Non-trivial solutions for Schrödinger-Poisson systems involving critical nonlocal term and potential vanishing at infinity

Liuyang Shao

doi:10.1515/math-2019-0091

Artikel Open Access

Non-trivial solutions for Schrödinger-Poisson systems involving critical nonlocal term and potential vanishing at infinity

Liuyang Shao

Veröffentlicht/Copyright: 18. Oktober 2019

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Aus der Zeitschrift Open Mathematics Band 17 Heft 1

Abstract

The present study is concerned with the following Schrödinger-Poisson system involving critical nonlocal term

−△u+V(x)u−l(x)ϕ|u|3u=ηK(x)f(u), in R3,−△ϕ=l(x)|u|5, in R3, (1.1)

where the potential V(x) and K(x) are positive continuous functions that vanish at infinity, and l(x) is bounded, nonnegative continuous function. Under some simple assumptions on V, K, l and f, we prove that the problem (1.1) has a non-trivial solution.

Keywords: Schrödinger-Poisson system; variational methods; critical nonlocal term; vanishing potential

MSC 2010: 35B09; 35J20

1 Introduction and main results

The aim of this paper is to investigate the existence of non-trivial solutions for the following Schrödinger-Poisson system involving critical nonlocal term and potential vanishing at infinity

−△u+V(x)u−l(x)ϕ|u|3u=ηK(x)f(u), in R3,−△ϕ=l(x)|u|5, in R3, (1.1)

where V, K ∈ C(ℝ³, ℝ), f ∈ C(ℝ³ × ℝ, ℝ), l(x) is bounded, nonnegative continuous function, and V, K are nonnegative functions which can be vanishing at infinity. η > 0 is a parameter and 2^∗ := 6 is the critical Sobolev exponent. Similar problems have been widely investigated, and it is well known that they have a strong physical meaning, because they appear in quantum mechanics models and in semiconductor theory. In particular, systems like (1.1) have been introduced in [1] as a model describing solitary waves, for nonlinear stationary equations of Schrödinger type interacting with an electrostatic field, and are usually known as Schrödinger-Poisson systems. Indeed, in (1.1) the first equation is a nonlinear stationary Schrödinger equation that is coupled with a Poisson equation, to be satisfied by ϕ, meaning that the potential is determined by the charge of the wave function. For more details, we refer the readers to [2, 3, 4, 5] and the references therein.

In recent years, with the aid of variational methods, there has been increasing attention to problems like (1.1) on the existence and non-existence of positive solutions, positive ground states, multiple solutions, sign-changing solutions and so on. see for instance [6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21], and the references therein. In fact, the most of the above results focused on the subcritical nonlocal term. However, the system (1.1) with critical nonlocal term has only been studied in [22, 23, 24, 25].

In [22], A. Azzollini and P. d’Avenia firstly studied the following Schrödinger-Poisson system with critical nonlocal term

−△u=μu+pϕ|u|3u=f(x,u),x∈BR,−△ϕ=p|u|5,x∈BR,u=ϕ=0,on ∂BR. (1.2)

They proved that the existence and nonexistence results for system (1.1) depend on the value of λ.

In [23], Liu studied the following Schrödinger-Poisson system with critical nonlocal term

−△u+V(x)u−K(x)ϕ|u|3u=f(x,u),in R3,−△ϕ=K(x)|u|5,in R3. (1.4)

Under the condition V(x), K(x), f are asymptotically periodic, the author proved the system (1.4) has at least a positive solution by the mountain pass theorem and the concentration-compactness principle.

In [25], F. Li, Y. Li and J. Shi proved

−△u+bu−ϕ|u|3u=f(u),in R3,−△ϕ=|u|5,in R3, (1.3)

possesses at least one positive radially symmetric solution when b > 0 is a constant. To the best of our knowledge, there seems to be little progress on the existence of nontrivial solution for Schrödinger-Poisson systems involving critical nonlocal term and potential vanishing at infinity.

By the motivation of above work, In our article, we establish the existence of non-trivial solution for problem (1.1) with critical nonlocal term and potential vanishing at infinity. Firstly the critical growth causes a lack of compactness, and it is much more difficult to obtain the existence of non-trivial solutions. Secondly since V(x) is potential vanishing at infinity, which makes our studies more interesting. At last, we obtain a non-trivial solution by using the mountain pass theorem without (PS) condition.

