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Advances in Nonlinear Analysis

Editor-in-Chief: Radulescu, Vicentiu / Squassina, Marco


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Singularly perturbed Choquard equations with nonlinearity satisfying Berestycki-Lions assumptions

Xianhua Tang
  • Corresponding author
  • School of Mathematics and Statistics, Central South University, Changsha, Hunan 410083, P.R.China
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/ Sitong Chen
Published Online: 2019-06-01 | DOI: https://doi.org/10.1515/anona-2020-0007

Abstract

In the present paper, we consider the following singularly perturbed problem:

ε2u+V(x)u=εα(IαF(u))f(u),xRN;uH1(RN),

where ε > 0 is a parameter, N ≥ 3, α ∈ (0, N), F(t) = 0tf(s)ds and Iα : ℝN → ℝ is the Riesz potential. By introducing some new tricks, we prove that the above problem admits a semiclassical ground state solution (ε ∈ (0, ε0)) and a ground state solution (ε = 1) under the general “Berestycki-Lions assumptions” on the nonlinearity f which are almost necessary, as well as some weak assumptions on the potential V. When ε = 1, our results generalize and improve the ones in [V. Moroz, J. Van Schaftingen, T. Am. Math. Soc. 367 (2015) 6557-6579] and [H. Berestycki, P.L. Lions, Arch. Rational Mech. Anal. 82 (1983) 313-345] and some other related literature. In particular, we propose a new approach to recover the compactness for a (PS)-sequence, and our approach is useful for many similar problems.

Keywords: Choquard equation; Ground state solution; Pohožaev mainfold; Berestycki-Lions assumptions; Singularly perturbed

MSC 2010: 35J20; 35J62; 35Q55

1 Introduction

In this paper, we consider the following singularly perturbed nonlinear Choquard equation:

ε2u+V(x)u=εα(IαF(u))f(u),xRN;uH1(RN),(1.1)

where ε > 0 is a parameter, N ≥ 3, α ∈ (0, N) and Iα : ℝN → ℝ is the Riesz potential defined by

Iα(x)=ΓNα2Γα22απN/2|x|Nα,xRN{0},

F(t) = ∈ 0tf(s)ds, V : ℝN → ℝ and f : ℝ → ℝ satisfy the following basic assumptions:

  • (V1)

    V ∈ 𝓒(ℝN, [0, ∞)) and V := lim|x|→∞V(x) > 0;

  • (F1)

    f ∈ 𝓒(ℝ, ℝ) and there exists a constant 𝓒0 > 0 such that

    |f(t)t|C0|t|(N+α)/N+|t|(N+α)/(N2),tR;

  • (F2)

    F(t) = o(t(N+α)/N) as t → 0 and F(t) = o(t(N+α)/(N–2)) as |t| → ∞;

  • (F3)

    there exists s0 > 0 such that F(s0) ≠ 0.

Note that (F1)-(F3) were almost necessary and sufficient conditions and regarded as the Berestycki-Lions type conditions to Choquard equations, which were introduced by Moroz and Van Schaftingen in [22] for the study of (1.1) with ε = 1.

In recent years, semiclassical problems like (1.1), i.e. the parameter ε goes to zero, have received attention from the mathematical community. For small ε > 0, bound states are called semiclassical states, which describe a kind of transition from Quantum Mechanics to Newtonian Mechanics. There are some nice work on semiclassical states for (1.1). For example, for a special form of (1.1) with N = 3, α = 2 and F(s) = s2/2, by proving the uniqueness and non-degeneracy, of the ground states for the limit problem, Wei and Winter [37] constructed a family of solutions by a Lyapunov-Schmidt type reduction; Cingolani et al. [9] proved the existence of solutions concentrating around several minimum points of V by a global penalization method. Moroz and Van Schaftingen [24] developed a nonlocal penalization technique to show that problem (1.1) with F(s) = |s|p/p and p ≥ 2 has a family of solutions concentrating at the local minimum of V provided V satisfies some additional assumptions at infinity. However, for (1.1) with general nonlinearity F which only satisfies (F1)-(F3), there seem to be no results in the existing literature. One of main purpose of this paper is to deal with this case.

When ε = 1, (1.1) reduces to the nonlinear Choquard equation of the form:

u+V(x)u=(IαF(u))f(u),xRN;uH1(RN).(1.2)

which has been extensively studied by using variational methods, see [1, 2, 3, 8, 17, 20, 21, 22, 23, 24, 25, 29, 36] and references therein. In view of (F1), (F2) and Hardy-Littlewood-Sobolev inequality, for some p ∈ (2, 2*) and any ϵ > 0, one has

RN(IαF(u))F(u)dx=ΓNα2Γα22απN/2RNRNF(u(x))F(u(y))|xy|NαdxdyC1F(u)2N/(N+α)2ϵu22(N+α)/N+u22(N+α)/(N2)+Cϵup(N+α)p/N,uH1(RN).(1.3)

It is standard to check using (1.3) that under (V1), (F1) and (F2), the energy functional defined in H1(ℝN) by

J(u)=12RN|u|2+V(x)u2dx12RN(IαF(u))F(u)dx(1.4)

is continuously differentiable and its critical points correspond to the weak solutions of (1.2).

If the potential V(x) ≡ V, then (1.2) reduces to the following autonomous form:

u+Vu=(IαF(u))f(u),xRN;uH1(RN),(1.5)

its energy functional is as follows:

J(u)=12RN|u|2+Vu2dx12RN(IαF(u))F(u)dx.(1.6)

Problem (1.5) is a semilinear elliptic equation with a nonlocal nonlinearity. For N = 3, α = 2, V = 1 and F(t) = t2/2, it covers in particular the Choquard-Pekar equation

u+u=(I2u2)u,xR3;uH1(R3),(1.7)

introduced by Pekar [27] at least in 1954, describing the quantum mechanics of a polaron at rest. In 1976, Choquard [16] used (1.7) to describe an electron trapped in its own hole. In 1996, Penrose [19] proposed (1.7) as a model of self-gravitating matter. In this context (1.7) is usually called the nonlinear Schrödinger-Newton equation, see Moroz-Schaftingen [22].

If we let α → 0 in (1.5), then we can get the following limiting problem:

u+Vu=g(u),xRN;uH1(RN),(1.8)

where g = Ff. In the fundamental paper [4], Berestycki-Lions proved that (1.8) has a radially symmetric positive solution provided that g satisfies the following assumptions:

  • (G1)

    g ∈ 𝓒(ℝ, ℝ) is odd and there exists a constant 𝓒0 > 0 such that

    |g(t)|C01+|t|(N+2)/(N2),tR;

  • (G2)

    g(t) = o(t) as t → 0 and g(t) = o(t(N+2)/(N–2)) as t → +∞;

  • (G3)

    there exists s0 > 0 such that G(s0)>12Vs02, where G(t) = 0tg(s)ds.

To prove the above result, Berestycki-Lions [4] considered the following constrained minimization problem

minu22:uS,(1.9)

where

S=uH1(RN):RNG(u)12Vu2dx=1;(1.10)

they first showed that by the Pólya-Szegö inequality for the Schwarz symmetrization, the minimum can be taken on radial and radially nonincreasing functions. Then they showed the existence of a minimum ŵH1(ℝN) by the direct method of the calculus of variations. With the Lagrange multiplier Theorem, they concluded that ū(x) := ŵ(x/tŵ) with tw^=N22Nw^2 is a least energy solution of (1.8). By noting the one-to-one correspondence between 𝓢 and

PG:=uH1(RN){0}:N22u22+NV2u22NRNG(u)dx=0,

Jeanjean-Tanaka [13] proved that ū minimizes the value of the energy functional on the Pohožaev manifold for (1.8).

However, the approach of Berestycki-Lions [4] fails for nonlocal problem (1.5) due to the appearance of the nonlocal term. In [22], Moroz-Van Schaftingen proved firstly the existence of a least energy solution to (1.5) under (F1)-(F3). To do that, they employed a scaling technique introduced by Jeanjean [11] to construct a Palais-Smale sequence ((PS)-sequence in short) that satisfies asymptotically the Pohožaev identity (a Pohožaev-Palais-Smale sequence in short), where the information related to the Pohožaev identity helps to ensure the boundedness of (PS)-sequences, and then used a concentration compactness argument to overcome the difficulty caused by lack of Sobolev embeddings. Such an approach could be useful for the study of other problems where radial symmetry of solutions either fails or is not readily available. For more related results on nonlocal problems, we refer to [6, 17, 18, 26, 31, 38].

