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Advances in Nonlinear Analysis

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


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Existence and non-existence results for Kirchhoff-type problems with convolution nonlinearity

Sitong Chen / Binlin Zhang
  • Corresponding author
  • Department of Mathematics, Heilongjiang Institute of Technology, Harbin, 150050, P. R. China
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/ Xianhua Tang
Published Online: 2018-09-20 | DOI: https://doi.org/10.1515/anona-2018-0147

Abstract

This paper is concerned with the following Kirchhoff-type problem with convolution nonlinearity:

-(a+b3|u|2dx)Δu+V(x)u=(Iα*F(u))f(u),x3,uH1(3),

where a,b>0, Iα:3, with α(0,3), is the Riesz potential, V𝒞(3,[0,)), f𝒞(,) and F(t)=0tf(s)ds. By using variational and some new analytical techniques, we prove that the above problem admits ground state solutions under mild assumptions on V and f. Moreover, we give a non-existence result. In particular, our results extend and improve the existing ones, and fill a gap in the case where f(u)=|u|q-2u, with q(1+α/3,2].

Keywords: Kirchhoff-type problem; Choquard equation; convolution nonlinearity; ground state solutions

MSC 2010: 35J20; 35Q55

1 Introduction and main results

In this paper, we consider the following Kirchhoff-type problem with convolution nonlinearity:

-(a+b3|u|2dx)Δu+V(x)u=(Iα*F(u))f(u),x3,uH1(3),(1.1)

where a,b>0, Iα:3, with α(0,3), is the Riesz potential defined by

Iα(x)=Γ(3-α2)Γ(α2)2απ3/2|x|3-α,x3{0},

V𝒞(3,), f𝒞(,) and F(t)=0tf(s)ds.

Such a problem is often referred to as being nonlocal due to the appearance of the terms (3|u|2dx)Δu or (Iα*F(u))f(u) which implies that (1.1) is no longer a pointwise identity. In particular, if b=0, then (1.1) reduces to the following generalized Choquard equation:

-Δu+V(x)u=(Iα*F(u))f(u),uH1(3).(1.2)

When α=2, V(x)1 and f(u)=u, (1.2) is known as the Choquard–Pekar equation or the stationary nonlinear Hartree equation, which was introduced in 1954, in a work by Pekar [29] describing the quantum mechanics of a polaron at rest; for more details and applications, we refer to [19, 27]. For the case where V(x)1 and f(u)=|u|p-2u, (1.2) is known to have a solution if and only if 1+α/3<p<3+α (see [24, p. 457], [27, Theorem 1]; see also [11, Lemma 2.7]). As described Moroz and Van Schaftingen in [28], since the Hardy–Littlewood–Sobolev inequality [20] implies

3(Iα*h1)h2(x)dxC(α)h16/(3+α)h26/(3+α)for all h1,h2L6/(3+α),(1.3)

where 3+α and 1+α/3 are the upper and lower critical exponents, which appear as extensions of the exponents 6 and 2 for the corresponding local problem.

Inspired by [24, 27, 28], we introduce the following basic assumption on f:

  • (F1)

    f𝒞(,), f(t)=o(|t|α/3) as |t|0 and f(t)=o(|t|2+α) as |t|.

If we let α0 in (1.1), then it becomes formally the following Kirchhoff-type problem with local nonlinearity g=Ff:

-(a+b3|u|2dx)Δu+V(x)u=g(u),x3,uH1(3),(1.4)

which is related to the stationary analogue of the Kirchhoff equation

ρ2ut2-(P0h+E2L0L|ux|2dx)2ux2=0.(1.5)

Equation (1.5) is proposed by Kirchhoff [15] as an extension of the classical D’Alembert’s wave equation for free vibrations of elastic strings. Kirchhoff’s model takes into account the changes in length of the string produced by transverse vibrations. For more details on the physical aspects, we refer the readers to [2, 3, 7, 8, 26].

After Lions [21] proposed an abstract functional analysis framework to (1.4), it has received more and more attention from the mathematical community; there have been many works about the existence of nontrivial solutions to (1.4) and its fractional version by using variational methods, for example, see [4, 6, 9, 12, 13, 16, 17, 18, 22, 30, 31, 35, 34, 37, 40, 41, 42] and the references therein. A typical way to deal with (1.4) is to use the mountain-pass theorem. For this purpose, one usually assumes that g(t) is superlinear at t=0 and super-cubic at t=. In this case, if g further satisfies the monotonicity condition

  • (G1)

    g(t)/|t|3 is increasing for t{0},

via the Nehari manifold approach, He and Zou [13] obtained the first existence result on ground state solutions of (1.4). For the case where g(t) is not super-cubic at t=, Li and Ye [17] proved that (1.4), with special forms V=1 and g(u)=|u|p-2u for 3<p<6, has a ground state positive solution by using a minimizing argument on a new manifold that is defined by a condition which is a combination of the Nehari equation and the Pohoz̆aev equality. This idea comes from Ruiz [33], in which the nonlinear Schrödinger–Poisson system was studied. Later, by introducing another suitable manifold differing from [17], Guo [12] and Tang and Chen [37] improved the above result to (1.4), where V satisfies

  • (V1)

    V𝒞(3,[0,)) and V:=lim|y|V(y)V(x) for all x3,

  • (V2)’

    V𝒞1(3,) and there exists θ(0,1) such that V(x)xaθ2|x|2 for all x3{0},

and g satisfies

  • (G2)

    g𝒞1(+,) and g(t)t is increasing on (0,),

and

  • (G4)

    g𝒞(,) and g(t)t+6G(t)|t|t is nondecreasing on (-,0)(0,), where G(t)=0tg(s)ds.

respectively, and some standard growth assumptions.

Compared with (1.2) and (1.4), it is more difficult to deal with (1.1) which involves two nonlocal terms. In [23], Lü investigated the following special form of (1.1):

-(a+b3|u|2dx)u+(1+μg(x))u=(Iα*|u|q)|u|q-2u,x3,uH1(3),(1.6)

where q(2,3+α), μ>0 is a parameter and g(x) is a nonnegative steep potential well function. By using the Nehari manifold and the concentration compactness principle, Lü proved the existence of ground state solutions for (1.6) if the parameter μ is large enough. It is worth pointing out that the same result is not available in the case where q(1+α/3,2], even when g(x)=0, since both the mountain pass theorem and the Nehari manifold argument do not work. In fact, in this case, it is more difficult to get a bounded (PS) sequence and to prove that the (PS) sequence converges weakly to a critical point of the corresponding functional in H1(3). To the best of our knowledge, there seem to be no results dealt with this case in the literature. As for the related study of problem (1.1) involving the critical exponents, we refer to [25, 32] for more details.