Below, we assume that the pair (V, K) of continuous functions V, K : ℝ³ → ℝ belongs to 𝓚. Throughout the paper, (V, K) ∈ 𝓚 means that

V(x), K(x) > 0 for all x ∈ ℝ³ and K ∈ L^∞(ℝ³).
If {A_n}_n ⊂ ℝ³ is a sequence of Borel sets such that the Lebesgue measure meas(A_n) ≤ R, for all n ∈ ℕ and some R > 0, then

limR→+∞∫An∩BRc(0)K(x)dx=0, uniformly in n∈N.

Furthermore, one of the below conditions occurs:

K/V ∈ L^∞(ℝ³) or
there exists p₀ ∈ (2, 6) such that

K(x)V(x)6−p04→0 as |x|→+∞.

The hypotheses (VK₁)–(VK₄) on functions V and K were first introduced in [26] and characterized problem (1.1) as zero mass. Problems of zero mass have been studied by many authors, see for example, [27, 28, 29, 30, 31, 32] and references therein.

Finally, we assume the following growth conditions at the origin and at the infinity for the continuous function f : ℝ → ℝ:

limt→0f(t)t=0, if (VK₃) holds; or limt→0f(t)|t|p0−1<∞, if (VK₄) holds.
f has a “quasicritical growth”, namely, limt→+∞f(t)t5=0.
There exists a θ ∈ (2, 2^∗) such that

0≤θF(t)≤tf(t) for all t∈R,

where F(u) = ∫0u f(s)ds.

Furthermore, we make the following hypotheses on the function l(x).

There exists x₀, such that

l(x0)=supx∈R3l(x).
For x close to x₀ we have

l(x)=l(x0)+O(|x−x0|), as x→x0.

Now we state our main results as follows.

Theorem 1.1

Suppose that (V, K) ∈ 𝓚, and f satisfies (f₁), (f₂), (f₃), l(x) satisfies (l₁)–(l₂).

If θ ∈ (1, 3], for sufficiently lange η > 0, then system (1.1) has at least one non-trivial solution.
If θ ∈ (3, 5), for any η > 0, then system (1.1) has at least one non-trivial solution.

Notation. In this paper we make use of the following notations: C will denote various positive constants; the strong (respectively weak) convergence is denoted by → (respectively ⇀); o(1) denotes o(1) → 0 as n → ∞, B_ρ(0) denotes a ball centered at the origin with radius ρ > 0.

The remainder of this paper is organized as follows. In Section 2, some preliminary results are presented. In Section 3, we give the proof of our main results.

2 Variational setting and preliminaries

Let us consider the space

E=u∈D1,2(R3):∫R3V(x)|u|2dx<+∞

endowed with the norm

∥u∥E2=∫R3|∇u|2dx+∫R3V(x)|u|2dx.

Recall that a weak solution of problem (1.1) is a function u ∈ E such that

∫R3∇u∇φdx+∫R3V(x)uφdx−∫R3l(x)ϕ|u|4φdx−η∫R3K(x)f(u)φdx=0, (2.1)

for all φ ∈ E.

Then, the weak solutions of (1.1) are the critical points of the energy functional defined on E by

J(u):=12∫R3(|∇u|2+V(x)|u|2)dx−110∫R3l(x)ϕu|u|5dx−η∫R3K(x)f(u)dx, (2.2)

where F(u) = ∫0u f(s)ds. More precisely, J ∈ C¹(E, ℝ) and its differential J′ : E → E′ is defined as

〈J′(u),v〉=∫R3(|∇u||∇v|+V(x)uv)dx−∫R3l(x)ϕu|u|4vdx−η∫R3K(x)f(u)vdx, (2.3)

for all v ∈ E, where E′ is the dual space of E.

We define the Lebesgue space LKp (ℝ^N) composed by all measurable functions u : ℝ^N → ℝ such that

LKp(R3)=u:R3→R|u is measurable and ∫R3K(x)|u|pdx<+∞,

endowed with norm

∥u∥LKp(R3):=∫R3K(x)|u|pdx1p,

and we will state, without proof, two important results of Alves and Souto (see [[26], Lemmas 2.1 and 2.2]).

