# Multiple Solutions for the Asymptotically Linear Kirchhoff Type Equations onRN

Authors: Yu Duan; Chun-Lei Tang
Journal: International Journal of Differential Equations
 (2016)

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## Abstract

The multiplicity of positive solutions for Kirchhoff type equations depending on a nonnegative parameterλ on RN is proved by using variational method. We will show that if the nonlinearities are asymptotically linear at infinity and λ>0 is sufficiently small, the Kirchhoff type equations have at least two positive solutions. For the perturbed problem, we give the result of existence of three positive solutions.


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## Body

## 1. Introduction and Main Results

The purpose of this article is to investigate the multiplicity of positive solutions to the following nonlocal Kirchhoff type equations:(1)a+λ∫RN∇u2+Vxu2dx-Δu+Vxu=qxfu+hxinRN,where N≥3, a is a positive constant, λ>0 is a parameter, and f:R→R is a continuous function.In recent years, the following Kirchhoff type equation(2)-a+b∫RN∇u2dxΔu+Vxu=fx,uinRN,has been studied by many researchers under variant assumptions on V and f. Problem (2) is often referred to as nonlocal problem because of the appearance of the term (∫RN∇u2dx)Δu which implies that (2) is no longer a pointwise identity. This causes some mathematical difficulties which make the study of (2) particularly interesting. Problem (2) arises in an interesting physical context. Indeed, replacing RN by a smooth bounded domain Ω⊂RN and setting V(x)=0, then problem (2) becomes the following Kirchhoff type Dirichlet problem:(3)-a+b∫Ω∇u2dxΔu=fx,u,inΩ,u=0,on∂Ω,which is related to the stationary analogue of the Kirchhoff equation(4)utt-a+b∫Ω∇u2dxΔu=fx,uthat was presented by Kirchhoff [1] as a generalization of the well-known d’Alembert’s equation (5)ρ∂2u∂t2-P0h+E2L∫0L∂u∂x2dx∂2u∂x2=fx,ufor free vibrations of elastic strings. Kirchhoff’s model takes into account the changes in length of the string produced by transverse vibrations. The readers can learn some early classical research of Kirchhoff equations from [2, 3]. However, (4) received great attention only after Lions [4] proposed an abstract framework to the problem. Some interesting results for problem (4) can be found in [5–7] and the references therein. There have been many works about the existence and multiplicity of nontrivial solutions to problem (3) using variational methods (see [8–18] and the references therein). Nevertheless, the problems they studied were based on a bounded domain of Ω⊂RN. Very recently, some authors had studied the Kirchhoff type equation on the whole space RN. Many solvability conditions with f near zero and infinity for problem (2) have been considered, such as the superlinear case (see [19–28]); the asymptotically linear case (see [29, 30]); the sublinear case (see [31–33]).Particularly, the following Kirchhoff type problem has been studied widely by some authors under various conditions onf and V:(6)a+λ∫RN∇u2+Vxu2dx-Δu+Vxu=fx,u,inRN.When f(x,u)=up-2u, p∈(2,2∗), Huang and Liu [34] considered (6) and studied existence and nonexistence of positive solution by variational methods; they also discussed the energy doubling property of nodal solutions by Nehari