that G‡(u_∗) = κ T‡(u) in H⁻¹(ℝ^N), that is, in the weak sense, u_∗ satisfies

κ−Δu+V(x)u=N+αN(Iα∗[Q(x)|u|N+αN])Q(x)|u|αN−1u,x∈RN.

Obviously, κ = N+αN c_∗ > 0 and by virtue of the maximum principle, u_∗ is positive. To remove the multiplier, let

uθ(⋅)=θu∗(⋅),θ=c∗−N2α,

then u_θ is a weak solution of problem (1.8) and

I(uθ)=12T(uθ)−N2(N+α)G(uθ)=12θ2T(u∗)−N2(N+α)θ2(N+α)NG(u∗)=12θ2−N2(N+α)θ2(N+α)Nc∗=α2(N+α)c∗−Nα>0.

Finally, we show that u_θ is a ground state solution of problem (1.8). In fact, for any nontrivial solution u of problem (1.8), we can see that T(u) = G(u) > 0 and

I(u)=α2(N+α)G(u).

Let

τ=1T(u)>0,uτ(⋅)=τu(⋅),

then T(u_τ) = 1 and

c∗≥G(uτ)=τ2(N+α)NG(u).

It follows that

G(u)≤c∗[T(u)]N+αN=c∗[G(u)]N+αN

and then

I(u)≥α2(N+α)c∗−Nα.

The proof is complete. □

Proposition 2.1

(Pohozăev Identity). If u ∈ H¹(ℝ^N) is a solution of problem (1.8), then the following Pohozăev identity holds

N−22∫RN|∇u|2dx+12∫RNNV(x)+〈x,∇V(x)〉u2dx=N2∫RNIα∗[Q(x)|u|N+αN]Q(x)|u|N+αNdx+NN+α∫RNIα∗[Q(x)|u|N+αN]〈x,∇Q(x)〉|u|N+αNdx.

Proof

The proof is similar to [20, Proposition 11] and [22, Theorem 3]. We omit the details here. □

Completion of Proof of Theorem 1.2

Proof

For any solution u ∈ H¹(ℝ^N) of problem (1.8), using u as a test function, we have

∫RN(|∇u|2+V(x)|u|2)dx=∫RNIα∗[Q(x)|u|N+αN]Q(x)|u|N+αNdx

Thanks to Proposition 2.1,

∫RN−|∇u|2+12〈x,∇V(x)〉u2dx=NN+α∫RNIα∗[Q(x)|u|N+αN]〈x,∇Q(x)〉|u|N+αNdx.

Then by (VQ) and Hardy’s inequality, if u is nontrivial,

∫RN|∇u|2dx<(N−2)24∫RN|u(x)|2|x|2dx≤∫RN|∇u|2dx,

which is a contradiction. The proof is complete. □

3 Proof of Theorem 1.3

3.1 Eigenvalues and eigenfunctions

Consider the eigenvalue problem

−Δu+u=λν2+|x|2u,u∈H1(RN). (3.1)

It was proven in [7] that problem (3.1) admits a sequence of eigenvalues {λ_n} with finite multiplicity and associated eigenfunctions {φ_n}, such that

0<μν=λ1<λ2≤⋯≤λn≤⋯→+∞,n→∞

and for any i, j ∈ ℕ and i ≠ j,

∫RN∇φi∇φj+φiφj=0,∫RN1ν2+|x|2|φi|2dx=1.

Moreover, for some C_n, δ_n > 0,

|φn(x)|+|∇φn(x)|≤Cnexp(−δn|x|),x∈RN.

For any n ∈ ℕ, we have the orthogonal decomposition H¹(ℝ^N) = E⁻ ⊕ E⁺, where

E−=span⁡{φ1,φ2,⋯,φn},E+=span⁡{φn+1,φn+2,⋯}¯.