Proposition 2.1

Assume (V, K) ∈ 𝓚 holds, E is compactly embedded in LKp (ℝ^N) for every p ∈ (2, 6). If (VK₄) holds, E is compactly embedded in LKp (ℝ^N).

Proposition 2.2

Suppose that f satisfies (f₁) and (f₂) and (V, K) ∈ 𝓚, Let {v_n} be such that v_n⇀ v in E. Then,

∫R3K(x)F(vn)dx→∫R3K(x)F(v)dx (2.4)

and

∫R3K(x)f(vn)vndx→∫R3K(x)f(v)vdx. (2.5)

Lemma 2.3

[25, Lemma 2.1] For every u ∈ L⁶(ℝ³), there exists a unique ϕ_u ∈ D^1,2(ℝ³) which is the solution of

−△ϕ=|u|5, in R3,

here ϕ_u can be expressed by the from

ϕu(x)=∫R3|u(y)|5|x−y|dy

Moreover,

∥ϕu∥D1,2(R3)2=∫R3ϕu|u|5;
ϕ_u(x) > 0 for x ∈ ℝ³;
for any θ > 0, ϕ_{u_θ} = θ²(ϕ_u)_θ, where u_θ(⋅) = u(⋅/θ);
for any t > 0, ϕ_tu = t⁵ϕ_u;
for any u ∈ L⁶(ℝ³),

∥ϕu∥D1,2(R3)≤S−12|u|65,∫R3ϕu|u|5≤S−1|u|610,

where S is defined in (1.6);
if u_n ⇀ u in L⁶(ℝ³) and u_n → u a.e in ℝ³ as n → ∞, then ϕ_{u_n} → ϕ_u in D^1,2(ℝ³).

Lemma 2.4

[25, Lemma 2.3] If u_n⇀ u in L⁶(ℝ³), a, e in ℝ³, then as n → ∞,

|un|5−|un−u|5−|u|5→0 in L65(R3),ϕun−ϕun−u−ϕu→0, in D1,2(R3),∫R3ϕun|un|5dx−∫R3ϕun−u|un−u|5dx−∫R3ϕu|u|5dx→0,

and

|un|3un−|un−u|(un−u)−|u|3u→0, in D1,2(R3).

3 Proof of Theorem 1.1

To prove Lemma 3.2, we need the following results.

Lemma 3.1

Suppose that (V, K) ∈ 𝓚 hold. Then for p ∈ [2, 6], there is C > 0 such that

∥u∥Lkp(R3)≤C∥u∥E, ∀ u∈E.

Proof

First we suppose that (VK₂) holds. The proof is trivial if p = 2 or 6. Now we prove that the embedding is true for p ∈ (2, 6) under the assumption (VK₃). For fixed p ∈ (2, 6), define m=(6−p)4, and hence p = 2m + (1 – m)6. so we have that

∫R3K(x)|u|pdx=∫R3|u|2m|u|(1−m)6dx≤∫R3|K(x)|1m|u|2dxm∫R3|u|6dx1−m≤supx∈R3|K(x)||V(x)|m∫R3V(x)u2dxm∫R3|u|6dx1−m≤Csupx∈R3|K(x)||V(x)|m∫R3V(x)u2dxm∫R3|∇u|2dx3(1−m)≤Csupx∈R3|K(x)||V(x)|m∫R3(|∇u|2+V(x)u2)dxm+3(1−m)=Csupx∈R3|K(x)||V(x)|msupx∈R3∫R3|∇u|2+V(x)|u|2dxp2.

Since K(x) ∈ L^∞(ℝ³) and K/V ∈ L^∞(ℝ³), we have that

∥u∥LKp(R3)≤C∥u∥E, for p∈(2,6).