manifold; Wu et al. [35] gave a total description on the positive solutions to (6), and they make an observation on the sign-changing solutions. The results of [34], respectively, complement the corresponding results of [25, 36]. Li and Ye [25] showed that problem (6) has no nontrivial solution provided f(x,u)=up-2u, p∈(2,3) when λ>0 is sufficiently large. If V(x)=b, Liu et al. [37] studied the existence of a positive solution for problem (6) involving subcritical growth, which unifies and sharply improves the results of [36]. Fan and Liu [38] studied (6) with concave-convex nonlinearities and showed that problem (6) has at least two positive solutions for λ>0 sufficiently small. When f(x,u) is asymptotically linear with respect to u at infinity, Ye and Yin [39] studied (6) and proved the existence of positive solution for λ sufficiently small and the nonexistence result for λ sufficiently large. When V(x)=V(x), f(x,u)=f(u) is asymptotically linear with respect to u at infinity; Li and Sun [40] showed the existence, nonexistence, and multiplicity to (6) in radial space Hr1(RN). When the nonlinearities f is sublinear or local sublinear, [41, 42] considered the existence and multiplicity of nontrivial solutions to problem (6). Recently, some authors extend problem (6) to the p-Kirchhoff elliptic equations (see, e.g., [43–49] and the references therein). In all works for (6) mentioned above except for [39, 40], we found that the nonlinearities f are superlinear, sublinear, or local sublinear. To the best of our knowledge, there is little information on the multiplicity of solution for (6) with the nonlinearities f satisfying the asymptotically linear condition at infinity. In this paper, we will try to study multiplicity of positive solutions for problem (1) when f is asymptotically linear at infinity.In order to reduce our statements, we make the following assumptions:(V)V(x)∈C(RN,R) satisfies infRN⁡V(x)≥V0>0, where V0 is a constant.(F1)f∈C(R,R+), limt→0⁡f(t)/t=0, and f(t)≡0 for all t≤0.(F2) There exists l∈(0,+∞) such that limt→∞⁡f(t)/t=l.(F3) There exists α∈(N/2,+∞) such that q(x)∈Lα(RN).(F4)l/a>μ∗≔inf⁡∫RN∇u2+V(x)u2dx:u∈H,∫RNq(x)u2dx=1, where H will be given below.(F5)0≤h∈L2(RN).Before stating our main results, we give several notations. Set(7)H1RN≔u∈L2RN:∇u∈L2RNwith the usual norm(8)uH1=∫RN∇u2+u2dx1/2.Let (9)H=u∈H1RN:∫RN∇u2+Vxu2dx<+∞with the inner product and the norm(10)u,v=∫RN∇u·∇v+Vxuvdx,u=u,u1/2.Since V(x) satisfies (V), it is easy to see that ∥·∥H1 is equivalent to ·. Obviously, the embedding H↪Ls(RN) is continuous for any s∈[2,2∗]. We denote by ∥·∥p the usual Lp(RN) norm.Define the functionalIλ,Jλ:H→R by(11)Iλu=12au2+14λu4-∫RNqxFudx,Jλu=Iλu-∫RNhxu+dx,u∈H,where F(t)=∫0tf(s)ds. Clearly, by the assumptions imposed on f, g, and h, we know that Iλ(u) and Jλ(u) are well defined on H, and Iλ,Jλ∈C1(H,R) with the derivative given by (12)Iλ′u,v=a+λu2∫RN∇u·∇v+Vxuvdx-∫RNqxfuvdx,Jλ′u,v=Iλ′u,v-∫RNhxv+dx,u,v∈H.It is standard to verify that the weak solutions of (1) correspond to the critical points of the functional Jλ. Our first result for (1) without h(x) is as follows.Theorem 1.
Assume thatN≥3, a is a positive constant, and λ>0 is a parameter. If the conditions (V) and (F1)–(F4) hold, then there exists λ∗>0 such that, for any λ∈(0,λ∗), problem (1) has at least two positive solutions.Remark 2.