Motivated by the above-mentioned papers, we shall deal with the existence of ground state solutions for (1.1) under (V1) and (F1). It is standard to check, according to (1.3), that under (V1) and (F1), the energy functional defined in H1(3) by

Φ(u)=123[a|u|2+V(x)u2]dx+b4(3|u|2dx)2-123(Iα*F(u))F(u)dx(1.7)

is continuously differentiable and its critical points correspond to the weak solutions of (1.1). We say a weak solution to (1.1) is a ground state solution if it minimizes the functional Φ among all nontrivial weak solutions.

In addition to (F1), we also need the following assumptions on f:

  • (F2)

    lim|t|F(t)|t|1-α=,

  • (F3)

    the function |t|α[f(t)t+(3+α)F(t)]/t is nondecreasing on (-,0)(0,+).

Remark 1.1.

(F3) is weaker than the following assumption, which is easier to verify:

  • (F4)

    the function |t|αf(t) is nondecreasing on (-,0)(0,+).

It is easy to see that there are many functions which satisfy (F1), (F2) and (F4). In addition, there are some functions which satisfy (F3), but not (F4), for example,

f(t)=α+2(2α+1)(2α+3)|t|αt+(α+1)|t|α-1t+α|t|α-2tsint+|t|αcost.

To overcome the lack of compactness of Sobolev embeddings in unbounded domains, different from [23] in which a steep potential well was considered, we assume that V satisfies (V1) and the decay condition:

  • (V2)

    V𝒞1(3,) and either of the following cases holds:

    • (A1)(a)

      V(x)xa2|x|2 for all x3{0},

    • (A2)(b)

      max{V(x)x,0}3/2(34)1/3π2a.

Now we are in a position to state the first main result.

Theorem 1.2.

Assume that V and f satisfy (V1), (V2) and (F1)(F3). Then problem (1.1) has a ground state solution u^H1(R3).

Next, we further provide a minimax characterization of the ground state energy. To this end, we introduce a new monotonicity condition on V as follows:

  • (V3)

    V𝒞1(3,) and there exists θ[0,1) such that

    t4V(tx)+V(tx)(tx)+θa2t2|x|2

    is nonincreasing on (0,+) for every x3{0}.

Similar to [12], we define the Pohoz̆aev functional related with (1.1):

𝒫(u)=a2u22+123[3V(x)+V(x)x]u2dx+b2u24-3+α23(Iα*F(u))F(u)dx.(1.8)

It is well known that any solution u of (1.1) satisfies 𝒫(u)=0. Motivated by [17, 37], we define the Nehari–Pohoz̆aev manifold of Φ by

:={uH1(3){0}:J(u):=12Φ(u),u+𝒫(u)=0}.(1.9)

Then every nontrivial solution of (1.1) is contained in . Our second main result is as follows.

Theorem 1.3.

Assume that V and f satisfy (V1), (V3) and (F1)(F3). Then problem (1.1) has a ground state solution u¯H1(R3) such that

Φ(u¯)=infΦ=infuH1(3){0}maxt>0Φ(t1/2ut)>0,

where and in the sequel ut(x):=u(x/t).

Applying Theorem 1.3 to the “limiting problem” of (1.1):

-(a+b3|u|2dx)Δu+Vu=(Iα*F(u))f(u),x3,uH1(3),(1.10)

similar to (1.7) and (1.9), we define

Φ(u):=123(a|u|2+Vu2)dx+b4(3|u|2dx)2-123(Iα*F(u))F(u)dx(1.11)

and

:={vH1(3){0}:J(u)=12(Φ)(u),u+𝒫(u)=0},

where 𝒫(u)=0 is the Pohoz̆ave type identity related with (1.10). Then we have the following corollary.

Corollary 1.4.

Assume that (F1)(F3) hold. Then problem (1.10) has a ground state solution u¯H1(R3) such that

Φ(u¯)=infΦ=infuH1(3){0}maxt>0Φ(t1/2ut)>0.

In the last part of this paper, we give a non-existence result for the following special form of (1.1):

-(a+b3|u|2dx)Δu+u=(Iα*|u|q)|u|q-2u,x3,uH1(3),(1.12)

where q>1.

Theorem 1.5.

If 1<q<1+α/3 or q3+α, then problem (1.12) does not admit any nontrivial solution.

Remark 1.6.

There are indeed functions which satisfy (V1)(V3). An example is given by V(x)=V1-A|x|2+1, where V1>1 and 0<A<a/8 are two positive constants. Our results extend and improve the previous results on (1.1) in the literature, which are new even when VV>0. In particular, Theorems 1.2 and 1.3 fill a gap on (1.1) in the case where f(u)=|u|q-2u, with q(1+α/3,2].

Remark 1.7.

Letting α0, our results cover the ones in [12, 17, 37], which dealt with (1.4) that can be considered as a limiting problem of (1.1) when α0. In fact, if α=0, then (V2)’ and (G2) imply case (i) of (V2) and (F3), respectively. Moreover, since (G4) and (F3) imply that G(t)/|t|t and |t|αF(t)/t are nondecreasing on (-,0)(0,+), respectively (see Lemma 2.3), one can see that (F3) reduces to (G4) when α=0.

To prove Theorem 1.2, we will use Jeanjean’s monotonicity trick [14], that is, an approximation procedure to obtain a bounded (PS)-sequence for Φ, instead of starting directly from an arbitrary (PS)-sequence. More precisely, firstly, for λ[1/2,1], we consider a family of functionals Φλ:H1(3) defined by

Φλ(u)=123(a|u|2+V(x)u2)dx+b4u24-λ23(Iα*F(u))F(u)dx.(1.13)

These functionals have a mountain pass geometry, and we denote the corresponding mountain pass levels by cλ. Moreover, Φλ has a bounded (PS)-sequence {un(λ)}H1(3) at level cλ for almost every λ[1/2,1]. Secondly, we use the global compactness lemma to show that the bounded sequence {un(λ)} converges weakly to a nontrivial critical point of Φλ. To do this, we have to establish the following strict inequality:

cλ<inf𝒦λΦλ,(1.14)

where Φλ is the associated limited functional defined by

Φλ(u)=123(a|u|2+Vu2)dx+b4u24-λ23(Iα*F(u))F(u)dx,(1.15)

and

𝒦λ:={wH1(3){0}:(Φλ)(w)=0}.