Next, we suppose that (VK₄) holds. Using the same argument as above, we define m0=(6−p0)4, and hence p₀ = 2m₀ + (1 – m₀)6 so that we have

∫R3K(x)|u|p0dx=∫R3K(x)|u|2m0|u|(1−m0)6dx≤∫R3|K(x)|1m0|u|2dxm0∫R3|u|6dx1−m0≤supx∈R3|K(x)||V(x)|m0∫R3V(x)|u|2dxm0∫R3|u|6dx1−m0≤Csupx∈R3|K(x)||V(x)|m0∫R3(|∇u|2+V(x)|u|2)dxp02.

From (VK₃) we deduce that |K(x)||V(x)|m0∈L∞(R3). It follows from the above inequality that

∥u∥LKp0(R3)≤C∥u∥E.

we complete the proof.□

The functional J satisfies the mountain pass geometry.

Lemma 3.2

The functional J satisfies the following conditions:

There exist ρ and α > 0 such that J(u) ≥ α with |u|_E = ρ.
There exists e ∈ B_ρ(0) with J(e) < 0.

Proof

Now, we distinguish two case.
1. We suppose that (VK₃) is true. For any ε > 0, it follows from (f₁) and (f₂) that there exists C_ε > 0 such that
  
  F(u)≤ε2|u|2+Cε|u|6, for all u∈E. (3.1)
  
  Thus, by (3.1) and Lemma 3.1, we get that
  
  ∫R3K(x)F(u)dx≤ε2∫R3K(x)|u|2dx+Cε∫R3K(x)|u|6dx≤ε2∥u∥E2+Cε∥u∥E6.
  
  Hence, in view of Lemma 2.3, we obtain
  
  J(u)=12∫R3|∇u|2+V(x)|u|2dx−110∫R3l(x)ϕu|u|5dx−η∫R3K(x)F(u)dx≥12∥u∥E2−C∥u∥E10−ε2∥u∥E2−Cε∥u∥E6=(1−ε2)∥u∥E2−C∥u∥E10−Cε∥u∥E6.
  
  So, taking ε=12, there exists enough small |u|_E = ρ, such that J(u) ≥ α.
2. We suppose that (VK₄) holds. By (f₁) and (f₂), there exist Cε′>0 and Cε″>0 such that
  
  F(u)≤Cε′|u|p0+Cε″|u|6, for all u∈E.
  
  Therefore
  
  J(u)=12∫R3|∇u|E2+V(x)u2dx−110∫R3l(x)ϕu|u|5dx−η∫R3K(x)F(u)dx≥12∥u∥E2−C∥u∥E6−C∥u∥E10−C∥u∥Ep0.
  
  The same as Case 1, we can take |u|_E = ρ such that J(u) ≥ α.
For every t > 0 we obtain

J(tu)=t22∥u∥E2−t1010∫R3K(x)ϕu|u|5dx−η∫R3K(x)F(tu)dx.

From (f₃), we obtain J(tu) → –∞ as t → +∞, so it is satisfies (ii). We complete the proof.□

As a consequence of Lemma 3.2, we can find a (PS) sequence of the functional J(u) at the level

c:=infγ∈Γmaxt∈[0,1]J(γ(t))>0, (3.2)

where the set of paths is defined as

J:={γ∈C([0,1],H1(R3)):γ(0)=0,J(γ(1))<0}.

Lemma 3.3

Let {u_n} be a (PS)_c sequence for J. Then {u_n} is bounded in E.

Proof

Let {u_n} ⊂ E be a (PS)_c sequence for J, that is

J(un)→c and J′(un)→0 as n→+∞.

Therefore, From (f₃), we have

c+1+∥un∥E≥J(un)−1θ〈J′(un),un〉=12−1θ∥un∥E2+ηθ∫R3K(x)(f(un)un−θF(un))dx+1θ−110∫R3l(x)ϕu|u|5dx≥12−1θ∥un∥E2,

which implies that {u_n} is bounded in E.