Compared with the works mentioned above except [39, 40], where the nonlinearities f are superlinear, sublinear, or local sublinear, here we consider problem (1) with asymptotically linear nonlinearities. So, our problem is different and extend the abovementioned results to some extent.Remark 3.
In [39], the authors only studied the existence of positive solutions. In this paper, we give multiplicity results when the potential V is different from the conditions of V in [39] and our method is simpler than that used in [39]. When V(x)=V(x) satisfied some assumptions, Li and Sun [40] showed the existence, nonexistence, and multiplicity of radial solutions. Here, we get multiplicity results in nonradial space.Remark 4.
Indeed, it is not difficult to find some functionsf(t), V(x), and q(x) such that the conditions of Theorem 1 are satisfied. For example, for any fixed R0>0, let(13)ft=R0t21+t,t>0,0,t≤0,qx=11+x,0≤x≤R0,∈LαBR0c0,x≥R0.Choosing V(x)=1, it is easy to know that (V) and (F1)–(F3) are satisfied for any R0>0. Moreover, l=R0 in (F2) and F(t)=R0t2-2t/2+ln⁡(1+t). To verify the condition (F4), we have to choose a special R0>0. Indeed, for R>0, take ψ∈C0∞(RN,[0,1]) such that (14)ψx=1,x≤R,0,x≥2R,and ∇ψx≤C/R for all x∈RN, where C>0 is a constant independent of x. Because of supp⁡ψ⊂B2R, thus for R0>2R, we have (15)∫RN∇ψ2+Vxψ2dx∫RNqxψ2dx=∫x≤2R∇ψ2+Vxψ2dx∫x≤Rqxψ2dx≤∫x≤2R1+C2/R2dx∫x≤R1/1+xdx≤C′R+1+R+1C′C2R2,where C′ is a constant independent of R. So, choosing R sufficiently large such that (R+1)C2/R2≤1, the condition (F4) holds for R0=aC′(R+3).In our second result, we consider the case of the perturbed Kirchhoff equations; that is,h(x)≠0, and we obtain the following result.Theorem 5.
Assume thatN≥3, a is a positive constant, and λ>0 is a parameter. If the conditions (V) and (F1–F5) hold, then there exist two constants λ~>0 and m0>0 such that, for any λ∈(0,λ~), problem (1) has at least three positive solutions when h2<m0.Remark 6.
In the aforementioned papers, the nonlinearities satisfyf(x,0)=0. Indeed, this condition is not necessary. Here, the nonlinearity may not be 0 at zero because of f(x,0)=h(x)≥0.In order to obtain our results, we have to overcome various difficulties. On the one hand, it is well known that Sobolev embeddingH↪Lp(RN) is continuous but not compact for p∈[2,2∗), and then it is usually difficult to prove that a minimizing sequence or a Palais-Smale sequence is strongly convergent if we seek solutions of problem (1) by variational methods. To overcome this difficulty, we make full use of integrability of potential function q(x) and perturbation h(x). On the other hand, as we all know, the (PS) sequence is bounded if the nonlinearity satisfies a variant of Ambrosetti-Rabinowitz type condition ((AR) in short) or 4-superlinearity. However, for the asymptotically linear case of problem (1), we can adopt a simple method to verify the boundedness of (PS) sequence. The conditions (F1) and (F2) are crucial to obtain the boundedness of (PS) sequence.This paper is organized as follows. We give some previous results and prove Theorem1 in Section 2. Section 3 is devoted to giving the proof of Theorem 5. Throughout this paper, C and Ci are used in various places to denote distinct constants.