A classical way to obtain (1.14) is to find a positive function wλ𝒦λ such that Φλ(wλ)=inf𝒦λΦλ when nonconstant potential V(x)V. However, it seems to be impossible to obtain the wλ mentioned above only under (F1)(F3). So the usual arguments cannot be applied here to prove (1.14). To overcome this difficulty, we follow a strategy introduced in [38], that is, we first show that there exists u¯ such that

u¯,Φ(u¯)=infΦ,(1.16)

and then, by means of the translation invariance for u¯ and the crucial inequality

Φλ(u)Φλ(t1/2ut)+1-t44Jλ(u)+a(1-t2)24u22for all uH1(3) and t>0

established in Lemma 3.3, we can find λ¯[1/2,1) such that

cλ<mλ:=infλΦλfor all λ(λ¯,1](1.17)

(see Lemma 3.5), where

λ={uH1(3){0}:Jλ(u)=12(Φλ)(u),u+𝒫λ(u)=0}

and 𝒫λ(u)=0 is the corresponding Pohoz̆ave type identity. In particular, any information on sign of u¯ is not required in our arguments. Finally, we choose two sequences {λn}(λ¯,1] and {uλn}H1(3){0} such that λn1 and Φλn(uλn)=0, and by using (1.17) and the global compactness lemma, we get a nontrivial critical point u¯ of Φ.

We would like to mention that in the proof of Theorem 1.2, a crucial step is to prove (1.16), which is a corollary of Theorem 1.3. Inspired by [5, 36, 37], we shall prove Theorem 1.3 by following this scheme:

  • (A1)

    we verify and establish the minimax characterization of m:=infΦ>0,

  • (A2)

    we prove that m is achieved,

  • (A3)

    we show that the minimizer of Φ on is a critical point.

Although we mainly follow the procedure of [36], we have to face many new difficulties due to the mutual competing effect between (3|u|2dx)Δu and (IαF(u))f(u). These difficulties enforce the implementation of new ideas and techniques. More precisely, in step 1, we first establish a key inequality, namely,

Φ(u)Φ(t1/2ut)+1-t44J(u)+a(1-θ)(1-t2)24u22for all uH1(3),t>0,(1.18)

in Lemma 2.5, where some more careful analyses on the convolution nonlinearity are introduced, see Lemmas 2.12.4; then we construct a saddle point structure with respect to the fibre {t1/2ut:t>0}H1(3) for uH1(3){0}, see Lemma 2.8; finally, based on these constructions, we obtain the minimax characterization of m, see Lemma 2.9. In step 2, we first choose a minimizing sequence {un} of Φ on , and show that {un} is bounded in H1(3); then, with the help of the key inequality (1.18) and a concentration-compactness argument, we prove that there exist u^H1(3){0} and t^>0 such that unu^ in H1(3), up to translations and extraction of a subsequence, and t^1/2u^t^ is a minimizer of infΦ, see Lemmas 2.14 and 2.15. This step is most difficult since there is no global compactness and not any information on Φ(un). Finally, in step 3, inspired by [38, Lemma 2.13], we use the key inequality (1.18), the deformation lemma and intermediary theorem for continuous functions, which overcome the difficulty that may not be a 𝒞1-manifold of H1(3), due to the lack of the smoothness of f(u), see Lemma 2.15.

Throughout the paper we make use of the following notations:

  • H1(3) denotes the usual Sobolev space equipped with the inner product and norm

    (u,v)=3(uv+uv)dx,u=(u,u)1/2for all u,vH1(3).

  • Ls(3) (1s<) denotes the Lebesgue space with the norm us=(3|u|sdx)1/s.

  • For any uH1(3){0}, ut(x):=u(x/t) for t>0.

  • For any x3 and r>0, Br(x):={y3:|y-x|<r}.

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

The rest of the paper is organized as follows. In Section 2, we study the existence of ground state solutions for (1.1) by using the Nehari–Pohoz̆aev manifold , and give the proof of Theorem 1.3. In Section 3, based on Jeanjean’s monotonicity trick, we consider the existence of ground state solutions for (1.1), and complete the proof of Theorem 1.2. In Section 4, we study the non-existence of solutions for problem (1.12) and present the proof of Theorem 1.5.

2 Proof of Theorem 1.3

In this section, we give the proof of Theorem 1.3. To this end, we give some useful lemmas. Since V(x)V satisfies (V1)(V3), all conclusions on Φ are also true for Φ in this paper. For (1.4), we always assume that V>0. First, by a simple calculation, we can verify the following lemma.

Lemma 2.1.

Assume that (V1) and (V3) hold. Then one has

4t4[V(x)-V(tx)]-(1-t4)V(x)x-θa(1-t2)22|x|2for all t0,x3{0}.(2.1)

Lemma 2.2.

Assume that (F1) and (F3) hold. Then for all s0 and tR,

g(s,t):=4s(3+α)/2F(s1/2t)-4F(t)+(1-s2)[f(t)t+(3+α)F(t)]0.(2.2)

Proof.

It is evident that (2.2) holds for all s>0 and t=0. For t0, it follows from (F3) that

ddsg(s,t)=2s|t|1-α{|s1/2t|α-1[f(s1/2t)s1/2t+(3+α)F(s1/2t)]|t|α-1[f(t)t+(3+α)F(t)]}{0,s1,0,0s<1,

which implies that g(s,t)g(1,t)=0 for all s0 and t(-,0)(0,+). ∎

Lemma 2.3.

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

|t|αF(t)t is nondecreasing on (-,0)(0,+).(2.3)

Proof.

By (F1) and (2.2), one has

g(0,t)=f(t)t+(α-1)F(t)0for all t.(2.4)

Since

ddt(|t|αF(t)t)=|t|α-2[f(t)t+(α-1)F(t)],

(2.3) follows from (2.4). ∎

Lemma 2.4.

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

h(t,u):=3{t3+α(Iα*F(t1/2u))F(t1/2u)+1-t44(Iα*F(u))f(u)u-(3+α)t4+1-α4(Iα*F(u))F(u)}dx0for all t>0,uH1(3).(2.5)

Proof.