We would like to point out that the approach used in [22] is only valid for autonomous equations, it does not work any more for nonautonomous equation (1.2) with V ≠ constant, since one could not construct a Pohožaev-Palais-Smale sequence as Moroz-Van Schaftingen did in [22]. Thus new techniques are required for the study of the nonautonomous equation (1.2) with f satisfying (F1)-(F3), which is another focus of this paper.

In view of [22, Theorem 3], every solution u of (1.5) satisfies the following Pohožaev type identity:

P(u):=N22u22+NV2u22N+α2RN(IαF(u))F(u)dx=0.(1.11)

Therefore, the following set

M:=uH1(RN){0}:P(u)=0

is a natural constraint for the functional 𝓘. Moreover, the least energy solution u0 obtained in [22] satisfies 𝓘(u0) ≥ inf𝓜𝓘. A natural question is whether there exists a solution ū ∈ 𝓜 such that

J(u¯)=infMJ.(1.12)

In the first part of this paper, motivated by [4, 7, 13, 22, 33, 35], we shall develop a more direct approach to obtain a ground state solution for (1.2) which has minimal “energy” 𝓘 in the set of all nontrivial solutions, moreover, this solution also minimizes the value of 𝓘 on the Pohožaev manifold associated with (1.2), under (F1)-(F3), (V1) and the following two additional conditions on V:

  • (V2)

    V(x) ≤ V for all x ∈ ℝN;

  • (V3)

    V ∈ 𝓒1(ℝN, ℝ) and there exists θ ∈ [0, 1) such that tNV(tx)+V(tx)(tx)tα+(N2)3θ4tα+2|x|2 is nonincreasing on (0, ∞) for all x ∈ ℝN ∖ {0}.

To state our first result, we define a functional on H1(ℝN) as follows:

P(u):=N22u22+12RN[NV(x)+V(x)x]u2dxN+α2RN(IαF(u))F(u)dx,(1.13)

which is associated with the Pohožaev identity 𝓟(u) = 0 of (1.2), see Lemma 3.2. Let

M:=uH1(RN){0}:P(u)=0.(1.14)

Our first result is as follows.

Theorem 1.1

Assume that V and f satisfy (V1)-(V3) and (F1)-(F3). Then problem (1.2) has a solution ūH1(ℝN) such that 𝓘(ū) = inf𝓜𝓘 = infuΛ maxt>0 𝓘(ut) > 0, where

ut(x):=u(x/t)andΛ:=uH1(RN):RN(IαF(u))F(u)dx>0.

Corollary 1.2

Assume that f satisfies (F1)-(F3). Then problem (1.5) has a solution ūH1(ℝN) such that 𝓘(ū) = inf𝓜 𝓘 = infuΛ maxt>0 𝓘(ut) > 0.

With the help of the Pohožaev type identity (1.11) established in [22], we easily prove that the solution ū obtained in Corollary 1.2 is also the least energy solution for (1.5). More precisely, we have the following theorem:

Theorem 1.3

Assume that f satisfies (F1)-(F3). Then problem (1.5) has a solution ūH1(ℝN) such that

J(u¯)=infMJ=infJ(u):uH1(RN){0}isasolutionof(1.5).

Remark 1.4

(V3) is a mild condition. In fact, V satisfies (V3) if the following assumption holds:

(V3′) V ∈ 𝓒1(ℝN, ℝ) and tNV(tx)+V(tx)(tx)tα is nonincreasing on(0, ∞) for all x ∈ ℝN.

There are indeed many functions which satisfy (V1)-(V3). For example

  1. V(x) = ab|x|2+1 with a > b and αNa + (α + 2)(N – 2)3 > [(N – 2)(α + 2) + 2(α + 4)]b > 0;

  2. V(x) = ab|x|α+1 with a ≥ (2 + α/N)b > 0;

  3. V(x) = abe|x|α with a > b > 0.

Remark 1.5

We point out that, as a consequence of Theorem 1.1, the least energy value m := inf𝓜𝓘 has a minimax characterization m = infuΛ maxt>0𝓘(ut) which is much simpler than the usual characterizations related to the Mountain Pass level.

In the second part of this paper, we are interested in the existence of the least energy solutions for (1.2) under (F1)-(F3). In this case, we can replace (V3) by the following weaker decay assumption on ∇V:

(V4) V ∈ 𝓒1(ℝN, ℝ) and there exist θ′ ∈ (0, 1) and ≥ 0 such that

V(x)x(N2)22|x|2,0<|x|<R¯;θαV(x),|x|R¯.

In this direction, we have the following theorem.

Theorem 1.6

Assume that V and f satisfy (V1), (V2), (V4) and (F1)-(F3). Then problem (1.2) has a solution ūH1(ℝN) such that 𝓘(ū) = inf𝓚𝓘, where

K:=uH1(RN){0}:J(u)=0.

Remark 1.7

(V1), (V2) and (V4) are satisfied by a very wide class of potentials. For example, V(x)=ab1+|x|β satisfies (V1) and (V4) for β > 0 and αa > (α + β)b > 0.

Applying Theorem 1.6 to the following perturbed problem:

u+[Vεh(x)]u=(IαF(u))f(u),xRN;uH1(RN),(1.15)

where V is a positive constant and the function h ∈ 𝓒1(ℝN, ℝ) verifies:

  • (H1)

    h(x) ≥ 0 for all x ∈ ℝN and lim|x|→∞ h(x) = 0;

  • (H2)

    supx∈ℝN[–∇h(x) ⋅ x] < ∞.

    Then we have the following corollary.

Corollary 1.8

Assume that h and f satisfy (H1), (H2) and (F1)-(F3). Then there exists a constant ε̂ > 0 such that problem (1.15) has a least energy solution ūεH1(ℝN) ∖ {0} for all 0 < εε̂.

In the last part of the present paper, we consider the singularly perturbed nonlinear Choquard equation (1.1), and prove the existence of semiclassical ground state solutions for (1.1) under weaker assumptions on V:

  • (V5)

    0 < V(x0) := minx∈ℝN V(x) < V for some x0 ∈ ℝN;

  • (V6)

    V ∈ 𝓒1(ℝN, ℝ) and there exists θ″ ∈ (0, 1) such that

    V(x)xθαV(x),xRN.

Condition (V5) was introduced by Rabinowitz in [28]. Our last result is as follows.

Theorem 1.9

Assume that V and f satisfy (V5), (V6) and (F1)-(F3). Then there exists a constant ε0 > 0 determined by terms of N, V and F (see Lemma 4.2) such that problem (1.1) has a least energy solution ūεH1(ℝN) ∖ {0} for all 0 < εε0.

Remark 1.10

(V5) is weaker than (V2). There are many functions that satisfy (V5) and (V6) but do not satisfy (V2). For example, V(x) = abcos|x|β1+|x|β satisfies (V1), (V5) and (V6) for β > 0 and αa > (α + β)b > 0.

Remark 1.11

Our approach could be applied to deal with many equations, such as Schrödinger equations, see [7]. In the existing literature, Schrödinger equations were considered by many authors (for example [4, 5, 10, 11, 13]).

To prove Theorem 1.1, we shall divide our arguments into three steps: i). Choosing a minimizing sequence {un} of 𝓘 on 𝓜, which satisfies

J(un)m:=infMJ,P(un)=0.(1.16)

Then showing that {un} is bounded in H1(ℝN). ii). With a concentration-compactness argument and “the least energy squeeze approach”, showing that {un} converges weakly to some ūH1(ℝN) ∖ {0}. And then showing that ū ∈ 𝓜 and 𝓘(ū) = inf𝓜𝓘. iii). Showing that ū is a critical point of 𝓘. Of them, Step ii) is the most difficult due to lack of global compactness and adequate information on 𝓘′(un). To avoid relying radial compactness, we establish a crucial inequality related to 𝓘(u), 𝓘(ut) and 𝓟(u) (Lemma 2.2), it plays a crucial role in our arguments, see Lemmas 2.7, 2.11, 2.13, 3.5, 4.2. With the help of this inequality, we then can complete Step ii) by using Lions’ concentration compactness, the least energy squeeze approach and some subtle analysis. Moreover, such an approach could be useful for the study of other problems where radial symmetry of bounded sequence either fails or is not readily available.

