We study the existence of infinitely many solutions for a class of modified Schrödinger-Kirchhoff-type equations by the dual method and the nonsmooth critical point theory.

1. Introduction and Main Result

In this paper, we study the following modified Schrödinger-Kirchhoff-type equations of the form(1)-a+b∫RN∇u2dxΔu-auΔu2+Vxu=gx,u,x∈RN,where a>0, b≥0, N≥3, g∈C(RN×R,R), and V∈C(RN,R).

When auΔu2=0, (1) is reduced to the following Kirchhoff-type problem:(2)-a+b∫RN∇u2dxΔu+Vxu=gx,u,x∈RN.If V(x)=0, problem (2) is related to the stationary analogue of the Kirchhoff equation(3)utt-a+b∫RN∇u2dxΔu=gx,uproposed by Kirchhoff in [1] as an existence of the classical D’Alembert’s wave equation for free vibrations of elastic string. Kirchhoff’s model takes into account the changes for free vibrations of elastic strings. Recently, there have been many papers concerned with the Kirchhoff-type problems by variational methods; see [2–8] and the references therein. Many studies of them are concentrated on a bounded smooth domain Ω of RN(N=1,2,3); it is well known that the embedding H1(RN)↪Lt(RN)(2≤t<2∗) is not compact. Hence, if we look for solution by variational methods, it is very difficult to prove (PS) condition. Moreover, in order to check the (PS) condition or some of its variants, one has to impose certain conditions.

When b=0 and a=1, (1) is reduced to the following modified nonlinear Schrödinger equation:(4)-Δu-uΔu2+Vxu=gx,u,x∈RN.Solutions of (4) are standing waves of the following quasilinear Schrödinger equations: (5)iψt+Δψ-Vxψ+kΔαψ2α′ψ2ψ+gx,ψ=0,x∈RN,where k is a real constant and α and g are real functions. Equations (5) are derived as model of several physical phenomena, such as [9, 10]. Many achievements had been obtained on the existence of ground states, infinitely many solutions, and soliton solutions for (4), by a dual approach, Nehari method, and the minimax methods in critical point theory, applying the perturbation approach and the Lusternik-Schnirelmann category theory; see [11–18]. Recently, K. Wu and X. Wu [19] obtained infinitely many small energy solutions of (1) by applying Clark’s theorem to a perturbation functional. In (1), set g(x,u)=αk(x)|u|p-2u+βu2(2∗)-1, in which α and β are real parameters, 2∗=2N/(N-2)(N≥3); Liang and Shi [20] obtained soliton solutions of (1) by using the concentration-compactness principle and minimax methods.

In this paper, we transform (1) to another equation with a continuous energy functional in some Banach space. We obtain the existence of multiple solution for problem (1) via using nonsmooth critical point theory and using some new techniques. Throughout this paper, the main ideas used here come from Colin and Jeanjean [12] and Liu et al. [14].

We need the following several notations. Let (6)H1RN≔u∈L2RN:∇u∈L2RNwith the inner product(7)u,vH1=∫RN∇u·∇v+uvdxand the norm(8)uH12=u,uH1=∫RN∇u2+u2dx.Let(9)HV1RN≔u∈H1RN:∫RNVxu2dx<∞and the norm(10)uHV12=∫RN∇u2+Vxu2dx.

Let the following assumption (V) hold:

V∈C(RN,R), V0∶=infx∈RNV(x)>0, and lim|x|→∞V(x)=∞.

Moreover, we need the following assumptions:

Let g∈C(RN×R,R); g(x,t) is odd in t.

There exist constants C>0 and 4<p<22∗ such that(11)gx,t≤C1+tp-1

for all (x,t)∈RN×R, where 2∗=2N/(N-2) is the Sobolev critical exponent.

g(x,t)=o(|t|) uniformly in x as |t|→0.

There exists μ>4 such that μG(x,t)≤g(x,t)t,∀(x,t)∈RN×R, and infx∈RN,t=1G(x,t)>0, where G(x,t)∶=∫0tg(x,s)ds.

The main result of this paper is as follows.

Theorem 1.

Assume that (V) and (g1)–(g4) are satisfied. Then problem (1) has a sequence {un} of solutions such that J(un)→∞ as n→∞.

Throughout the paper, → and ⇀ denote the strong and weak convergence, respectively. C, c, Ci, and ci express distinct constants. For 1≤s<∞, the usual Lebesgue space is endowed with the norm (12)us≔∫RNusdx1/s.

The paper is organized as follows. In Section 2, we reformulate our problem into a new one which has an associated functional well defined in a suitable space and present some preliminary results. In Section 3, we introduce some notions and results of nonsmooth critical point theory and we show that the functional I satisfies (PS)C condition. In Section 4, we complete the proof of Theorem 1.