Note that (F1) and (2.3) imply

F(t)0for all t(2.6)

and

Iα*(tα/2F(t1/2u)t1/2)-Iα*F(u){0,t1,0,0<t<1.(2.7)

By (F3), (2.6) and (2.7), we have

ddth(t,u)=3{t2+α(Iα*F(t1/2u))[f(t1/2u)t1/2u+(3+α)F(t1/2u)]-t3(Iα*F(u))[f(u)u+(3+α)F(u)]}dx=t33|u|1-α{(tα/2F(t1/2u)t1/2)|t1/2u|α-1[f(t1/2u)t1/2u+(3+α)F(t1/2u)]-(Iα*F(u))|u|α-1[f(u)u+(3+α)F(u)]}dx{0,t1,0,0<t<1,

which implies that h(t,u)h(1,u)=0 for all uH1(3). This shows that (2.4) holds. ∎

By (1.7) and (1.8), one has

J(u)=au22+123[4V(x)+V(x)x]u2dx+bu24-123(Iα*F(u))[f(u)u+(3+α)F(u)]dx.(2.8)

Lemma 2.5.

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

Φ(u)Φ(t1/2ut)+1-t44J(u)+a(1-θ)(1-t2)24u22for all uH1(3),t>0.(2.9)

Proof.

According to the Hardy inequality, we have

u22143u2|x|2dxfor all uH1(3).(2.10)

Note that

Φ(t1/2ut)=at22u22+t423V(tx)u2dx+bt44u24-t3+α23(Iα*F(t1/2u))F(t1/2u)dxfor all uH1(3),t>0.(2.11)

Thus, by (1.7), (2.4), (2.8) and (2.11), one has

Φ(u)-Φ(t1/2ut)=a(1-t2)2u22+123[V(x)-t4V(tx)]u2dx+b(1-t4)4u24+123[t3-α(Iα*F(t1/2u))F(t1/2u)-(Iα*F(u))F(u)]dx=1-t44J(u)+a(1-t2)24u22+183{4t4[V(x)-V(tx)]-(1-t4)V(x)x}u2dx+123{t3+α(Iα*F(t1/2u))F(t1/2u)+1-t44(Iα*F(u))f(u)u-(3+α)t4+1-α4(Iα*F(u))F(u)}dx1-t44J(u)+a(1-θ)(1-t2)24u22.

This shows that (2.9) holds. ∎

Note that

J(u)=au22+2Vu22+bu24-123(Iα*F(u))[f(u)u+(3+α)F(u)]dx.(2.12)

From Lemma 2.5, we have the following two corollaries.

Corollary 2.6.

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

Φ(u)Φ(t1/2ut)+1-t44J(u)+a(1-t2)24u22for all uH1(3),t>0.(2.13)

Corollary 2.7.

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

Φ(u)=maxt>0Φ(t1/2ut)for all u.

Letting t0, (2.9) and (2.13) imply

Φ(u)14J(u)+a(1-θ)4u22for all uH1(3)(2.14)

and

Φ(u)14J(u)+a4u22for all uH1(3).

Lemma 2.8.

Assume that (V1), (V3) and (F1)(F3) hold. Then for any uH1(R3){0}, there exists a unique tu>0 such that tu1/2utuM.

Proof.

Let uH1(3){0} be fixed and define a function ζ(t):=Φ(t1/2ut) on (0,). Clearly, by (2.8) and (2.11), we have

ζ(t)=0at2u22+t423[4V(tx)+V(tx)tx]u2dx+bt4u24-t3+α23(Iα*F(t1/2u))[f(t1/2u)t1/2u+(3+α)F(t1/2u)]dx=0J(t1/2ut)=0t1/2ut.(2.15)

Note that (F1) implies that for any ε>0, there exists Cε>0 such that

|f(t)t|+|F(t)|ε|t|1+3/α+Cε|t|3+αfor all t.(2.16)

By (2) and (2.16), we have limt0+ζ(t)=0 and ζ(t)>0 for t>0 small. By (F3), one has

tα/2[4f(t1/2u)t1/2u+(3+α)F(t1/2u)]t1/24f(u)u+(3+α)F(u)for all t1.(2.17)

From (F2), (2.6), (2) and (2.17), we can deduce that ζ(t)<0 for t large. Therefore, there exists some t^=tu>0 such that ζ(t^)=0 and t^1/2ut^.

Next we claim that tu is unique for any uH1(3){0}. In fact, for any given uH1(3){0}, let t^1,t^2>0 be such that ζ(t^1)=ζ(t^2)=0. Then J(t^11/2ut^1)=J(t^21/2ut^2)=0. Jointly with (2.9), we have

Φ(t^11/2ut^1)Φ(t^21/2ut^2)+(1-θ)a(t^12-t^22)24t^12u22(2.18)

and

Φ(t^21/2ut^2)Φ(t^11/2ut^1)+(1-θ)a(t^22-t^12)24t^22u22.(2.19)

Both (2.18) and (2.19) imply that t^1=t^2. Therefore, tu>0 is unique for any uH1(3){0}. ∎

Combining Corollary 2.7 with Lemma 2.8, we have the following lemma.

Lemma 2.9.

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

infuΦ(u)=m=infuH1(3){0}maxt>0Φ(t1/2ut).

Lemma 2.10.

Assume that (V1) and (V3) hold. Then there exist γ1,γ2>0 such that

γ1u2au22+123[4V(x)+V(x)x]u2dxγ2u2for all uH1(3).(2.20)

Proof.

Letting t=0 and t in (2.1), we have, respectively,

V(x)xθa2|x|2for all x3{0}(2.21)

and

4V(x)+V(x)x4V-θa2|x|2for all x3{0}.(2.22)

The last inequality in (2.20) follows from (V1), (2.10) and (2.21). By (2.10) and (2.22), we have

au22+123[4V(x)+V(x)x]u2dx(1-θ)au22+2Vu22min{(1-θ)a,2V}u2:=γ1u2for all uH1(3),

as desired. ∎

Lemma 2.11.

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

  • (i)

    there exists ρ>0 such that uρ for all u,

  • (ii)

    m=infuΦ(u)>0.

Proof.