Classically, in order to show the existence of solutions for (1.2), one compares the critical level with the one of (1.5) (i.e. the problem at infinity). To this end, it is necessary to establish a strict inequality similar to

maxt[0,1]J(y0(t))<infJ(u):uH1(RN){0}is a solution of(1.5)

for some path y0 ∈ 𝓒([0, 1], H1(ℝN)). Clearly, y0(t) > 0 is a natural requirement under (V1), which usually involves an additional assumption on f besides (F1)-(F3), such as f(t) is odd and f(t)t ≥ 0, see [22, Theorem 1.4]. We would like to point out that the above strict inequality is not used in our arguments, see Section 2. Our approach could be useful for the study of other problems where paths or the ground state solutions of the problem at infinity are not sign definite.

To prove Theorem 1.6, as in Jeanjean-Tanaka [13], for λ ∈ [1/2, 1] we consider the family of functionals 𝓘λ : H1(ℝN) → ℝ defined by

Jλ(u)=12RN|u|2+V(x)u2dxλ2RN(IαF(u))F(u)dx.(1.17)

These functionals have a Mountain Pass geometry, and denoting cλ the corresponding Mountain Pass levels. Corresponding to (1.17), we also let

Jλ(u)=12RN|u|2+Vu2dxλ2RN(IαF(u))F(u)dx.(1.18)

By Corollary 1.2, for every λ ∈ [1/2, 1], there exists a minimizer uλofJλonMλ, where

Mλ:=uH1(RN){0}:Pλ(u)=0(1.19)

and

Pλ(u)=N22u22+NVu22(N+α)λ2RN(IαF(u))F(u)dx.(1.20)

Let

A(u)=12RN|u|2+V(x)u2dx,B(u)=12RN(IαF(u))F(u)dx.

Then 𝓘λ(u) = A(u) – λB(u). Since B(u) is not sign definite, it prevents us from employing Jeanjean’s monotonicity trick [12]. More trouble, it is difficult to show the following key inequality

cλ<mλ:=infuMλJλ(u)=Jλ(uλ),λ[1/2,1](1.21)

due to the minimizer uλ being not positive definite.

Thanks to the work of Jeanjean-Toland [15], 𝓘λ still has a bounded (PS)-sequence {un(λ)} ⊂ H1(ℝN) at level cλ for almost every λ ∈ [1/2, 1]. Different from the arguments in the existing literature, by means of u1 and the key inequality established in Lemma 2.2, we can find a constant λ̄ ∈ [1/2, 1) and then prove directly the following inequality

cλ<mλ,λ(λ¯,1],(1.22)

see Lemma 3.5. In particular, it is not require any information on sign of u1 in our arguments. Applying (1.22) and a precise decomposition of bounded (PS)-sequences in [13], we can get a nontrivial critical point uλ of 𝓘λ which possesses energy cλ for almost every λ ∈ [λ̄, 1]. Finally, with a Pohožaev identity we proved that (1.2) admits a least energy solution under (V1), (V2), (V4) and (F1)-(F3).

Throughout the paper we make use of the following notations:

  • H1(ℝN) denotes the usual Sobolev space equipped with the inner product and norm

    (u,v)=RN(uv+uv)dx,|u=(u,u)1/2,u,vH1(RN);

  • Ls(ℝN) (1 ≤ s < ∞) denotes the Lebesgue space with the norm |u|s = (∫N|u|s dx)1/s;

  • For any uH1(ℝN) ∖ {0}, ut(x) := u(t–1x) for t > 0;

  • For any x ∈ ℝN and r > 0, Br(x) := {y ∈ ℝN : |yx| < r};

  • C1, C2, ⋯ denote positive constants possibly different in different places.

The rest of the paper is organized as follows. In Section 2, we give some preliminaries, and give the proofs of Theorems 1.1 and 1.3. Section 3 is devoted to finding a least energy solution for (1.2) and Theorem 1.6 will be proved in this section. In the last section, we show the existence of semiclassical ground state solutions for (1.1) and prove Theorem 1.9.

2 Ground state solutions for (1.2)

In this section, we give the proofs of Theorems 1.1 and 1.3. To this end, we give some useful lemmas. Since V(x) ≡ V satisfies (V1)-(V3), thus all conclusions on 𝓘 are also true for 𝓘. For (1.5), we always assume that V > 0. First, by a simple calculation, we can verify Lemma 2.1.

Lemma 2.1

The following two inequalities hold:

g(t):=2+α(N+α)tN2+(N2)tN+α>g(1)=0,t[0,1)(1,+),(2.1)

h(t):=α(N+α)tN+NtN+α>h(1)=0,t[0,1)(1,+).(2.2)

Moreover (V3) implies the following inequality holds:

α+NtN+αV(x)(N+α)tNV(tx)+tN+α1V(x)x(N2)2θ2+α(N+α)tN2+(N2)tN+α4|x|2,t0,xRN{0}.(2.3)

Lemma 2.2

Assume that (V1)-(V3), (F1) and (F2) hold. Then

J(u)J(ut)+1tN+αN+αP(u)+(1θ)g(t)2(N+α)u22,uH1(RN),t>0.(2.4)

Proof

According to Hardy inequality, we have

u22(N2)24RNu2|x|2dx,uH1(RN).(2.5)

Note that

J(ut)=tN22u22+tN2RNV(tx)u2dxtN+α2RN(IαF(u))F(u)dx.(2.6)

Thus, by (1.4), (1.13), (2.1), (2.3), (2.5) and (2.6), one has

J(u)J(ut)=1tN22u22+12RNV(x)tNV(tx)u2dx1tN+α2RN(IαF(u))F(u)dx=1tN+αN+αN22u22+12RN[NV(x)+V(x)x]u2dxN+α2RN(IαF(u))F(u)dx+2+α(N+α)tN2+(N2)tN+α2(N+α)u22+12RNα+NtN+αN+αV(x)tNV(tx)1tN+αN+αV(x)xu2dx1tN+αN+αP(u)+(1θ)g(t)2(N+α)u22.

This shows that (2.4) holds.□

From Lemma 2.2, we have the following two corollaries.

Corollary 2.3

Assume that (F1) and (F2) hold. Then

J(u)=J(ut)+1tN+αN+αP(u)+g(t)u22+Vh(t)u222(N+α),uH1(RN),t>0.(2.7)

Corollary 2.4

Assume that (V1)-(V3), (F1) and (F2) hold. Then for u ∈ 𝓜

J(u)=maxt>0J(ut).(2.8)

Lemma 2.5

Assume that (V1)-(V3) hold. Then there exist two constants y1, y2 > 0 such that

y1u2(N2)u22+RNNV(x)+V(x)xu2dxy2u2,uH1(RN).(2.9)

Proof

Let t = 0 and t → ∞ in (2.3), respectively, and using (V1), (V2), one has

V(x)xαV+(N2)2(2+α)θ4|x|2,xRN{0},(2.10)

NV(N2)3θ4|x|2NV(x)(N2)3θ4|x|2V(x)x,xRN{0}.(2.11)

By (2.10), (2.11) and V ∈ 𝓒1(ℝN, ℝ), there exists a constant M0 > 0 such that

|V(x)x|M0,xRN.(2.12)

From (2.3), one has

NV(x)+V(x)x(N2)3θ4|x|2+(N+α)tαV(tx)(N2)2(2+α)θ4|x|2V(x)x+αV(x)tNα,t>0,xRN{0}.(2.13)

By (V1), there exists R > 0 such that V(x)V2 for all |x| ≥ R and

(N2)2(2+α)θ4+M0+αVRN<(N+α)V4.(2.14)

It follows from (V1), (V2), (2.12), (2.13) and (2.14) that

NV(x)+V(x)x(N2)3θ4|x|2+(N+α)RαV(Rx)(N2)2(2+α)θ4|x|2V(x)x+αV(x)RNα(N2)3θ4|x|2+(N+α)RαV4,|x|1.(2.15)