□

Because of the appearance of the critical nonlocal term, we have to estimate the Mountain-pass value given by (3.2) carefully, To do it, we choose the extremal function Uε(x)=(3ε2)14(ε2+|x−x0|2)12 to solve –△u = u⁵ in ℝ³. Let φ ∈ C0∞ (ℝ³) be a cut-off function verifying that 0 ≤ φ(x) ≤ 1 for all x ∈ ℝ³, suppφ ⊂ B₂(x₀), and φ(x) ≡ 1 on B₁(x₀). Set V_ε = φU_ε, then thanks to the asymptotic estimates from [8], we have

∇vε|22=S32+O(ε), |vε|62=S12+O(ε)

and for s ∈ [2, 6)

|vε|ss=O(εs2), if s∈[2,3),O(ε32|logε|), if s=3,O(ε6−s2), if s∈(3,6),

where S denotes the best constant for the embedding D^1,2(ℝ³) ↪ L⁶(ℝ³), namely,

S:=infu∈D1,2(R3)∫R3|∇u|2dx,∫R3|u|6dx=1.

We define

Vmax:=maxx∈B2R(x0)V(x)

and

Kmin:=minx∈B2R(x0)K(x).

By the assumption (l₂) we also have

l(x0)∫B2R(x0)ϕ|u|5dx≤∫B2R(x0)l(x)ϕ|u|5dx.

Lemma 3.4

Suppose that (V, K) ∈ 𝓚, (f₁)–(f₃) hold, and l(x) satisfies (l₁)–((l₂). Then, there exists a u₀ ∈ E ∖ {0} such that

0<supt≥0J(tu0)<25S32|l(x)|L∞(R3)−12.

Proof

We now consider

I(tvε)=t22∫R3(|∇vε|2+V(x)|vε|2)dx−t1010∫R3l(x)ϕvε|vε|5dx−η∫R3K(x)F(tvε)dx.

By Lemma 3.1, we know that, there exists t_ε > 0 such that supt≥0 J(tv_ε) > 0 is attained and limt→∞ J(tv_ε) = –∞ for any ε > 0.

We suppose that there exists ϱ₁, ϱ₂ such that ϱ₁ < t_ε < ϱ₂ for small enough ε > 0. In fact, J(t_ε v_ε) = supt≥0 J(tv_ε), and hence dJ(tv_ε)/dt|_{t=t_ε} = 0, we obtain that

tε∫B2R(x0)(|∇vε|2+V(x)|vε|2)dx−η∫B2R(x0)K(x)f(tεvε)vεdx−tε10∫B2R(x0)l(x)ϕvε|vε|5dx=0. (3.3)

Now, we prove that t_ε → +∞ as ε_n → 0⁺ does not hold. By (3.3)

tεn∫B2R(x0)(|∇vεn|2+V(x)|vεn|2)dx≥tεn10∫B2R(x0)l(x)ϕvε|vε|5dx,

which is a contradiction when t_ε → +∞.

Similarly, we suppose that there is a sequence t͠_{ε_n} → 0 as ε_n → 0⁺. Firstly, if (VK₃) holds, from (f₁) and (f₂), for all δ > 0 there exists C_δ > 0 such that

∫R3K(x)f(t~εnvεn)vεndx≤δt~εn∫R3K(x)|vεn|2dx+Cδ(t~εn)5∫R3K(x)|vεn|6dx≤δCt~εn∫R3(|∇vεn|2+V(x)|vεn|2)dx+Cδ(t~εn)5∫R3K(x)|vεn|6dx.

Choosing δ=12C, it follows from (3.3) that

t~εn2∫R3(|∇vεn|2+V(x)|vεn|2)dx≤Cδ(t~εn)5∫R3K(x)|vεn|6dx+(t~εn)9∫R3l(x)ϕvn|vεn|5dx.

Next, we suppose that (VK₄) holds. By (f₁), (f₂), there is a constant C > 0, such that

∫R3K(x)f(t~εnvεn)dx≤(t~εn)p0−1∫R3K(x)|vεn|p0dx+C¯(t~εn)5∫R3K(x)|vεn|6dx.

It again follows from (3.3) that

t~εn∫R3(|∇vεn|2+V(x)|vεn|2)dx≤(t~εn)p0−1∫R3K(x)|vεn|p0dx+C¯(t~εn)5∫R3K(x)|vεn|6dx+(t~εn)9∫R3l(x)ϕvεn|vεn|5dx,

We arrive at a contradiction because p₀ > 2. So we complete the proof.