(i) Since J(u)=0 for u, by (1.3), (2.8), (2.16), (2.20) and the Sobolev embedding theorem, one has

γ1u2au22+123[4V(x)+V(x)x]u2dx+b(3|u|2dx)2=123(Iα*F(u))[f(u)u+(3+α)F(u)]dxC1(u2+2α/3+u6+2α),

which implies

uρ:=min{1,(γ12C1)3/2α}for all u.(2.23)

(ii) Let {un} be such that Φ(un)m. There are two possible cases. Case 1: infnNun2:=ρ1>0. In this case, by (2.14), one has

m+o(1)=Φ(un)=Φ(un)-14J(un)a(1-θ)4un22a(1-θ)4ρ12.

Case 2: infnNun2=0. By (2.23), passing to a subsequence, we have

un20,un212ρ+o(1).(2.24)

By (V1), there exists R>0 such that V(x)V/2 for |x|R. This implies

{|tx|R}V(tx)u2dxV2{|tx|R}u2dxfor all t>0,uH1(3).(2.25)

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

{|tx|<R}u2dx(4πR33t3)2/3({|tx|<R}u6dx)1/3(4πR33t3)2/3S-1u22for all t>0,uH1(3).(2.26)

Let

δ0=min{V,aS(34πR3)2/3}.(2.27)

By (1.3), (2.16) and the Sobolev embedding inequality, we have

3(Iα*F(u))F(u)dxC(α)F(u)6/(3+α)214δ0u22+2α/3+C2u66+2α14δ0u22+2α/3+C2S-(3+α)u26+2αfor all uH1(3).(2.28)

Let tn=un2-2. Then (2.24) implies that {tn} is bounded. Since J(un)=0, it follows from (2.9), (2.11), (2.24) and (2.28) that

m+o(1)=Φ(un)Φ(tn1/2(un)tn)=atn22un22+tn423V(tnx)un2dx+btn44un24-tn3+α23(Iα*F(tn1/2un))F(tn1/2un)dxaS2(34πR3)2/3tn4{|tnx|<R}un2dx+Vtn44{|tnx|R}un2dx-18δ0tn4+4/3αun22+2α/3-C22S3+αtn6+2αun26+2α18δ0tn4un22[2-(tn4un22)α/3]+o(1)=18δ0+o(1).

Cases 1 and 2 show that m=infuΦ(u)>0. ∎

Combining [1, Lemma 5.1], [10, Lemma 2.2], [36, Lemmas 2.7 and 2.8], and [39], we can obtain the following Brezis–Lieb type lemma.

Lemma 2.12.

Assume that (V1) and (F1) hold and V(x)xL(R3). If unu¯ in H1(R3), then, along a subsequence,

Φ(un)=Φ(u¯)+Φ(un-u¯)+b2u¯22(un-u¯)22+o(1),Φ(un),un=Φ(u¯),u¯+Φ(un-u¯),un-u¯+2bu¯22(un-u¯)22+o(1),J(un)=J(u¯)+J(un-u¯)+2bu¯22(un-u¯)22+o(1).

Lemma 2.13.

Assume that (V1), (V3) and (F1)(F3) hold. Then mm:=infuMΦ(u).

Proof.

In view of Lemmas 2.8 and 2.11, we have and m>0. Arguing indirectly, we assume that m>m. Let ε:=m-m. Then there exists uε such that

uεandm+ε2>Φ(uε).(2.29)

In view of Lemma 2.8, there exists tε>0 such that tε1/2(uε)tε. Thus, it follows from (V1), (1.7), (1.11), (2.13) and (2.29) that

m-ε2=m+ε2>Φ(uε)Φ(tε1/2(uε)tε)Φ(tε1/2(uε)tε)m.

This contradiction shows that mm. ∎

Lemma 2.14.

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

Proof.

In view of Lemmas 2.8 and 2.11, we have and m>0. Let {un} be such that Φ(un)m. Since J(un)=0, it follows from (2.14) that

m+o(1)=Φ(un)=Φ(un)-14J(un)a(1-θ)4un22.

This shows that {un2} is bounded. Next, we prove that {un} is also bounded. Arguing by contradiction, suppose that un2. By (1.3), (2.16) and the Sobolev embedding inequality, one has, for all uH1(3),

3(Iα*F(u))F(u)dxC(α)F(u)6/(3+α)2δ04(δ016m)α/3u22+2α/3+C3u66+2αδ04(δ016m)α/3u22+2α/3+C3S-(3+α)u26+2α,(2.30)

where δ0>0 is defined by (2.27). Let t~n=(16m/δ0un22)1/4, then t~n0. Since J(un)=0, it follows from (1.11), (2.9), (2.11), (2.25), (2.26) and (2.30) that

m+o(1)=Φ(un)Φ(t~n1/2(un)t~n)=at~n22un22+t~n423V(t~nx)un2dx+bt~n44un24-t~n3+α23(Iα*F(t~n1/2un))F(t~n1/2un)dxaS2(34πR3)2/3t~n4|t~nx|<Run2dx+Vt~n44{|t~nx|R}un2dx-δ08(δ016m)α/3t~n4+4/3αun22+2α/3-C32S3+αt~n6+2αun26+2α18δ0t~n4un22[2-(δ0t~n4un2216m)α/3]+o(1)=2m+o(1).

This contradiction shows that {un2} is bounded. Hence, {un} is bounded in H1(3). Passing to a subsequence, we have unu¯ in H1(3). Then unu¯ in Llocs(3) for 2s<6 and unu¯ a.e. in 3. There are two possible cases: (i) u¯=0 and (ii) u¯0. Case (i)  u¯=0, i.e., un0 in H1(R3). Then un0 in Llocs(3) for 2s<2* and un0 a.e. in 3. Using (V1) and (2.21), it is easy to show that

limn3[V-V(x)]un2dx=limn3V(x)xun2dx=0.(2.31)

From (1.7), (1.11), (2.8), (2.12) and (2.31), one can get

Φ(un)m,J(un)0.(2.32)

By (F1), for some p(1+α/3,3+α) and any ϵ>0, there exists Cϵ>0 such that

|f(t)t|+|F(t)|ϵ(|t|1+3/α+|t|3+α)+Cϵ|t|pfor all t.(2.33)

From (1.3), (2.12), (2.16), (2.32), (2.33) and Lemma 2.11 (i), one has

min{a,2V}ρ02min{a,2V}un2aun22+2Vun22+bun24=123(Iα*F(un))[f(un)un+(3+α)F(un)]dx+o(1)C4[ϵ(un21+α/3+u66+2α)+Cεun6p/(3+α)p]2+o(1).(2.34)