Making use of the Hölder inequality and the Sobolev inequality, we get

|x|<1u2dxωN(22)/2|x|<1|u|2dx2/2ωN2/NS1u22,(2.16)

where ωN denotes the volume of the unit ball of ℝN. Thus it follows from (2.5), (2.10), (2.11), (2.15) and (2.16) that

(N2)u22+RNNV(x)+V(x)xu2dx[N2+(2+α)θ]u22+(N+α)Vu22[N2+(2+α)θ+(N+α)V]u2:=y2u2,uH1(RN)(2.17)

and

(N2)u22+RNNV(x)+V(x)xu2dx=(N2)u22+|x|<1NV(x)+V(x)xu2dx+|x|1NV(x)+V(x)xu2dx(N2)u22(N2)3θ4RNu2|x|2dx+(N+α)RαV4|x|1u2dx(1θ)(N2)u22+(N+α)RαV4|x|1u2dx(1θ)(N2)2u22+(1θ)(N2)S2ωN2/N|x|<1u2dx+(N+α)RαV4|x|1u2dx(1θ)(N2)2u22+min(1θ)(N2)S2ωN2/N,(N+α)RαV4u22min(1θ)(N2)2,(1θ)(N2)S2ωN2/N,(N+α)RαV4u2:=y1u2,uH1(RN).(2.18)

Both (2.17) and (2.18) imply that (2.9) holds.□

To show 𝓜 ≠ ∅, we define a set Λ as follows:

Λ=uH1(RN):RN(IαF(u))F(u)dx>0.(2.19)

Lemma 2.6

Assume that (V1)-(V3) and (F1)-(F3) hold. Then Λ ≠ ∅ and

uH1(RN){0}:P(u)0orP(u)0Λ.(2.20)

Proof

In view of the proof of [22, The proof of Claim 1 in Proposition 2.1], (F3) implies Λ ≠ ∅. Next, we have two cases to distinguish:

  1. uH1(ℝN) ∖ {0} and 𝓟(u) ≤ 0, then (1.11) implies uΛ.

  2. uH1(ℝN) ∖ {0} and 𝓟(u) ≤ 0, then it follows from (1.13), (2.5) and (2.11) that

    N+α2RN(IαF(u))F(u)dx=P(u)N22u2212RNNV(x)+V(x)xu2dxN22u22+(N2)3θ8RNu2|x|2dx(1θ)(N2)2u22<0,

    which implies uΛ.□

Lemma 2.7

Assume that (V1)-(V3) and (F1)-(F3) hold. Then for any uΛ, there exists a unique tu > 0 such that utu ∈ 𝓜.

Proof

Let uΛ be fixed and define a function ζ(t) := 𝓘(ut) on (0, ∞). Clearly, by (1.13) and (2.6), we have

ζ(t)=0N22tN2u22+tN2RN[NV(tx)+V(tx)(tx)]u2dx(N+α)tN+α2RN(IαF(u))F(u)dx=0P(ut)=0utM.(2.21)

It is easy to verify, using (V1), (V2), (F1), (2.6) and the definition of Λ, that limt→0 ζ(t) = 0, ζ(t) > 0 for t > 0 small and ζ(t) < 0 for t large. Therefore maxt∈(0,∞) ζ(t) is achieved at tu > 0 so that ζ′(tu) = 0 and utu ∈ 𝓜.

Next we claim that tu is unique for any uΛ. In fact, for any given uΛ, let t1, t2 > 0 such that ut1, ut2 ∈ 𝓜. Then 𝓟(ut1) = 𝓟(ut2) = 0. Jointly with (2.4), we have

Jut1Jut2+t1N+αt2N+α(N+α)t1N+αPut1+(1θ)g(t2/t1)2(N+α)ut122=Jut2+(1θ)t1N2g(t2/t1)2(N+α)u22(2.22)

and

Jut2Jut1+t2N+αt1N+α(N+α)t2N+αPut2+(1θ)g(t1/t2)2(N+α)ut222=Jut1+(1θ)t2N2g(t1/t2)2(N+α)u22.(2.23)

(2.1), (2.22) and (2.23) imply t1 = t2. Therefore, tu > 0 is unique for any uΛ.□

Corollary 2.8

Assume that (F1)-(F3) hold. Then for any uΛ, there exists a unique tu > 0 such that utu ∈ 𝓜.

From Corollary 2.4, Lemmas 2.6, 2.7 and Corollary 2.8, we have 𝓜 ≠ ∅, 𝓜 ≠ ∅ and the following lemma.

Lemma 2.9

Assume that (V1)-(V3) and (F1)-(F3 hold. Then

infuMJ(u):=m=infuΛmaxt>0J(ut).

The following lemma is a known result which can be proved by a standard argument(see [32]).

Lemma 2.10

Assume that (V1), (F1) and (F2) hold. If unū in H1(ℝN), then

J(un)=J(u¯)+J(unu¯)+o(1)(2.24)

and

P(un)=P(u¯)+P(unu¯)+o(1).(2.25)

Lemma 2.11

Assume that (V1)-(V3) and (F1)-(F3) hold. Then

  1. there exists ρ0 > 0 such that |u| ≥ ρ0, ∀ u ∈ 𝓜;

  2. m = infu∈𝓜 𝓘(u) > 0.

Proof

(i). Since 𝓟(u) = 0 for all u ∈ 𝓜, by (1.3), (1.13), (2.9) and Sobolev embedding theorem, one has

y12u2N22u22+12RN[NV(x)+V(x)x]u2dx=N+α2RN(IαF(u))F(u)dxu2(N+α)/N+C1u2(N+α)/(N2),(2.26)

which implies

uρ0:=min1,y12(1+C1)N/2α,uM.(2.27)

(ii). Let {un} ⊂ 𝓜 be such that 𝓘(un) → m. There are two possible cases:

1) infn∈ℕ ∥∇un2 > 0 and 2) infn∈ℕ ∥∇un2 = 0.

  • Case 1)

    infn∈ℕ ∥∇un2 := ϱ0 > 0. In this case, from (2.4) with t → 0, we have

    m+o(1)=J(un)(1θ)(2+α)2(N+α)un22(1θ)(2+α)2(N+α)ϱ02.

  • Case 2)

    infn∈ℕ ∥∇ un2 = 0. In this case, by (2.27), passing to a subsequence, one has

    un20,un212ρ0.(2.28)

    By (1.3) and the Sobolev inequality, one has for all uH1(ℝN),

    RN(IαF(u))F(u)dxC2u22(N+α)/N+u22(N+α)/(N2)C2u22(N+α)/N+S(N+α)/(N2)u22(N+α)/(N2).(2.29)

    By (V1), there exists R > 0 such that V(x)V2 for |x| ≥ R. This implies

    |tx|RV(tx)u2dxV2|tx|Ru2dx,t>0,uH1(RN).(2.30)

    Making use of the Hölder inequality and the Sobolev inequality, we get

    |tx|<Ru2dxωNRNtN(22)/2|tx|<Ru2dx2/2ωN2/NR2t2S1u22,t>0,uH1(RN).(2.31)

    Let

    δ0=minV,SR2ωN2/N(2.32)

    and

    tn=δ04C21/αun22/N.(2.33)

    Then (2.28) implies {tn} is bounded. Thus it follows from (2.6), (2.8), (2.28), (2.29), (2.30), (2.31), (2.32) and (2.33) that

    m+o(1)=J(un)J(un)tn=tnN22un22+tnN2RNV(tnx)un2dxtnN+α2RN(IαF(un))F(un)dxS2R2ωN2/NtnN|tnx|<Run2dx+14VtnN|tnx|Run2dx12C2tnN+αun22(N+α)/NC22S(N+α)/(N2)tnN+αun22(N+α)/(N2)14δ0tnNun2212C2tnN+αun22(N+α)/N+o(1)=14tnNun22δ02C2tnαun22α/N+o(1)=δ08δ04C2N/α+o(1).(2.34)

    Cases 1) and 2) show that m = infu∈𝓜 𝓘(u) > 0.□

Lemma 2.12

Assume that (V1)-(V3) and (F1)-(F3) hold. Then mm.