## 2. Proof of Theorem1

In the following, we give some lemmas which are important to prove our main result.Lemma 7.
Suppose that(V) and (F1)–(F3) hold; then, Iλ(u) is coercive on H.Proof.
By(F1) and (F2), we see that f(s)/s is bounded in R. So, setting L0=sups∈R⁡f(s)/s, thus L0∈(0,+∞) and for any s∈R(16)0≤fss≤L0.Then,(17)Fs≤L02s2,∀s∈R.Because of α∈(N/2,+∞), we have(18)2<2αα-1<2∗.Furthermore, by (17), (18), (F3), and the Hölder and Sobolev inequalities, we deduce that for any u∈H(19)∫RNqxFudx≤L02∫RNqxu2dx≤L02∫RNqαxdx1/α∫RNu2α/α-1dxα-1/α≤L02∫RNqαxdx1/αu2α/α-12≤C1L02u2.Thus, we obtain (20)Iλu=12au2+14λu4-∫RNqxFudx≥12au2+14λu4-C1L02u2,which shows that Iλ(u) is coercive on H.It follows from Lemma7 that Iλ is bounded from below on H and thus we may define cλ≔infH⁡Iλ.Lemma 8.
Assume that(V) and (F1–F4) are satisfied; then, Iλ(u) satisfies the (PS) condition.Proof.
Suppose that{un}⊂H is the (PS) sequence for the functional Iλ; that is,(21)Iλunis  bounded,Iλ′un⟶0asn⟶∞.By Lemma 7, the sequence {un} is bounded in H. Going if necessary to a subsequence, we can assume that un⇀u weakly in H for some u∈H. Now, we begin to prove un→u strongly in H. As we all know, it is sufficient to show that un→u as n→∞. By (21), we see that (22)o1=Iλ′un,un-u=a+λun2un,un-u-∫RNqxfunun-udx.So, we have(23)a+λun2un,un-u=o1+∫RNqxfunun-udx.Thus, to show that 〈un,un-u〉=o(1) is equivalent to proving that(24)∫RNqxfunun-udx=o1.By (F3), for any ε>0, there exists Rε>0 such that(25)∫x≥Rqαxdx<εα,∀R≥Rε.By (16), (18), (25), (F3), and the Sobolev and Hölder inequalities, we have (26)∫RNqxfunun-udx≤∫RNqxun-ufundx≤L0∫x≤Rqxun-uundx+L0∫x≥Rqxun-uundx≤L0qLαBR0unL2α/α-1BR0un-uL2α/α-1BR0+L0qLαBRc0un-uL2α/α-1BRc0unL2α/α-1BRc0≤o1+C2qLαBRc0un-uun≤o1+C3ε.This implies(27)∫RNqxfunun-udx=o1.So, un,un-u→0. It is easy to see that u,un-u→0. Hence, un-u,un-u→0; that is, un→u strongly in H.Proof of Theorem1.
The proof of this theorem is divided into two steps.
Step 1. In this step, we will show that problem (1) has a mountain pass solution.
By(F3), we see that α>N/2, and thus 2∗(α-1)/α>2. So, we may choose a constant β∈(2,2∗(α-1)/α] such that βα/α-1≤2∗. For any ε>0, it follows from (F1) and (F2) that there exists Cε>0 and β∈(2,2∗(α-1)/α] such that (28)fs≤εs+Cεsβ-1,∀s∈R,and, then,(29)Fs≤ε2s2+Cεβsβ,∀s∈R.Using (29), (F3), and Sobolev and Hölder inequalities, we deduce that for any u∈H(30)∫RNqxFudx≤ε2∫RNqxu2dx+Cεβ∫RNqxuβdx≤ε2∫RNqαxdx1/α∫RNu2α/α-1dxα-1/α+Cεβ∫RNqαxdx1/α∫RNuβα/α-1dxα-1/α≤ε2∫RNqαxdx1/αu2α/α-12+Cεβ∫RNqαxdx1/αuβα/α-1β≤C4ε2u2+C5Cεβuβ.Thus, we obtain(31)Iλu=12au2+14λu4-∫RNqxFudx≥a-C4ε2u2-C5Cεβuβ.So, fixing ε∈(0,a/C4) and letting u=ρ>0 sufficiently small, it is easy to see that there exists a constant α>0 such that(32)Iλuu=ρ≥α>0.
By(F4), there is v∈H such that v≥0, ∫RNq(x)v2dx=1, and μ∗≤v2<l/a. Combining (F2) with Fatou’s lemma, we deduce that (33)limt→+∞⁡I0tvt2=a2v2-limt→∞⁡∫RNqxFtvt2dx≤12av2-l<0,which implies that there exist e∈H with e>ρ such that I0(e)<0. Since Iλ(e)→I0(e) as λ→0+, we see that there exists λ∗>0 such that Iλ∗(e)<0, and then(34)Iλe<Iλ∗e<0for all λ∈(0,λ∗). From (32), (34), and Mountain Pass Theorem, there is a sequence {un}⊂H such that (35)Iλunis  bounded,Iλ′un⟶0,asn⟶∞.Using Lemma 8, we know Iλ satisfies (PS)-condition. So, by Theorem  2.2 in [50], Iλ possess a critical point v0 with Iλ(v0)≥α>0. Setting v-=max⁡{0,-v}, since a+λv02〈v0,v0-〉-∫RNq(x)f(v0)v0-dx=0, then by (F1) we have (36)v0-=0,which implies v0≥0 a.e. in RN. By the strong maximum principle, v0 is positive on H and Iλ(v0)>0.
Step 2. Problem (1) has a global minimum; that is, there exists a positive function u0∈H such that Iλ′(u0)=0 and cλ=infH⁡Iλ=Iλ(u0)<0.
From Lemmas7 and 8, we know that Iλ(u) is bounded from below and satisfies (PS)-condition, and then by Theorem  4.4 in [51], cλ=infHIλ is a critical value of Iλ; that is, there exists a function u0∈H such that Iλ′(u0)=0 and Iλ(u0)=cλ. In view of (34), we know cλ=Iλ(u0)≤Iλ(e)<0, which implies that u0≠0, and using the same arguments as in Step 1, it is easy to know that u0 is positive.
Because ofIλ(v0)>0>Iλ(u0), we get two different critical points v0,u0>0; that is, problem (1) has two positive solutions, and then the proof of Theorem 1 is completed.