Using (2) and Lions’ concentration compactness principle [39, Lemma 1.21], we can prove that there exist δ>0 and a sequence {yn}3 such that B1(yn)|un|2dx>δ. Let u^n(x)=un(x+yn). Then we have u^n=un and

J(u^n)=o(1),Φ(u^n)m,B1(0)|u^n|2dx>δ.(2.35)

Therefore, there exists u^H1(3){0} such that, passing to a subsequence,

{u^nu^in H1(3),u^nu^in Llocs(3) for all s[1,6),u^nu^,a.e. on 3.(2.36)

Let wn=u^n-u^. Then (2.36) and Lemma 2.12 yield

Φ(u^n)=Φ(u^)+Φ(wn)+b2u^22wn22+o(1)(2.37)

and

J(u^n)=J(u^)+J(wn)+2bu^22wn22+o(1).(2.38)

For uH1(3), we let

Ψ(u):=Φ(u)-14J(u)=a4u22+183(Iα*F(u))[f(u)u+(α-1)F(u)]dx.(2.39)

From (1.11), (2.12),(2.35), (2.37), (2.38) and (2.39), one has

Ψ(wn)=m-Ψ(u^)+o(1),J(wn)-J(u^)+o(1).(2.40)

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

Φ(u^)=m,J(u^)=0.(2.41)

Next, we assume that wn0. In view of Lemma 2.8, there exists tn>0 such that tn1/2(wn)tn. We claim that J(u^)0. Otherwise, if J(u^)>0, then (2.40) implies J(wn)<0 for large n. From (1.11), (2.12), (2.13), (2.40) and Lemma 2.13, we obtain

m-Ψ(u^)+o(1)=Ψ(wn)=Φ(wn)-14J(wn)Φ(tn1/2(wn)tn)-tn34J(wn)+a(1-tn2)24wn22mmfor large n,

which is a contradiction due to Ψ(u^)>0. Hence, J(u^)0. In view of Lemma 2.8, there exists t>0 such that t1/2u^t. By (1.11), (2.4), (2.12), (2.13), (2.32), (2.35), (2.39), the weak semicontinuity of norm, Fatou’s lemma and Lemma 2.13, we have

m=limnΨ(u^n)Ψ(u^)=Φ(u^)-14J(u^)Φ(t1/2u^t)-t44J(u^)+a(1-t2)24u^22m-t44J(u^)+a(1-t2)24u^22m,

which implies that (2.41) holds too. In view of Lemma 2.8, there exists t^>0 such that t^1/2u^t^. Moreover, it follows from (V1), (1.7), (1.11), (2.13) and (2.41) that

mΦ(t^1/2u^t^)Φ(t^1/2u^t^)Φ(u^)=m.

This shows that m is achieved at t^1/2u^t^. Case (ii)  u¯0. In this case, analogous to the proof of (2.41), by using Φ and J instead of Φ and J, we can deduce that

Φ(u¯)=m,J(u¯)=0.

Hence, the proof is complete. ∎

Lemma 2.15.

Assume that (V1), (V3) and (F1)(F3) hold. If u¯M and Φ(u¯)=m, then u¯ is a critical point of Φ.

Proof.

Following the idea of [38, Lemma 2.13], we use the deformation lemma and intermediary theorem for continuous functions to prove this lemma. Assume that Φ(u¯)0. Then there exist δ>0 and ϱ>0 such that

u-u¯3δΦ(u)ϱ.

By [37, equation (2.47)], one has limt1t1/2u¯t-u¯=0. Thus, there exists δ1>0 such that

|t1/2-1|<δ1t1/2u¯t-u¯<δ.

In view of (2), (2.16) and (2.17), there exist T1(0,1) and T2(1,) such that

J(T11/2u¯T1)>0,J(T21/2u¯T2)<0.

The rest of the proof is similar to the proof of [38, Lemma 2.13]. Indeed, we can obtain the desired conclusion by using

Φ(t1/2u¯t)Φ(u¯)-a(1-θ)(1-t2)24u¯22=m-a(1-θ)(1-t2)24u¯22for all t>0

and

ε:=min{a(1-θ)(1-T12)212u¯22,a(1-θ)(1-T22)212u¯22,1,ϱδ8}

instead of [38, (2.40) and ε], respectively. ∎

Proof of Theorem 1.3.

In view of Lemmas 2.9, 2.14 and 2.15, there exists u¯ such that

Φ(u¯)=m=infuH1(3){0}maxt>0Φ(t1/2ut),Φ(u¯)=0.

This shows that u¯ is a ground state solution of (1.1). ∎

3 Proof of Theorem 1.2

In this section, we give the proof of Theorem 1.2. Without loss of generality, we consider that V(x)V.

Proposition 3.1 ([14]).

Let X be a Banach space and let KR+ be an interval. We consider a family {Iλ}λK of C1-functionals on X of the form

λ(u)=A(u)-λB(u)for all λK,

where B(u)0 for all uX, and such that either A(u)+ or B(u)+ as u. We assume that there are two points v1,v2 in X such that

cλ:=infγΓmaxt[0,1]λ(γ(t))>max{λ(v1),λ(v2)},where Γ={γ𝒞([0,1],X):γ(0)=v1,γ(1)=v2}.

Then, for almost every λK, there is a bounded (PS)cλ sequence for Iλ, that is, there exists a sequence such that

  • (i)

    {un(λ)} is bounded in X,

  • (ii)

    λ(un(λ))cλ,

  • (iii)

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

Moreover, cλ is nonincreasing and left continuous on λ[1/2,1].

Lemma 3.2 ([12]).

Assume that (V1), (V2) and (F1) hold. Let u be a critical point of Φλ in H1(R3). Then we have the following Pohoz̆ave type identity:

𝒫λ(u):=a2u22+123[3V(x)+V(x)x]u2dx+b2u24-3+α2λ3(Iα*F(u))F(u)dx=0.(3.1)

We set Jλ(u):=12Φλ(u),u+𝒫λ(u). Then

Jλ(u)=au22+123[4V(x)+V(x)x]u2dx+bu24-λ23(Iα*F(u))[f(u)u+(3+α)F(u)]dx(3.2)

for λ[1/2,1]. Correspondingly, we also let

Jλ(u)=au22+2Vu22+bu24-λ23(Iα*F(u))[f(u)u+(3+α)F(u)]dx(3.3)

for λ[1/2,1]. Set

λ={uH1(3){0}:Jλ(u)=0},mλ=infuλΦλ(u).