Proof

Arguing indirectly, we assume that m > m. Let ε := mm. Then there exists uε such that

uεMandm+ε2>J(uε).(2.35)

In view of Lemmas 2.6 and 2.7, there exists tε > 0 such that (uε)tεM. Thus, it follows from (V1), (V2), (1.4), (1.6), (2.7) and (2.35) that

m+ε2>J(uε)J(uε)tεJ(uε)tεm.

This contradiction shows the conclusion of Lemma 2.12 is true.□

Lemma 2.13

Assume that (V1)-(V3) and (F1)-(F3) hold. Then m is achieved.

Proof

In view of Lemma 2.11, we have m > 0. Let {un} ⊂ 𝓜 be such that 𝓘(un) → m. Since 𝓟(un) = 0, then it follows from (2.4) with t → 0 that

m+o(1)=J(un)(1θ)(2+α)2(N+α)un22.(2.36)

This shows that {∥∇ un2} is bounded. Next, we prove that {∥un2} is also bounded. Arguing by contradiction, suppose that ∥un2 → ∞. By (1.3) and the Sobolev inequality, one has

RN(IαF(u))F(u)dxδ04δ016mα/Nu22(N+α)/N+C3u22(N+α)/(N2)δ04δ016mα/Nu22(N+α)/N+C3S(N+α)/(N2)u22(N+α)/(N2),uH1(RN),(2.37)

where δ0 is given by (2.32). Let

t^n=16mδ01/Nun22/N.(2.38)

Then n → 0. Thus it follows from (2.6), (2.8), (2.30), (2.31), (2.32), (2.37) and (2.38) that

m+o(1)=J(un)J(un)t^n=t^nN22un22+t^nN2RNV(t^nx)un2dxt^nN+α2RN(IαF(un))F(un)dxS2R2ωN2/Nt^nN|t^nx|<Run2dx+14Vt^nN|t^nx|Run2dx18δ0δ016mα/Nt^nN+αun22(N+α)/NC32S(N+α)/(N2)t^nN+αun22(N+α)/(N2)14δ0t^nNun22δ08δ016mα/Nt^nN+αun22(N+α)/N+o(1)=δ04t^nNun22112δ0t^nNun2216mα/N+o(1)=2m+o(1).(2.39)

This contradiction shows that {∥un2} is also bounded. Hence, {un} is bounded in H1(ℝN). Passing to a subsequence, we have unū in H1(ℝN). Then unū in Llocs(ℝN) for 2 ≤ s < 2* and unū a.e. in ℝN. There are two possible cases: i). ū = 0 and ii). ū ≠ 0.

  • Case i)

    ū = 0, i.e. un ⇀ 0 in H1(ℝN). Then un → 0 in Llocs(ℝN) for 2 ≤ s < 2* and un → 0 a.e. in ℝN. By (V1) and (V3), it is easy to show that (for a detailed proof, see Lemma 2.2 of [34])

    limnRN[VV(x)]un2dx=limnRNV(x)xun2dx=0.(2.40)

    From (1.4), (1.6), (1.11), (1.13) and (2.40), one can get

    J(un)m,P(un)0.(2.41)

    From Lemma 2.11 (i), (1.11) and (2.41), one has

    min{N2,NV}ρ02min{N2,NV}un2(N2)un22+NVun22=(N+α)RN(IαF(un))F(un)dx+o(1).(2.42)

    Using (1.3), (2.42) and Lions’ concentration compactness principle [39, Lemma 1.21], we can prove that there exist δ > 0 and a sequence {yn} ⊂ ℝN such that ∫B1(yn)|un|2 dx > δ. Let ûn(x) = un(x + yn). Then we have ∥ûn∥ = ∥un∥ and

    J(u^n)m,P(u^n)=o(1)0,B1(0)|u^n|2dx>δ.(2.43)

    Therefore, there exists ûH1(ℝN) ∖ {0} such that, passing to a subsequence,

    u^nu^,inH1(RN);u^nu^,inLlocs(RN),s[1,2);u^nu^,a.e. onRN.(2.44)

    Let wn = ûnû. Then (2.44) and Lemma 2.10 yield

    J(u^n)=J(u^)+J(wn)+o(1)(2.45)

    and

    P(u^n)=P(u^)+P(wn)+o(1).(2.46)

    Set

    Ψ0(u)=(2+α)u22+αVu222(N+α).(2.47)

    From (1.6), (1.11), (2.43), (2.45) and (2.46), one has

    Ψ0(wn)=mΨ0(u^)+o(1),P(wn)=P(u^)+o(1).(2.48)

    If there exists a subsequence {wni} of {wn} such that wni = 0, then going to this subsequence, we have

    J(u^)=m,P(u^)=0.(2.49)

    Next, we assume that wn ≠ 0. We claim that 𝓟(û) ≤ 0. Otherwise, if 𝓟(û) > 0, then (2.48) implies 𝓟(wn) < 0 for large n. In view of Lemma 2.6 and Corollary 2.8, there exists tn > 0 such that (wn)tn ∈ 𝓜 for large n. From (1.6), (1.11), (2.7), (2.47) and (2.48), we obtain

    mΨ0(u^)+o(1)=Ψ0(wn)=J(wn)1N+αP(wn)J(wn)tntnNN+αP(wn)mtnNN+αP(wn)m,

    which implies 𝓟(û) ≤ 0 due to mm and Ψ0(û) > 0. Since û ≠ 0 and 𝓟(û) ≤ 0, in view of Lemma 2.6 and Corollary 2.8, there exists > 0 such that û ∈ 𝓜. From (1.6), (1.11), (2.7), (2.47), (2.43) and the weak semicontinuity of norm, one has

    m=limnJ(u^n)1N+αP(u^n)=limnΨ0(u^n)Ψ0(u^)=J(u^)1N+αP(u^)Ju^t^t^NN+αP(u^)mt^NN+αP(u^)mt^NN+αP(u^)m,

    which implies (2.49) holds also. In view of Lemmas 2.6 and 2.7, there exists > 0 such that û ∈ 𝓜, moreover, it follows from (V1), (V2), (1.4), (1.6), (2.49) and Corollary 2.3 that

    mJ(u^t~)J(u^t~)J(u^)=m.

    This shows that m is achieved at û ∈ 𝓜.

  • Case ii)

    ū ≠ 0. Let vn = unū. Then Lemma 2.10 yields

    J(un)=J(u¯)+J(vn)+o(1)(2.50)

    and

    P(un)=P(u¯)+P(vn)+o(1).(2.51)

    Set

    Ψ(u)=2+α2(N+α)u22+12(N+α)RN[αV(x)(V(x),x)]u2dx.(2.52)

    Then it follows from (2.5) and (2.10) that

    (2+α)u22+RN[αV(x)(V(x),x)]u2dx(2+α)u22(2+α)(N2)2θ4RNu2|x|2dx(1θ)(2+α)u22,uH1(RN).(2.53)

    Since 𝓘(un) → m and 𝓟(un) = 0, then it follows from (1.4), (1.13), (2.50), (2.51) and (2.52) that

    Ψ(vn)=mΨ(u¯)+o(1),P(vn)=P(u¯)+o(1).(2.54)

    If there exists a subsequence {vni} of {vn} such that vni = 0, then going to this subsequence, we have

    J(u¯)=m,P(u¯)=0,(2.55)

    which implies the conclusion of Lemma 2.13 holds. Next, we assume that vn ≠ 0. We claim that 𝓟(ū) ≤ 0. Otherwise 𝓟(ū) > 0, then (2.54) implies 𝓟(vn) < 0 for large n. In view of Lemmas 2.6 and 2.7, there exists tn > 0 such that (vn)tn ∈ 𝓜 for large n. From (1.4), (1.13), (2.4), (2.52) and (2.54), we obtain

    mΨ(u¯)+o(1)=Ψ(vn)=J(vn)1N+αP(vn)J(vn)tntnNN+αP(vn)mtnNN+αP(vn)m,

    which implies 𝓟(ū) ≤ 0 due to Ψ(ū) > 0. Since ū ≠ 0 and 𝓟(ū) ≤ 0, in view of Lemmas 2.6 and 2.7, there exists > 0 such that ū ∈ 𝓜. From (1.4), (1.13), (2.4), (2.52), (2.53) and the weak semicontinuity of norm, one has

    m=limnJ(un)1N+αP(un)=limnΨ(un)Ψ(u¯)=J(u¯)1N+αP(u¯)Ju¯t¯t¯NN+αP(u¯)mt¯NN+αP(u¯)m,

    which implies (2.55) also holds. □

Lemma 2.14

Assume that (V1)-(V3) and (F1)-(F3) hold. If ū ∈ 𝓜 and 𝓘(ū) = m, then ū is a critical point of 𝓘.