3.2 Energy estimates

For any ε > 0, set

uε(x)=εN2U(εx),

where U is given in Section 2 above for λ = ν. Similarly, we have

∫RN(Iα∗|uε|N+αN)|uε|N+αN=c∞

and

mε:=∫RN|∇uε|2+Vμ,ν(x)|uε|2=1+aμ,νε2+o(ε2),asε→0

where

aμ,ν:=Cν−2[N2(N−2)4(N+1)−μ]∫RN1|x|2(1+|x|2)Ndx

and C > 0 is independent of ε, μ, ν.

From now on, we assume

μ>maxN2(N−2)4(N+1),μν

and for some n ∈ ℕ, we assume μ ∈ [λ_n, λ_n+1) when N ≥ 4 or μ ∈ (λ_n, λ_n+1) when N = 3. Define

E^(vε):={w∈H1(RN):w=tvε+v,t≥0,v∈E−}

where vε:=uεmε and the energy functional

J(u)=12∫RN(|∇u|2+Vμ,ν(x)|u|2)dx−N2(N+α)∫RNIα∗[Qβ(x)|u|N+αN]Qβ(x)|u|N+αNdx.

Lemma 3.1

For ε > 0 small enough, there holds that

supw∈E^(vε)J(w)<α2(N+α)c∞−Nα,

provided when N = 3, one of the followings holds

α = 0, ν_α = 0, 32 < α < 3;
0 < α < min { 96−α , 2}, ν_α > 0
α > 2, ν_α > − 1, 32 < α < 3;

and when N ≥ 4, one of the followings holds
α = 0, ν_α = 0;
0 < α < 2, ν_α > 0;
α > 2, ν_α > − 1.

Proof

For any v ∈ E⁻ and t > 0,

J(tvε+v)=t22∫RN|∇vε|2+Vμ,ν|vε|2+t∫RN∇vε⋅∇v+Vμ,νvεv+12∫RN|∇v|2+Vμ,ν|v|2−N2(N+α)∫RN(Iα∗[Qβ(x)|tvε+v|N+αN])Qβ(x)|tvε+v|N+αN. (3.2)

Noting that v ∈ E⁻, ∫_ℝ^N |∇v|² + V_μ,ν |v|² ≤ 0 and then

J(tvε+v)≤t22+t∫RN∇vε⋅∇v+Vμ,νvεv−N2(N+α)∫RN(Iα∗[Qβ(x)|tvε+v|N+αN])Qβ(x)|tvε+v|N+αN. (3.3)

Since |∇vε|≤C1εN2+1 and |∇ v| ∈ L^r(ℝ^N) for any r > 0,

∫RN∇vε⋅∇v+Vμ,νvεv≤∥∇vε∥∞∥∇v∥L1+(1+μ)∥vε∥∞∥v∥1.

Recalling that it is proven in [18] that

∥u∥∗:=∫RN(Iα∗|u|N+αN)|u|N+αNN2(N+α),u∈L2(RN),

is a norm in L²(ℝ^N). Thanks to the fact that dim(E⁻) = n and all the norms are equivalent in E⁻, we get that

∫RN∇vε⋅∇v+Vμ,νvεv≤C2εN2∥v∥. (3.4)

In the following, we estimate the convolution term

G(tvε+v):=∫RN(Iα∗[Qβ(x)|tvε+v|N+αN])Qβ(x)|tvε+v|N+αN.

By [7, Lemma 3.5], for any x ∈ ℝ^N, ε > 0, v ∈ E⁻ and t > 0,

|tvε+v|N+αN−|tvε|N+αN−|v|N+αN≤C3|tvε||v|αN, (3.5)

where C₃ = 2^(N−α)/N (N + α)/N. It follows that

∫RN(Iα∗[Qβ(x)|tvε+v|N+αN])Qβ(x)|tvε+v|N+αN≥t2(N+α)N∫RN(Iα∗[Qβ(x)|vε|N+αN])Qβ(x)|vε|N+αN+2tN+αN∫RN(Iα∗[Qβ(x)|vε|N+αN])Qβ(x)|v|N+αN+∫RN(Iα∗[Qβ(x)|v|N+αN])Qβ(x)|v|N+αN−2C3∫RN(Iα∗[Qβ(x)(|tvε|N+αN+|v|N+αN)])Qβ(x)|tvε||v|αN:=J1+J2+J3+J4. (3.6)