By Corollary 2.6, we have the following lemma.

Lemma 3.3.

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

Φλ(u)Φλ(t1/2ut)+1-t44Jλ(u)+a(1-t2)24u22for all uH1(3),t>0.(3.4)

In view of Theorem 1.2, Φ1=Φ has a minimizer u0 on 1=, i.e.,

u1,(Φ1)(u)=0andm1=Φ1(u).(3.5)

Since (1.10) is autonomous, V𝒞(3,) and V(x)V but V(x)V, there exist x¯3 and r¯>0 such that

V-V(x)>0,|u(x)|>0for a.e. x, with |x-x¯|r¯.(3.6)

Lemma 3.4.

Assume that (V1), (V2) and (F1)(F3) hold. Then

  • (i)

    there exists T>0 independent of λ such that Φλ(T1/2(u)T)<0 for all λ[1/2,1] ;

  • (ii)

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

    cλ:=infγΓmaxt[0,1]Φλ(γ(t))κ0>max{Φλ(0),Φλ(T1/2(u)T)},

    where

    Γ={γ𝒞([0,1],H1(3)):γ(0)=0,γ(1)=T1/2(u)T};

  • (iii)

    cλ and mλ are nonincreasing on λ[1/2,1].

The proof of Lemma 3.4 is standard, so we omit it.

Lemma 3.5.

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

Proof.

It is easy to see that Φλ(t1/2(u)t) is continuous on t(0,). Hence, for any λ[1/2,1), we can choose tλ(0,T) such that Φλ(tλ1/2(u)tλ)=maxt(0,T]Φλ(t1/2(u)t). By (2.3) and (2.7), one has

(Iα*F(tλ1/2u)tλ(1-α)/2)F(tλ1/2u)tλ(1-α)/2(Iα*F(T1/2u)T(1-α)/2)F(T1/2u)T(1-α)/2.(3.7)

Set

γ0(t)={(tT)1/2(u)(tT)for t>0,0for t=0.

Then γ0Γ, defined by Lemma 3.4 (ii). Moreover,

Φλ(tλ1/2(u)tλ)=maxt[0,1]Φλ(γ0(t))cλ.(3.8)

Let

ζ0:=min{3r¯/8(1+|x¯|),1/4}.(3.9)

Then it follows from (3.6) and (3.9) that

|x-x¯|r¯2ands[1-ζ0,1+ζ0]|sx-x¯|r¯.(3.10)

Let

λ¯:=max{12,1-(1-ζ0)4mins[1-ζ0,1+ζ0]3[V-V(sx)]|u|2dxT3+α3(Iα*F(T1/2u))F(T1/2u)dx,   1-a(1-θ)ζ02u222T3+α3(Iα*F(T1/2u))F(T1/2u)dx}.(3.11)

Then it follows from (3.6) and (3.10) that 1/2λ¯<1. We have two cases to distinguish: Case (i)  tλ[1-ζ0,1+ζ0]. By (1.13), (1.15), (3.4), (3.7)–(3.11) and Lemma 3.4 (iii), we have

mλm1=Φ1(u)Φ1(tλ1/2(u)tλ)=Φλ(tλ1/2(u)tλ)-1-λ2tλ3+α3(Iα*F(tλ1/2u))F(tλ1/2u)dx+tλ423[V-V(tλx)]|u|2dxcλ-1-λ2T3+α3(Iα*F(T1/2u))F(T1/2u)dx+(1-ζ0)42mins[1-ζ0,1+ζ0]3[V-V(sx)]|u|2dx>cλfor all λ(λ¯,1].

Case (ii)  tλ(0,1-ζ0)(1+ζ0,T]. Since VV(x) for all x3, it follows from (1.13), (1.15), (3.4), (3.5), (3.7), (3.8), (3.11) and Lemma 3.4 (iii) that

mλm1=Φ1(u)Φ1(tλ1/2(u)tλ)+a(1-θ)(1-tλ2)24u22=Φλ(tλ1/2(u)tλ)-1-λ2tλ3+α3(Iα*F(tλ1/2u))F(tλ1/2u)dx+tλ423[V-V(tλx)]|u|2dx+a(1-θ)(1-tλ2)24u22cλ-1-λ2T3+α3(Iα*F(T1/2u))F(T1/2u)dx+a(1-θ)ζ024u22>cλfor all λ(λ¯,1].

In both cases, we obtain cλ<mλ for λ(λ¯,1]. ∎

Lemma 3.6.

Assume that (V1), (V2) and (F1)(F3) hold. Let {un} be a bounded (PS)cλ sequence for Φλ with λ[1/2,1]. Then there exist a subsequence of {un}, still denoted it by {un}, and u0H1(R3) such that

  • (i)

    Aλ2:=limnun22 exists, unuλ in H1(3) and λ(uλ)=0 ;

  • (ii)

    wk0 and (λ)(wk)=0 for 1kl ;

  • (iii)

    we have

    c+bAλ44=λ(uλ)+k=1lλ(wk)

    and

    Aλ2=uλ22+k=1lwk22,(3.12)

    where

    λ(u)=a+bAλ223|u|2dx+123V(x)u2dx-λ23(Iα*F(u))F(u)dx(3.13)

    and

    λ(u)=a+bAλ223|u|2dx+V23u2dx-λ23(Iα*F(u))F(u)dx.(3.14)

We agree that in the case l=0, the above holds without wk.

Lemma 3.7.

Assume that (V1), (V2) and (F1)(F3) hold. Then, for almost every λ(λ¯,1], there exists uλH1(R3){0} such that

Φλ(uλ)=0,Φλ(uλ)=cλ.

Proof.

Lemma 3.4 implies that Φλ(u) satisfies the assumptions of Proposition 3.1, with X=H1(3) and λ=Φλ. So, for almost every λ[1/2,1], there exists a bounded sequence {un(λ)}H1(3) (for simplicity, we denote it by {un} instead of {un(λ)}) such that

Φλ(un)cλ>0,Φλ(un)0.