Proof

Similar to the proof of [7, Lemma 2.12], we can conclude above conclusion by using

Ju¯tJ(u¯)(1θ)g(t)2(N+α)u¯22=m(1θ)g(t)2(N+α)u¯22,t>0(2.56)

and

ε:=min(1θ)g(0.5)5(N+α)u¯22,(1θ)g(1.5)5(N+α)u¯22,1,ϱδ8.

instead of [7, (2.35) and ε], respectively.

Proof of Theorem 1.1

In view of Lemmas 2.9, 2.13 and 2.14, there exists ū ∈ 𝓜 such that

J(u¯)=m=infuΛmaxt>0J(ut)>0,J(u¯)=0.

This shows that ū is a nontrivial solution of (1.2). □

Proof of Theorem 1.3

Let

K:=uH1(RN){0}:(J)(u)=0,m^:=infuKJ(u).

On the one hand, in view of Corollary 1.2, there exists ū ∈ 𝓜 such that 𝓘(ū) = m and (𝓘)′(ū) = 0. This shows that 𝓚 ≠ ∅ and m. On the other hand, if w ∈ 𝓚, then it follows from (1.11) (i.e. [22, Theorem 3]) that w ∈ 𝓜. Thus, 𝓘(w) ≥ m for all w ∈ 𝓚, which yields that m. Therefore, = m.

3 The least energy solutions for (1.2)

In this section, we give the proof of Theorem 1.6.

Proposition 3.1

[15] Let X be a Banach space and let J ⊂ ℝ+ be an interval, and

Φλ(u)=A(u)λB(u),λJ,

be a family of 𝓒1-functional on X such that

  1. either A(u) → +∞ or B(u) → +∞, asu∥ → ∞;

  2. B maps every bounded set of X into a set ofbounded below;

  3. there are two points v1, v2 in X such that

    c~λ:=infyΓ~maxt[0,1]Φλ(y(t))>max{Φλ(v1),Φλ(v2)},(3.1)

where

Γ~=yC([0,1],X):y(0)=v1,y(1)=v2.

Then, for almost every λJ, there exists a sequence {un(λ)} such that

  1. {un(λ)} is bounded in X;

  2. Φλ(un(λ)) → cλ;

  3. Φλ(un(λ)) → 0 in X*, where X* is the dual of X.

Similar to the proof of [22, Theorem 3], we can prove the following lemma.

Lemma 3.2

Assume that (V1), (F1) and (F2) hold. Let u be a critical point of 𝓘λ in H1(ℝN), then we have the following Pohožaev type identity

Pλ(u):=N22u22+12RNNV(x)+V(x)xu2dx(N+α)λ2RN(IαF(u))F(u)dx=0.(3.2)

By Corollary 2.3, we have the following lemma.

Lemma 3.3

Assume that (F1) and (F2) hold. Then

Jλ(u)=Jλut+1tNNPλ(u)+g(t)u22+Vh(t)u222(N+α),uH1(RN),t>0,λ0.(3.3)

In view of Corollary 1.2, J1=J has a minimizer u10onM1=M, i.e.

u1M1,(J1)(u1)=0andm1=J1(u1),(3.4)

where mλ is defined by (1.21). Since (1.5) is autonomous, V ∈ 𝓒(ℝN, ℝ) and V(x) ≤ V but V(x) ≢ V, then there exist ∈ ℝN and > 0 such that

VV(x)>0,|u1(x)|>0a.e.|xx¯|r¯.(3.5)

By (V1), we have Vmax := maxx∈ℝN V(x) ∈ (0, ∞). Let

Jλ(u)=12RN|u|2+Vmaxu2dxλ2RN(IαF(u))F(u)dx.(3.6)

Then it follows from (2.6) and (3.4) that there exists T > 0 such that

I1/2(u1)t<0,tT.(3.7)

Lemma 3.4

Assume that (V1) and (F1)-(F3) hold. Then

  1. Jλ(u1)T<0 for all λ ∈ [0.5, 1];

  2. there exists a positive constant κ0 independent of λ such that for all λ ∈ [0.5, 1],

    cλ:=infyΓmaxt[0,1]Jλ(y(t))κ0>maxJλ(0),Jλ(u1)T,

    where

    Γ=yC([0,1],H1(RN)):y(0)=0,y(1)=(u1)T;(3.8)

    1. cλ is bounded for λ ∈ [0.5, 1];

    2. mλ is non-increasing on λ ∈ [0.5, 1];

    3. lim supλλ0 cλcλ0 for λ0 ∈ (0.5, 1].

Since mλ=Jλ(uλ)andRN(IαF(uλ))F(uλ)dx>0, then the proof of (i)-(iv) in Lemma 3.4 is standard, (v) can be proved similar to [12, Lemma 2.3], so we omit it.

Lemma 3.5

Assume that (V1), (V2) and (F1)-(F3) hold. Then there exists λ̄ ∈ [1/2, 1) such that cλ < mλ for λ ∈ (λ̄, 1].

Proof

It is easy to see that Jλ(u1)t is continuous on t ∈ (0, ∞). Hence for any λ ∈ [1/2, 1], we can choose tλ ∈ (0, T) such that Jλ(u1)tλ=maxt[0,T]Jλ(u1)t. Setting

y0(t)=(u1)(tT),fort>0,0,fort=0.(3.9)

Then y0Γ defined by Lemma 3.4 (ii). Moreover

Jλ(u1)tλ=maxt[0,1]Jλy0(t)cλ.(3.10)

Since P(u1)=0, then RN(IαF(u1))F(u1)dx>0.

Let

ζ0:=min{3r¯/8(1+|x¯|),1/4}.(3.11)

Then it follows from (3.5) and (3.11) that

|xx¯|r¯2ands[1ζ0,1+ζ0]|sxx¯|r¯.(3.12)

Let

λ¯:=max12,1(1ζ0)Nmins[1ζ0,1+ζ0]RNVV(sx)|u1|2dxTN+αRN(IαF(u1))F(u1)dx,1min{g(1ζ0),g(1+ζ0)}u122+Vmin{h(1ζ0),h(1+ζ0)}u122(N+α)TN+αRN(IαF(u1))F(u1)dx.(3.13)

Then it follows from (2.1), (2.2), (3.5) and (3.12) that 1/2 ≤ λ̄ < 1. We have two cases to distinguish:

  • Case i)

    tλ ∈ [1 − ζ0, 1 + ζ0]. From (1.17), (1.18), (3.3)-(3.10), (3.12), (3.13) and Lemma 3.4 (iv), we have

    mλm1=J1(u1)J1(u1)tλ=Jλ(u1)tλ(1λ)tλN+α2RN(IαF(u1))F(u1)dx+tλN2RN[VV(tλx)]|u1|2dxcλ(1λ)TN+α2RN(IαF(u1))F(u1)dx+(1ζ0)N2mins[1ζ0,1+ζ0]RNVV(sx)|u1|2dx>cλ,λ(λ¯,1].

  • Case ii)

    tλ ∈ (0, 1 − ζ0) ∪ (1 + ζ0, T]. From (1.17), (1.18), (2.1), (2.2), (3.3), (3.4), (3.10), (3.13) and Lemma 3.4 (iv), we have

    mλm1=J1(u1)=J1(u1)tλ+g(tλ)u122+Vh(tλ)u1222(N+α)=Jλ(u1)tλ(1λ)tλN+α2RN(IαF(u1))F(u1)dx+tλN2RN[VV(tλx)]|u1|2dx+g(tλ)u122+Vh(tλ)u1222(N+α)cλ(1λ)TN+α2RN(IαF(u1))F(u1)dx+min{g(1ζ0),g(1+ζ0)}u122+Vmin{h(1ζ0),h(1+ζ0)}u1222(N+α)>cλ,λ(λ¯,1].