Obviously, J₂ ≥ 0. So combing (3.3)-(3.6), we have

J(tvε+v)≤12t2+C2tεN2∥v∥−N2(N+α)(J1+J3+J4). (3.7)

Similarly as above, as ε → 0,

t−2(N+α)NJ1≥c∞−N+αNc∞aμ,νε2+o(ε2),ifβ=0,νβ=0;c∞+bβ,ανβεβ+o(εβ),if0<β<2,νβ>0;c∞−N+αNc∞aμ,νε2+o(ε2),ifβ>2,νβ>−1.

Due to the equivalence of norms in E−,J3≥C4∥v∥2(N+α)N. Thanks to the Hardy–Littlewood–Sobolev inequality and Q_α ∈ L^∞(ℝ^N),

|J4|≤C5t(∥tvε∥L2+∥v∥L2)N+αN(∫RN|vε|2NN+α|v|2αN+α)N+α2N.

Then noting that |vε(x)|≤C6εN2,x∈RN and E−⊂L2αN+α(RN),

∫RN|vε|2NN+α|v|2αN+α≤C7εN2N+α∫RN|v|2αN+α.

Thanks to dim(E⁻) < ∞, it is easy to check that

∫RN|v|2αN+α≤C8∥v∥2αN+α,for allv∈E−,

which implies that

|J4|≤C9(tN+αN∥v∥αN+∥v∥N+2αN)tεN2.

Let C10=NC42(N+α) (independent of v), by Young’s inequality the following hold:

C2tεN2∥v∥≤C103∥v∥2(N+α)N+C11t2(N+α)N+2αεN(N+α)N+2α, (3.8)

C9∥v∥αNt2N+αNεN2≤C103∥v∥2(N+α)N+C12t2(N+α)NεN(N+α)2N+α, (3.9)

C9∥v∥N+2αNtεN2≤C103∥v∥2(N+α)N+C13t2(N+α)NεN+α, (3.10)

from which we obtain

|J4|≤2C103∥v∥2(N+α)N+C12t2(N+α)NεN(N+α)2N+α+C13t2(N+α)NεN+α.

Then it follows from (3.7)-(3.8) that

J(tvε+v)≤12t2−N2(N+α)J1+C11t2(N+α)N+2αεN(N+α)N+2α+N2(N+α)t2(N+α)NC12εN(N+α)2N+α+C13εN+α. (3.11)

In particular, (3.11) implies that there exist 0 < t_∗ < t^∗(independent of ε) such that, for all v ∈ E⁻, ε > 0 small,

J(tvε+v)≤α4(N+α)c∞−Nα,t∈(0,t∗)∪(t∗,∞). (3.12)

If μ ∈ (λ_n, λ_n+1), there exists a constant C₁₄ such that

∫RN|∇v|2+Vμ,ν≤−C14∥v∥2,for allv∈E−.

Similarly as above, by (3.2) we have in place of (3.7) that

J(tvε+v)≤12t2−C142∥v∥2+C2tεN2∥v∥−N2(N+α)(J1+J3+J4).

We now use again estimates (3.8) and (3.10) and (3.9) replaced by the following

C9∥v∥αNt2N+αNεN2≤C14∥v∥2+C15t2(2N+α)2N−αεN22N−α,

to get the following estimate

J(tvε+v)≤12t2−N2(N+α)J1+C11t2(N+α)N+2αεN(N+α)N+2α+N2(N+α)C15t2(2N+α)2N−αεN22N−α+C13t2(N+α)NεN+α. (3.13)

α = ν_α = 0 or α > 2, ν_α ≥ − 1.