By Lemma 3.6, there exist a subsequence of {un}, still denoted by {un}, and uλH1(3) such that Aλ2:=limnun22 exists, unuλ in H1(3) and λ(uλ)=0, and there exist l{0} and w1,,wlH1(3){0} such that (λ)(wk)=0 for 1kl,

cλ+bAλ44=λ(uλ)+k=1lλ(wk)

and

Aλ2=uλ22+k=1lwk22.

Since λ(uλ)=0, we have the following Pohoz̆aev identity:

𝒫~λ(uλ):=a+bAλ22uλ22+123[3V(x)+V(x)x]uλ2dx-3+α2λ3(Iα*F(u))F(u)dx=0.(3.15)

If case (i) of (V2) holds, then it follows from the Hardy inequality that

3V(x)xu2dxa23u2|x|2dx2au22for all uH1(3).(3.16)

If case (ii) of (V2) holds, then it follows from the Sobolev embedding inequality that

3V(x)xu2dx(3|max{V(x)x,0}|3/2dx)2/3(3u6dx)1/3max{V(x)x,0}3/2Su222au22for all uH1(3),(3.17)

where S=infuH1(3){0}u22/u62=(34)1/3π2. It follows from (2.4), (3.13), (3.15) and either of (3.16) and (3.17) that

λ(uλ)=λ(uλ)-14[12(λ)(uλ),uλ+𝒫~λ(uλ)]=a+bAλ24uλ22-183V(x)xuλ2dx+λ83(Iα*F(uλ))[f(uλ)uλ+(α-1)F(uλ)]dxbAλ24uλ22.(3.18)

Since (λ)(wk)=0, we have 𝒫~λ(wk)=0. Thus, from (3.3), (3.12), (3.14) and (3.15), it follows that

0=12(λ)(wk),wk+𝒫~λ(wk)=(a+bAλ2)wk22+2Vwk22-λ23(Iα*F(wk))[f(wk)wk+(3+α)F(wk)]dxJλ(wk).(3.19)

Since wkH1(3){0}, in view of Lemma 2.8, there exists tk>0 such that tk1/2(wk)tkλ. From (1.15), (2.13), (3.3),(3.14) and (3.19), one has

λ(wk)=λ(wk)-14[12(λ)(wk),wk+𝒫~λ(wk)]=a+bAλ24wk22+λ83(Iα*F(uλ))[f(uλ)uλ+(α-1)F(uλ)]dx=bAλ24wk22+Φλ(wk)-14Jλ(wk)bAλ24wk22+Φλ(tk1/2(wk)tk)-tk44Jλ(wk)bAλ24wk22+mλ.(3.20)

It follows from (2.4), (3.12), (3.18) and (3.20) that

cλ+bAλ44=λ(uλ)+k=1lλ(wk)lmλ+bAλ24[uλ22+k=1lwk22]lmλ+bAλ44for all λ(λ¯,1],

which, together with Lemma 3.5, implies that l=0 and λ(uλ)=cλ+bAλ44. Hence, unuλ in H1(3) and Φλ(uλ)=cλ. ∎

Lemma 3.8.

Assume that (V1), (V2) and (F1)(F3) hold. Then there exists u¯H1(R3){0} such that

Φ(u¯)=0,Φ(u¯)=c1>0.(3.21)

Proof.

In view of Lemma 3.7, there exist two sequences {λn}(λ¯,1] and {uλn}H1(3), denoted by {un}, such that

λn1,Φλn(un)=0,Φλn(un)=cλn.(3.22)

By Lemma 3.4 (iii), (1.13), (3.1), (3.22) and either of (3.16) and (3.17), one has

c1/2cλn=Φλn(un)-13+α𝒫λn(un)=12(3+α){a(2+α)un22+3[αV(x)-V(x)x]un2dx}+b(1+α)4(3+α)un24α2(3+α)3V(x)un2dx+b(1+α)4(3+α)un24.

This shows that {un} is bounded in H1(3). In view of the proof of Lemma 3.7, we can show that there exists u¯H1(3){0} such that (3.21) holds. ∎

Proof of Theorem 1.2.

Let

𝒦:={uH1(3){0}:Φ(u)=0},m^:=infu𝒦Φ(u).

Then Lemma 3.8 shows that 𝒦 and m^c1. For any u𝒦, (2.8), (3.2) and Lemma 3.2 imply 𝒫(u)=0. By (3.16) or (3.17), we have Φ(u)=Φ(u)-13𝒫(u)>0 for any u𝒦, and so m^0. Let {un}𝒦 be such that

Φ(un)=0,Φ(un)m^.

In view of Lemma 3.5, m^c1<m1. Arguing as in the proof of Lemma 3.8, we can prove that there exists u^H1(3){0} such that

Φ(u^)=0,Φ(u^)=m^.

This shows that u^H1(3) is a ground state solution of (1.1). ∎

4 Proof of Theorem 1.5

In this section, we give the proof of Theorem 1.5. In view of (1.7), the energy functional corresponding to (1.12) is defined in H1(3) by

Φ^(u)=123[a|u|2+u2]dx+b4(3|u|2dx)2-12q23(Iα*|u|q)|u|qdx.

From Lemma 3.2, if Φ^(u)=0, then u satisfies the following Pohoz̆aev type identity:

P^(u):=a2u22+32u22+b2u24-3+α2q23(Iα*|u|q)|u|qdx=0.(4.1)

Proof of Theorem 1.5.

Let vH1(3) be a solution to (1.12). Then

Φ^(v),v=av22+v22+bv24-1q3(Iα*|v|q)|v|qdx=0.(4.2)

By (4.1) and (4.2), one has

0=P^(v)-3+α2qΦ^(v),v=q-(3+α)2qav22+3q-(3+α)2qv22+q-(3+α)4qbv24.(4.3)

If 1<q<1+α/3 or q3+α, then (4.3) implies v=0. This completes the proof. ∎

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About the article

Received: 2018-06-28

Accepted: 2018-06-29

Published Online: 2018-09-20

Published in Print: 2019-03-01


Funding Source: National Natural Science Foundation of China

Award identifier / Grant number: 11571370

Award identifier / Grant number: 11871199

Xianhua Tang was partially supported by the National Natural Science Foundation of China (No. 11571370). Binlin Zhang was partially supported by the National Natural Science Foundation of China (No. 11871199).


Citation Information: Advances in Nonlinear Analysis, Volume 9, Issue 1, Pages 148–167, ISSN (Online) 2191-950X, ISSN (Print) 2191-9496, DOI: https://doi.org/10.1515/anona-2018-0147.

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