In both cases, we obtain that cλ < mλ for λ ∈ (λ̄, 1].

Lemma 3.6

[14] Assume that (V1) and (F1)-(F3) hold. Let {un} be a bounded (PS) - sequence for 𝓘λ, for λ ∈ [1/2, 1]. Then there exists a subsequence of {un}, still denoted by {un}, an integer l ∈ ℕ ∪ {0}, a sequence {ynk} and wkH1(ℝ3) for 1 ≤ kl, such that

  1. unu0 with Jλ(u0)=0;

  2. wk ≠ 0 and (Jλ)(wk)=0 for 1 ≤ kl;

  3. unu0k=1lwk(+ynk)0;

  4. Jλ(un)Jλ(u0)+i=1lJλ(wi);

where we agree that in the case l = 0 the above holds without wk.

Lemma 3.7

Assume that (V1) and (V4) hold. Then there exists y3 > 0 such that

(2+α)u22+RNαV(x)V(x)xu2dxy3u2,uH1(RN).(3.14)

Proof

From (V1), (V4) and (2.5), we have

(2+α)u22+RNαV(x)V(x)xu2dx=(2+α)u22(N2)22RNu2|x|2dx+RNαV(x)V(x)x+(N2)22|x|2u2dxαu22+(1θ)αRNV(x)u2dxy3u2

for some y3 > 0 due to (V1). □

Lemma 3.8

Assume that (V1), (V2), (V4) and (F1)-(F3) hold. Then for almost every λ ∈ (λ̄, 1], there exists uλH1(ℝN) ∖ {0} such that

Jλ(uλ)=0,Jλ(uλ)=cλ.(3.15)

Proof

Under (V1), (V2) and (F1)-(F3), Lemma 3.4 implies that 𝓘λ(u) satisfies the assumptions of Proposition 3.1 with X = H1(ℝN), Φλ = 𝓘λ and J = (λ̄, 1]. So for almost every λ ∈ (λ̄, 1], there exists a bounded sequence {un(λ)} ⊂ H1(ℝN) (for simplicity, we denote the sequence by {un} instead of {un(λ)}) such that

Jλ(un)cλ>0,Jλ(un)0.(3.16)

By Lemmas 3.2 and 3.6, there exist a subsequence of {un}, still denoted by {un}, uλH1(ℝN), an integer l ∈ ℕ ∪ {0}, and w1, …, wlH1(ℝN) ∖ {0} such that

unuλinH1(RN),Jλ(uλ)=0,(3.17)

(Jλ)(wk)=0,Jλ(wk)mλ,1kl(3.18)

and

cλ=Jλ(uλ)+k=1lJλ(wk).(3.19)

Since Jλ(uλ)=0, then it follows from Lemma 3.2 that

Pλ(uλ)=N22uλ22+12RNNV(x)+V(x)xuλ2dx(N+α)λ2RN(IαF(uλ))F(uλ)dx=0.(3.20)

Since ∥un∥ ↛ 0, we deduce from (3.18) and (3.19) that if uλ = 0 then l ≥ 1 and

cλ=Jλ(uλ)+k=1lJλ(wk)mλ,

which contradicts with Lemma 3.5. Thus uλ ≠ 0. It follows from (1.17), (3.14) and (3.20) that

Jλ(uλ)=Jλ(uλ)1N+αPλ(uλ)=2+α2(N+α)uλ22+12(N+α)RNαV(x)V(x)xuλ2dxy32(N+α)uλ2>0.(3.21)

From (3.19) and (3.21), one has

cλ=Jλ(uλ)+k=1lJλ(wk)>lmλ.(3.22)

By Lemma 3.5, we have cλ < mλ for λ ∈ (λ̄, 1], which, together with (3.22), implies that l = 0 and 𝓘λ(uλ) = cλ.

Lemma 3.9

Assume that (V1), (V2), (V4) and (F1)-(F3) hold. Then there exists ūH1(ℝN) ∖ {0} such that

J(u¯)=0,0<J(u¯)c1.(3.23)

Proof

In view of Lemmas 3.4 (iii) and 3.8, there exist two sequences {λn} ⊂ (λ̄, 1] and {uλn} ⊂ H1(ℝN) ∖ {0}, denoted by {un}, such that

λn1,cλnc,Jλn(un)=0,Jλn(un)=cλn.(3.24)

Then it follows from (3.24) and Lemma 3.2 that 𝓟λn(un) = 0. From (1.17), (3.14), (3.20), (3.24) and Lemma 3.4 (iii), one has

C4cλn=Jλn(un)1N+αPλn(un)=2+α2(N+α)un22+12(N+α)RNαV(x)V(x)xun2dxy32(N+α)un2.(3.25)

This shows that {∥un∥} is bounded in H1(ℝN). In view of Lemma 3.4 (v), we have limn→∞ cλn = c*c1. Hence, it follows from (1.17) and (3.24) that

J(un)c,J(un)0.(3.26)

This shows that {un} satisfies (3.16) with cλ = c*. In view of the proof of Lemma 3.8, we can show that there exists ūH1(ℝN) ∖ {0} such that (3.23) holds.

Proof of Theorem 1.6

Let := infu∈𝓚𝓘(u). Then Lemma 3.9 shows that 𝓚 ≠ ∅ and c1. For any u ∈ 𝓚, Lemma 3.2 implies 𝓟(u) = 𝓟1(u) = 0. Hence it follows from (3.21) that 𝓘(u) = 𝓘1(u) > 0 for all u ∈ 𝓚, and so ≥ 0. Let {un} ⊂ 𝓚 such that

J(un)=0,J(un)m^.(3.27)

In view of Lemma 3.5, c1 < m1. By a similar argument as in the proof of Lemma 3.8, we can prove that there exists ūH1(ℝN) ∖ {0} such that

J(u¯)=0,J(u¯)=m^.(3.28)

This shows that ū is a least energy solution of (1.2). □

4 Semiclassical states for (1.1)

In this section, we give the proof of Theorem 1.9. From now on we assume without loss of generality that x0 = 0, that is V(0) < V. Performing the scaling u(x) = v(ε x) one easily sees that problem (1.1) is equivalent to

u+Vε(x)u=(IαF(u))f(u),xRN;uH1(RN),(4.1)

where Vε(x) = V(ε x). The energy functional associated to problem (4.1) is given by

Jε(u)=12RN|u|2+Vε(x)u2dx12RN(IαF(u))F(u)dx.(4.2)

As in Section 3, we also define, for λ ∈ [1/2, 1] and ε ≥ 0, the family of functionals Jλε : H1(ℝN) → ℝ as follows

Jλε(u)=12RN|u|2+Vε(x)u2dxλ2RN(IαF(u))F(u)dx.(4.3)

Since V ∈ 𝓒(ℝN, ℝ), V(0) < V and u1H1(ℝN) ∖ {0}, then there exist > 0 and R0 > 0 such that

VV(x)>14VV(0),|x|r^,(4.4)

VV(0)+43NVmaxV|x|>R0|u1|2dx12VV(0)u122(4.5)

and

TNVmaxV|x|>R0|u1|2dxmin{g(1/2),g(3/2)}u122+Vmin{h(1/2),h(3/2)}u1222(N+α).(4.6)

Similar to Lemma 3.4, we can prove the following lemma.

Lemma 4.1

Assume that (V1) and (F1)-(F3) hold. Then

  1. Jλε(u1)T<0 for all λ ∈ [0.5, 1] and ε ≥ 0;

  2. there exists a positive constant κ̂0 independent of λ and ε ≥ 0 such that for all λ ∈ [0.5, 1] and ε ≥ 0,

    cλε:=infyΓmaxt[0,1]Jλε(y(t))κ^0>maxJλε(0),Jλε(u1)T,

    where Γ is defined by (3.8);

  3. cλε is bounded for λ ∈ [0.5, 1] and ε ≥ 0.