Since 0 < ϱ₁ < t_ε < ϱ₂ < ∞, together with the definitions of V_max and K_min, we have

J(tvε)=t22∫R3(|∇vε|2+V(x)|vε|2)dx−t1010∫R3l(x)ϕvε|vε|5dx−η∫R3K(x)F(tvε)dx≤tε22∫R3(|∇vε|2+V(x)|vε|2)dx−tε1010∫R3l(x)ϕvε|vε|5dx−η∫R3K(x)F(tεvε)dx≤tε22∫R3|∇vε|2dx+tε22Vmax(x)∫R3|vε|2dx−tε1010∫R3l(x)ϕvε|vε|5dx−ηKmin(x)∫R3F(tεvε)dx.

We define

h(t):=t22∫R3|∇vε|2dx−t1010∫R3l(x)ϕvε|vε|5dx.

By some elementary calculations, we obtain

maxt≥0h(t)=4(12∫R3|∇vε|2dx)545(12∫R3l(x)ϕvε|vε|5dx)14=25(∫R3|∇vε|2dx)54(∫R3l(x)ϕvε|vε|5dx)14.

The Poisson equation –△ϕ_{v_ε} = |v_ε|⁵ and Cauchy’s inequality give

∫R3|vε|6dx=∫R3∇ϕvε|∇vε|dx≤12|l(x)|∞∫R3|∇ϕvε|2dx+|l(x)|∞2∫R3|∇vε|2dx=12|l(x)|∞∫R3ϕvε|vε|5dx+|l(x)|∞2∫R3|∇vε|2dx.

This implies that

∫R3l(x)ϕvε|vε|5dx≥2|l(x)|∞∫R3l(x)|vε|6dx−|l(x)|∞2∫R3|∇vε|2dx=|l(x)|∞2S32+O(ε).

As a consequence of the above fact, one has

maxt≥0h(t)≤25(S25+O(ε))54(|l(x)|∞2S32+O(ε))14=25|l(x)|∞S32+O(ε).

On the other hand, from (f₃), we obtain that F(s) ≥ Cs^θ, for s > 0. Hence,

∫B2R(x0)F(tεvε)dx≥C∫B2R(x0)(tεvε)θdx≥Cϱ1θ∫BR(x0)(vε)θdx=Cϱ1θO(εθ+12), if θ∈[1,2),Cϱ1θO(ε32|logε|), if θ=2,Cϱ1θO(ε5−θ2), if θ∈(2,5). (3.4)

We have maxt≥0J(tvε)=J(tεvε) at the beginning, that is,

maxt≥0J(tvε)=tε22∫R3|∇vε|2+V(x)|vε|2dx−tε1010∫R3l(x)ϕvε|vε|5dx−η∫R3K(x)F(tvε)dx≤h(tε)+Vmax2∫R3|vε|2dx−η∫R3K(x)F(tvε)dx≤25|l(x)|∞S32+CO(ε)−η∫R3K(x)F(tvε)dx.

Using (3.4), we have

CO(ε)−η∫R3K(x)F(tvε)dx≤CO(ε)−Cηϱ1θO(εθ+12), if θ∈[1,2),Cηϱ1θO(ε32|logε|), if θ=2,Cηϱ1θO(ε5−θ2), if θ∈(2,5).

If (1, 2) and η=ε−12, then 12<θ+12−12<1 and hence for small enough ε > 0

CO(ε)−η∫R3K(x)F(tvε)dx≤CO(ε)−Cϱ1θε−12O(εθ+12)<0,

when η > 0 is enough large.

If θ = 2 and η=ε−12, then for small enough ε > 0

CO(ε)−η∫R3K(x)F(tvε)dx≤CO(ε)−Cϱ1θε−12O(ε32)|logε|<0,

when η > 0 is enough large.

If θ ∈ (2, 3] and η=ε−12, then 12≤5−θ2−12<1 and hence for small enough ε > 0

CO(ε)−η∫R3K(x)F(tvε)dx≤CO(ε)−Cϱ1θε−12O(ε5−θ2)<0,

when η > 0 is enough large.

If θ ∈ (3, 5) then 0<5−θ2<1 and for small enough ε > 0

CO(ε)−η∫R3K(x)F(tvε)dx≤CO(ε)−Cηϱ1θO(ε5−θ2)<0,

for any η > 0.