## 3. Proof of Theorem5

First, we need the following lemmas which are important to prove Theorem5.Lemma 9.
Suppose that(V) and (F1)–(F5) hold; then, Jλ(u) is coercive on H.Proof.
The proof is similar to the proof of Lemma7, so we omit it here.Lemma 10.
Assume that(V) and (F1)–(F5) are satisfied; then, Jλ(u) satisfies the (PS) condition.Proof.
By Lemma8, we only need to show ∫RNh(x)(un-u)dx=o(1). By (F5), for the above-given ε>0, there exists Rε′>0 such that(37)∫x≥Rh2xdx<ε2,∀R≥Rε′.By (37) and the Sobolev and Hölder inequalities, we have (38)∫RNhxun-udx≤∫x≤Rhxun-udx+∫x≥Rhxun-udx≤hL2BR0un-uL2BR0+hL2BRc0un-uL2BRc0≤o1+C′ε;that is(39)∫RNhxun-udx=o1.Using (27) and (39), we obtain (40)∫RNqxfunun-udx+∫RNhxun-udx=o1.So, un,un-u→0. It is easy to see that u,un-u→0. Hence, un-u,un-u→0; that is, un→u strongly in H.Proof of Theorem5.
The proof of this theorem is divided into four steps.
Step 1. In this step, we will show that problem (1) has a positive mountain pass solution.
Set(41)Iλ∗u=12au2+14λ∗u4-∫RNqxFudx,where λ∗ is given in Theorem 1. By Lemma 7, we known that Iλ∗(u) is coercive on H. So we can define (42)cλ∗≔infH⁡Iλ∗.Using (34), we have cλ≤cλ∗<0. By (31), we know (43)Jλu=12au2+14λu4-∫RNqxFudx-∫RNhxu+dx≥a-C4ε2u2-C5Cεβuβ-1V0h2u≥ua-C4ε2u-C5Cεβuβ-1-1V0h2.So, choosing ε=a/2C4 and setting (44)gt=a4t-C5Cεβtβ-1for t≥0, we see that there exists a constant ρ1>0 sufficiently small such that 0<ρ1<δ≔min⁡{ρ,a2-λ∗cλ∗-a/λ∗} and g(ρ1)>0, where β∈(2,2∗(α-1)/α] and ρ is given by (32). Taking m1≔V0/2g(ρ1), it then follows that there exists a constant α1≔1/2g(ρ1)ρ1>0 such that(45)Jλuu=ρ1≥α1for all h satisfying h2<m1.
Using the similar proof of (34), we can obtain that there exists a constant λ~>0 and a function e∈H with e>ρ1 such that(46)Jλe<0for all λ∈(0,λ~).
From (45), (46), and Mountain Pass Theorem, there is a sequence {un}⊂H such that (47)Jλunis  bounded,Jλ′un⟶0asn⟶∞.It follows from Lemma 10 that Jλ satisfies (PS)-condition. So, using Theorem  2.2 in [50], Jλ possess a critical point u1 with Jλ(u1)≥α>0 when h2<m1. Let u-=max⁡{0,-u}. Since a+λu12〈u1,u1-〉-∫RNq(x)f(u1)u1-dx-∫RNh(x)(u1-)+dx=0, then by (F1) and (F5) we