Lemma 4.2

Assume that (V1), (V5) and (F1)-(F3) hold. Then there exists λ̃ ∈ [1/2, 1) such that cλε<mλ for λ ∈ (λ̃, 1] and ε ∈ [0, ε0], where and in the sequel ε0 := r̂/R0 T.

Proof

For any ε ≥ 0, it is easy to see that Jλε(u1)t is continuous on t ∈ (0, ∞). Hence for any λ ∈ [1/2, 1] and ε ≥ 0, we can choose tλε ∈ (0, T) such that Jλε(u1)tλε=maxt[0,T]Jλε(u1)t. Setting y0(t) as in (3.9). Then y0Γ defined by (3.8). Moreover

Jλε(u1)tλε=maxt[0,1]Jλεy0(t)cλε.(4.7)

Since P(u1)=0,thenRN(IαF(u1))F(u1)dx>0. Let

λ~:=min12,1VV(0)u12283NRN(IαF(u1))F(u1)dx,1min{g(1/2),g(3/2)}u122+Vmin{h(1/2),h(3/2)}u1222(N+α)TNRN(IαF(u1))F(u1)dx.(4.8)

Then it follows from (2.1), (2.2) and (V5) that 1/2 ≤ λ̃ < 1. We have two cases to distinguish:

  • Case i)

    tλε ∈ [1/2, 3/2]. From (1.18), (3.3), (4.3)-(4.8) and Lemma 3.4 (iv), we have

    mλm1=J1(u1)J1(u1)tλε=Jλε(u1)tλε(1λ)(tλε)N2RN(IαF(u1))F(u1)dx+(tλε)N2RN[VVε(tλεx)]|u1|2dxcλε3N(1λ)2N+1RN(IαF(u1))F(u1)dx+VV(0)2N+3|x|R0|u1|2dx3NVmaxV2N+1|x|>R0|u1|2dx=cλε3N(1λ)2N+1RN(IαF(u1))F(u1)dx+VV(0)2N+3u122VV(0)+43NVmaxV2N+3|x|>R0|u1|2dxcλε3N(1λ)2N+1RN(IαF(u1))F(u1)dx+VV(0)2N+4u122>cλε,λ(λ~,1],ε[0,ε0].

  • Case ii)

    tλε ∈ (0, 1/2) ∪ (3/2, T). From (1.18), (3.3), (4.3)-(4.8) and Lemma 3.4 (iv), we have

    mλm1=J1(u1)J1(u1)tλε+g(tλε)u122+Vh(tλε)u1222(N+α)=Jλε(u1)tλε(1λ)(tλε)N2RN(IαF(u1))F(u1)dx+(tλε)N2RN[VVε(tλεx)]|u1|2dx+g(tλε)u122+Vh(tλε)u1222(N+α)cλε(1λ)TN2RN(IαF(u1))F(u1)dxTNVmaxV2|x|>R0|u1|2dx+min{g(1/2),g(3/2)}u122+Vmin{h(1/2),h(3/2)}u1222(N+α)cλε(1λ)TN2RN(IαF(u1))F(u1)dx+min{g(1/2),g(3/2)}u122+Vmin{h(1/2),h(3/2)}u1224(N+α)>cλε,λ(λ~,1],ε[0,ε0].

In both cases, we obtain that cλε<mλ for λ ∈ (λ̃, 1] and ε ∈ [0, ε0].

Lemma 4.3

Assume that (V1), (V5), (V6) and (F1)-(F3) hold. Then for every ε ∈ (0, ε0] and for almost every λ ∈ (λ̃, 1], there exists uλεH1(ℝN) ∖ {0} such that

(Jλε)(uλε)=0,Jλε(uλε)=cλε.(4.9)

Proof

For any fixed ε ∈ (0, ε0], under (V1) and (F1)-(F3), Lemma 4.1 implies that Jλε(u) satisfies the assumptions of Proposition 3.1 with X = H1(ℝN), J = [λ̃, 1] and Φλ=Jλε. So for almost every λ ∈ (λ̃, 1], there exists a bounded sequence {unε(λ)}H1(ℝN) (for simplicity, we denote the sequence by {unε} instead of {unε(λ)} such that

Jλε(unε)cλε>0,(Jλε)(unε)0.(4.10)

By Lemma 3.6, there exist a subsequence of {unε}, still denoted by {unε}, and uλεH1(ℝN), an integer l ∈ ℕ ∪ {0}, and w1, …, wlH1(ℝN) ∖ {0} such that

unεuλεinH1(RN),(Jλε)(uλε)=0,(4.11)

(Jλ)(wk)=0,Jλ(wk)mλ,1kl(4.12)

and

cλε=Jλε(uλε)+k=1lJλ(wk).(4.13)

Since (Jλε)(uλε)=0, then it follows from Lemma 3.2 that

Pλε(uλε):=N22uλε22+12RNNVε(x)+Vε(x)x(uλε)2dxNλRN(IαF(uλε))F(uλε)dx=0.(4.14)

Since unε0, we deduce from (3.12) and (4.13) that if uλ = 0 then l ≥ 1 and

cλε=Jλε(uλε)+k=1lJλ(wk)mλ,

which contradicts with Lemma 4.2. Thus uλε ≠ 0. It follows from (4.3), (4.14) and (V6) that

Jλε(uλε)=Jλε(uλε)1N+αPλε(uλε)=2+α2(N+α)uλε22+12(N+α)RNαVε(x)Vε(x)x(uλε)2dx12(N+α)(2+α)uλε22+(1θ)αV(0)uλε22>0.(4.15)

From (4.13) and (4.15), one has

cλε=Jλε(uλε)+k=1lJλ(wk)>lmλ.(4.16)

By Lemma 4.2, we have cλε<mλ for λ ∈ (λ̃, 1], which, together with (4.16), implies that l = 0 and Jλε(uλε)=cλε. □

Lemma 4.4

Assume that (V1), (V5), (V6) and (F1)-(F3) hold. Then for any ε ∈ (0, ε0], there exists ūεH1(ℝN) ∖ {0} such that (𝓘ε)′(ūε) = 0 and 𝓘ε(ūε) > 0.

Proof

In view of Lemma 4.3, for any fixed ε ∈ (0, ε0], there exist two sequences {λn} ⊂ [λ̃, 1] and {uλnε}H1(ℝN) ∖ {0}, denoted by {unε}, such that

λn1,cλnεcε,(Jλnε)(unε)=0,0<Jλnε(unε)=cλnε.(4.17)

Then it follows from (4.17) and Lemma 3.2 that Pλnε(unε)=0. From (V6), (4.3), (4.14), (4.17) and Lemma 4.1 (iii), one has

C6cλnε=Jλnε(unε)1N+αPλnε(unε)=2+α2(N+α)unε22+12(N+α)RNαVε(x)Vε(x)x(unε)2dx12(N+α)(2+α)uλε22+(1θ)αV(0)uλε22.(4.18)

This shows that {unε} is bounded in H1(ℝN). In view of (4.17), we have limncλnε=cε. Hence, it follows from (4.2) and (4.17) that

Jε(unε)cε,(Jε)(unε)0.

This shows that {unε} satisfies (4.10) with cλε=cε. In view of the proof of Lemma 4.3, we can show that there exists ūεH1(ℝN) ∖ {0} such that (𝓘ε)′(ūε) = 0 and 𝓘ε(ūε) > 0.

Proof of Theorem 1.9

By a similar argument as the proof of Theorem 1.6, we can prove Theorem 1.9 by using Lemmas 4.2, 4.3 and 4.4 instead of 3.5, 3.8 and 3.9, respectively, so, we omit it. □

Acknowledgement

This work was partially supported by the National Natural Science Foundation of China (11571370).

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About the article

Received: 2018-11-24

Accepted: 2018-12-19

Published Online: 2019-06-01

Published in Print: 2019-03-01


Citation Information: Advances in Nonlinear Analysis, Volume 9, Issue 1, Pages 413–437, ISSN (Online) 2191-950X, ISSN (Print) 2191-9496, DOI: https://doi.org/10.1515/anona-2020-0007.

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© 2020 X. Tang and S. Chen, published by De Gruyter. This work is licensed under the Creative Commons Attribution 4.0 Public License. BY 4.0

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