Consequently, we show that for θ ∈ (3, 5) with any η > 0, or θ ∈ (1, 3] with enough large η > 0

CO(ε)−η∫R3K(x)F(tvε)dx<0.

So, we can obtain that supt≥0J(tvε)<25S32|l(x)|L∞(R3)−12. □

Proof of Theorem 1.1

Proof. From Lemma 3.3, we know that {u_n}_n is bounded in E. Up to a subsequence, we have

un⇀u in E,un→u in Llocr(R3), for r∈[2,6),un→u a.e in R3. (3.5)

By (2.1) – (2.3), we get

J(un)=12∫R3(|∇un|2+V(x)|un|2)dx−η∫R3K(x)F(un)dx−110∫R3l(x)ϕun|un|5dx=c+o(1)

and

〈J′(un),un〉=∫R3(|∇un|2+V(x)|un|2)dx−η∫R3K(x)F(un)dx−∫R3l(x)ϕun|un|5dx=o(1).

Assuming v_n = u_n – u, in view of (3.5), Proposition 2.2, Lemma 2.4 and the Brézis-Lieb Lemma [33, 34],

J(un)=J(u)+12∫R3(|∇vn|2+V(x)|vn|2)dx−110∫R3l(x)ϕvn|vn|5dx (3.6)

and

〈J′(un),un〉=∫R3(|∇un|2+V(x)|un|2)dx−η∫R3K(x)f(un)undx−∫R3l(x)ϕun|un|5dx=∫R3|∇u|2+V(x)|u|2)dx−∫R3l(x)ϕu|u|5dx−η∫R3K(x)f(u)udx+∫R3(|∇vn|2+V(x)|vn|2)dx−∫R3l(x)ϕvn|vn|5dx+on(1). (3.7)

Since J′(u_n) → 0 as n → +∞ and by (3.5) again, we get

〈J′(un),u〉=∫R3(|∇u|2+V(x)u2)dx−∫R3l(x)ϕu|u|5dx−η∫R3K(x)f(u)udx. (3.8)

From (3.7) and (3.8), we obtain

∫R3|∇vn|2dx+∫R3V(x)|vn|2dx−∫R3l(x)ϕvn|vn|5dx→0 as n→+∞ (3.9)

and

J(u)=12∫R3(|∇u|2+V(x)|u|2)dx−110∫R3∫R3l(x)ϕu|u|5dx−η∫R3K(x)F(u)dx=12η∫R3K(x)f(u)udx−η∫R3K(x)F(u)dx+25∫R3l(x)ϕu|u|5dx≥0. (3.10)

Without loss of generality we can suppose that

∫R3(|∇vn|2+V(x)|vn|2)dx→l as n→∞ (3.11)

and from (3.9)

∫R3l(x)ϕvn|vn|5dx→l as n→∞. (3.12)

By estimate

∫R3l(x)ϕvn|vn|5dx≤|l(x)|∞2|vn|610S≤|l(x)|∞2∥vn∥10S6. (3.13)

Combining (3.10) – (3.13), which implies l≤|l(x)|∞2l5S6. Therefore, either l = 0 or l≥|l(x)|∞−12S32.

If l > 0, we have l≥|l(x)|∞−12S32. Taking the limit in (3.6) as n → +∞, and using (3.10), we obtain c₀ ≥ 25|l(x)|∞−12S32. On the other hand, from (3.2) and Lemma 3.4, we obtain c0≤25|l(x)|∞−12S32. We get a contradiction. This shows that l = 0. Thus

J(u)=c>0 and J′(u)=0,

i.e. u is a non-trivial solution of (1.1). We complete the proof.□

Acknowledgements

The authors thank the referees for valuable comments and suggestions which improved the presentation of this manuscript. This work is partially supported by the Fundamental Research Funds for the National Natural Science Foundation of China 11671403 and Guizhou University of Finance and Economics of 2018YJ20.

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Received: 2019-04-04

Accepted: 2019-08-16

Published Online: 2019-10-18

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https://doi.org/10.1515/math-2019-0091

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Schrödinger-Poisson system; variational methods; critical nonlocal term; vanishing potential

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