If N ≥ 4, then N(N+α)2N+α > 2. Moreover, it follows from (3.11) that, for all v ∈ E⁻, as α → 0, there holds that

J(tvε+v)≤12t2−N2(N+α)t2(N+α)Nc∞−N+αNc∞aμ,νε2−C12εN(N+α)2N+α+o(ε2)+o(εN(N+α)2N+α),t∈[t∗,t∗],

which implies that

J(tvε+v)≤12t2−N2(N+α)t2(N+α)Nc∞−N+αNc∞aμ,νε2+o(ε2),t∈[t∗,t∗].

Noting that

maxt≥0g(t)=g(c∞−N2α)=α2(N+α)c∞−Nα,

where

g(t)=12t2−N2(N+α)t2(N+α)Nc∞.

So thanks to (3.12) and a_μ,ν < 0 when μ > N2(N−2)4(N+1) , we have as ε > 0 small,

supw∈E^(vε)J(w)≤maxt≥0g(t)+C16c∞aμ,νε2+o(ε2)<α2(N+α)c∞−Nα,

where C₁₆ is independent of ε.

If N = 3, we have N(N+α)N+2α>2 and N22N−α>2 if 32 < α < 3. By (3.13), for all v ∈ E⁻ and t ∈ [t_∗, t^∗], as ε → 0,

J(tvε+v)≤12t2−N2(N+α)J1+C11t∗2(N+α)N+2αεN(N+α)N+2α+N2(N+α)C15t∗2α2N(2N−α)εN22N−α+C13εN+αt∗2(N+α)N=12t2−N2(N+α)c∞t2(N+α)N+12t∗2(N+α)Nc∞aμ,νε2+o(ε2).

Similarly as above, supw∈E^(vε)J(w)<α2(N+α)c∞−Nα for ε > 0 small.
0 < α < 2, ν_α > 0.

Similarly as above, if N ≥ 4, for all v ∈ E⁻, as ε → 0, there holds that

J(tvε+v)≤12t2−N2(N+α)t2(N+α)Nc∞+bβ,ανβεβ−C12εN(N+α)2N+α+o(εβ)+o(εN(N+α)2N+α),t∈[t∗,t∗].

Recalling that N(N+α)2N+α > 2 when N ≥ 4, which implies that

J(tvε+v)≤12t2−N2(N+α)t2(N+α)Nc∞+bβ,ανβεβ+o(εβ),t∈[t∗,t∗].

Similarly as above, we get that supw∈E^(vε)J(w)<α2(N+α)c∞−Nα, for ε > 0 small.

If N = 3, similarly as above, for all v ∈ E⁻ and t ∈ [t_∗, t^∗], as ε → 0,

J(tvε+v)≤12t2−N2(N+α)t2(N+α)Nc∞+bβ,ανβεβ+o(εβ)+N2(N+α)C15t∗4N+2α2N−αεN22N−α=12t2−N2(N+α)c∞t2(N+α)N−N2(N+α)t∗2(N+α)Nbβ,ανβεβ+N2(N+α)C15t∗4N+2α2N−αεN22N−α+o(εβ).

It follows that if α < N22N−α , for all v ∈ E⁻ and t ∈ [t_∗, t^∗], as ε → 0,

J(tvε+v)≤12t2−N2(N+α)c∞t2(N+α)N−N2(N+α)t∗2(N+α)Nbβ,ανβεβ+o(εβ).

Similarly as above, supw∈E^(vε)J(w)<α2(N+α)c∞−Nα for ε > 0 small. The proof is complete. □

3.3 Palais-Smale condition

Similarly to [7], we have the following compactness result, which plays a crucial role in finding nontrivial solutions of (1.8).

Lemma 3.2

The functional J satisfies the Palais-Smale condition in (-∞, c) if c<α2(N+α)c∞−Nα. Namely, if {u_m}_m∈ℕ ⊂ H¹(ℝ^N) satisfies

J(um)→c,J′(um)→0in H−1(RN), as m→∞

then up to a subsequence, there exists u ∈ H¹(ℝ^N) such that u_m → u strongly in H¹(ℝ^N), as m → ∞.