have (48)u1-=0,which implies u1≥0 a.e. in RN. So, by the strong maximum principle, u1 is positive on H.Step 2. In this step, we prove the existence of local minimum solution for problem (1).Sinceh∈L2(RN) and h≥0, we can choose a function ψ∈H such that (49)∫RNhxψ+dx>0.Hence, we obtain (50)limt→0⁡Jλtψt=limt→0⁡t2/2aψ2+t4/4λψ4-∫RNqxFtψdx-∫RNhxtψ+dxt≤limt→0⁡t2aψ2+t34λψ4-∫RNhxψ+dx=-∫RNhxψ+dx<0.Thus, we have (51)c1≔inf⁡Jλu:u∈B¯ρ1<0,where ρ1 is given by (45) and Bρ1={u∈H:u<ρ1}. By Ekeland’s variational principle, there exists a sequence {un}⊂B¯ρ such that (52)c1≤Jλun<c1+1n,Jλv≥Jλun-1nv-unfor all v∈B¯ρ1. Then, by a standard procedure, we can show that {un} is a bounded Palais-Smale sequence of Jλ. Therefore, Lemma 10 implies that there exists a function u2∈Bρ1 such that Jλ(u2)=c1<0 and Jλ′(u2)=0. Similarly, u2>0.Step 3. Problem (1) has a global minimum.It follows from Lemmas9 and 10 that Jλ(u) is bounded from below and satisfies the (PS) condition, so we may define c2≔infH⁡Jλ. Using Theorem  4.4 in [51], c2=infH⁡Jλ is a critical value of Jλ; that is, there exists a critical point u3∈H such that Jλ′(u3)=0 and Jλ(u3)=c2. By (46), c2=Jλ(u3)=infH⁡Jλ<0, which implies u3≠0. Similarly, u3>0.Step 4. u1, u2, and u3 are different from each other; that is, problem (1) has three positive solutions.SinceJλ(u1)>0>Jλ(u2) and Jλ(u1)>0>Jλ(u3), thus u1≠u2 and u1≠u3. Next, we claim that u2≢u3 and then the proof of Theorem 5 is completed. Using the proof of Theorem 1, we know Iλ(u0)=cλ<0 and Iλ(0)=0. Using (31), Iλ(u)>0 for all u<ρ. Then, (53)Iλu=12au2+14λu4-∫RNqxFudx=12au2+14λu4-∫RNqxFudx≤12au2+14λ∗u4<-14cλ∗uniformly in λ when u<δ≔min⁡{ρ,a2-λ∗cλ∗-a/λ∗}. So, when u<δ and h2<m2≔-V0cλ∗/4δ, we deduce that(54)Jλu=Iλu-∫RNhxu+dx≥14cλ∗-1V0h2u≥14cλ∗-1V0h2δ>12cλ∗.Since ρ1<δ, using (54), we obtain (55)c1=inf⁡Jλu:u∈B¯ρ1=Jλu2>12cλ∗,Jλu0=Iλu0-∫RNhxu0+dx≤Iλu0=cλ≤cλ∗,when h2<m2. Thus, we have (56)Jλu0≤cλ∗<12cλ∗<Jλu2.So, c2=Jλ(u3)=infH⁡Jλ≤Jλ(u0)<Jλ(u2); that is, u2≢u3. Set m0≔min⁡{m1,m2}. From the discussion above, we can obtain that problem (1) has three positive solutions when h2<m0.



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