Proof of Lemma 3.2

The proof is similar to [7, Lemma 3.7]. For the sake of completeness, we give just a sketch. Let {u_m}_m∈ℕ ⊂ H¹(ℝ^N) be a (P-S)_c sequence, namely

J(um)→c<α2(N+α)c∞−Nα,J′(um)→0in H−1(RN),as m→∞.

Then by a similar fashion, we know that the sequence {u_m}_m∈ℕ is bounded in H¹(ℝ^N) and up to a subsequence, there exists u ∈ H¹(ℝ^N) such that u_m ⇀ u weakly in H¹(ℝ^N) and almost everywhere in ℝ^N, as m → ∞. Let v_m = u_m − u, then by the weak convergence and a Brezis-Lieb type lemma [19, Lemma 2.4],

c+om(1)=J(u)+12∥vm∥2−N2(N+α)G∞(vm),om(1)=〈J′(u),u〉+∥vm∥2−G∞(vm). (3.14)

Since J′(u) = 0 in H⁻¹ (ℝ^N),

J(u)=J(u)−N2(N+α)〈J′(u),u〉=α2(N+α)∫RN|∇u|2+Vμ,ν|u|2≥0.

Suppose G_∞(v_m) → l ≥ 0, as m → ∞, then by (3.14) lim_m→∞ ∥v_m∥² = l. If l > 0, then

l+om(1)=G∞(vm)≤c∞∥vm∥22(N+α)N≤c∞l+om(1)N+αN,

which implies l≥c∞−Nα. Then by (3.14),

c≥α2(N+α)c∞−Nα,

which is a contradiction. Therefore l = 0 and the proof is complete. □

Proof of Theorem 1.3

Now, we are in position to prove Theorem 1.3. By virtue of [3, Theorem 2.4] due to P. Bartolo, V. Benci and D. Fortunato, similarly as that in [7], (1.8) admits at least one nontrivial solution u ∈ H¹(ℝ^N) with J(u)<α2(N+α)c∞−Nα. Let

K:={u∈H1(RN)∖{0}:J′(u)=0in H−1(RN)},

then K ≠ ∅ and

m:=infu∈KJ(u)<α2(N+α)c∞−Nα.

Similarly to [7], m > 0 and there exists u₀ ∈ H¹(ℝ^N) such that u₀ ∈ K and J(u₀) = m. The proof of Theorem 1.3 is complete. □

Acknowledgements

Zhisu Liu is supported by the NSFC (11626127), Natural Science Foundation of Hunan Province (2019JJ50490; 2020JJ3029), Scientific Research Foundation of Hunan Province Education Department (Grant No. 18C0455), and the Fundamental Research Funds for the Central Universities, China University of Geosciences (Wuhan, Grant number: CUGST2). J. Zhang was partially supported by NSFC (11871123) and Team Building Project for Graduate Tutors in Chongqing (JDDSTD201802).

Conflict of interest

Conflict of interest statement: Authors state no conflict of interest.

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Received: 2021-03-24

Accepted: 2021-04-13

Published Online: 2021-06-30

This work is licensed under the Creative Commons Attribution 4.0 International License.

Groundstates for Choquard type equations with weighted potentials and Hardy–Littlewood–Sobolev lower critical exponent

Abstract

1 Introduction and main results

Theorem 1.1

Theorem 1.2

Remark 1.1

Remark 1.2

Corollary 1.1

Theorem 1.3

2 Proof of Theorem 1.1-1.2

Lemma 2.1

Lemma 2.2

Proof

Lemma 2.3

Proof

Lemma 2.4

Proof

Proof

Proposition 2.1

Proof

Proof

3 Proof of Theorem 1.3

3.1 Eigenvalues and eigenfunctions

3.2 Energy estimates

Lemma 3.1

Proof

3.3 Palais-Smale condition

Lemma 3.2

Proof of Lemma 3.2

Proof of Theorem 1.3

Acknowledgements

References

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