2. The Dual Variational Framework and Preliminary Results

The energy functional corresponding to problem (1) is defined as follows: (13)Ju=a2∫RN∇u2dx+b4∫RN∇u2dx2+a∫RNu2∇u2dx+12∫RNVxu2dx-∫RNGx,udx.It should be pointed out that the main difficulty in treating this class of quasilinear equations in RN is the lack of compactness and the second-order nonhomogeneous term auΔu2 which prevents us from working directly in a classical function space. J is not defined in HV1(RN); thus we may not apply directly the variational method to study (1). To move these obstacles, we make the change of variable (see [12, 14]); that is, we consider v=f-1(u), where f is defined by(14)f′t=11+2f2t,on 0,+∞,ft=-f-t,on -∞,0.In order to prove our main result, we need some further properties of the function f.

Lemma 2.

The function f has the following properties:

f is uniquely defined, C∞, and invertible;

f′t≤1 for all t∈R;

ft≤t for all t∈R;

limt→0(ft/t)=1; limt→+∞(ft/t)=21/4;

ft≤21/4|t|1/2 for all t∈R;

(1/2)ft≤tf′(t)≤f(t), for all t≥0; f(t)<tf′(t)<(1/2)f(t) for all t<0;

there exists a positive constant C such that (15)ft≥Ct,ift≤1,Ct1/2,ift≥1;

ftf′t≤1/2<1 for all t∈R;

there exist positive constants C1, C2 such that (16)t≤C1ft+C2ft2∀t∈R;

f2(t) is convex and f2(λt)≤λ2f2(t) for all t∈R and λ≥1;

∀ξ>0, ∃C(ξ)>0, s. t. f2(ξt)≤C(ξ)f2(t);

f(t)/t is decreasing for t>0.

After this change of variables, the functional (17)Iv=Jfv=a2∫RN∇fv2dx+a∫RNf2v∇fv2dx+b4∫RN∇fv2dx2+12∫RNVxf2vdx-∫RNGx,fvdx=a2∫RN∇v2dx+b4∫RN∇fv2dx2+12∫RNVxf2vdx-∫RNGx,fvdxis well defined on(18)E=v∈H1RN:∫RNVxf2vdx<∞which is a Banach space endowed with the norm (19)v=∇v2+infλ>01λ1+∫RNVxf2λvdx≔∇v2+v.

A standard argument which is similar to that in [12] shows that if v∈E is a critical point of the functional I, then u=f(v)∈E and u is a weak solution of (1).

We have the following result with respect to the space E. Its proof can be found in [13, 14].

Proposition 3.

(1) There exists a positive C such that, for all v∈E, (20)∫RNVxf2vdx1+∫RNVxf2vdx1/2≤Cv.

(2) If vn→v in E, then(21)∫RNVxf2vn-f2vdx⟶0,∫RNVxfvn-fv2dx⟶0.

(3) If vn(x)→v(x) a.e. x∈RN and (22)∫RNVxf2vndx⟶∫RNVxf2vdx,then(23)vn-v⟶0,i.e.infλ>01λ1+∫RNVxf2λvn-vdx⟶0.

(4) If vn→v in E, then f(vn)→f(v) in Lr(RN), for 2≤r≤22∗.

(5) Under assumption (V), the embedding E↪Ls(RN) is compact for 2≤s<2∗, and the embedding E↪Ls(RN) is continuous for 2≤s≤2∗.

(6) The embedding E↪H1(RN) is continuous. Moreover, C0∞(RN) is dense in E.

Lemma 4.

Assume that conditions (V), (g2), and (g3) hold; then, one has for I the following assertions:

I:E→R is continuous.

For every v∈E and φ∈E∩L∞(RN), the derivation of I in the direction φ at v exists and will be denoted by(24)I′v,φ=a∫RN∇v∇φdx+b∫RNf′v2∇v2dx∫RNf′v2∇v∇φdx+∫RNf′vf′′v∇v2φdx+∫RNVxfvf′vφdx-∫RNgx,fvf′vφdx.

The map 〈v,φ〉→〈I′(v),φ〉 satisfies the following:

〈I′(v),φ〉 is linear in φ∈E∩L∞(RN);

〈I′(v),φ〉 is continuous in v; that is, if vn→v in E, then 〈I′(vn),φ〉→〈I′(v),φ〉 as n→∞.

Let {vn}⊂E be a sequence such that vn→v in E. By Proposition 3, we have (26)limn→∞∫RNVxf2vndx=∫RNVxf2vdx.Moreover, vn→v in Ls(RN) for 2≤s≤2∗, ∇vn→∇v in L2(RN), and, up to subsequence, vn(x)→v(x), ∇vn(x)→∇v(x) a.e. x∈RN.

∀ɛ>0, by (g2) and (g3), there exists Cɛ>0 such that(27)gx,t≤ɛt+Cɛtp-1,Gx,t≤ɛt2+Cɛtp,∀x,t∈RN×R.Thus, by Lemma 2 and (27), (28)Gx,fvn-Gx,fv≤ɛvn2+v2+Cvnp/2+vp/2.By Lemma A.1 of [2] and Lebesgue’s theorem, one has(29)limn→∞∫RNGx,fvndx=∫RNGx,fv.By Lemma 2 and Lebesgue’s theorem,(30)∫RN∇fvn2dx2-∫RN∇fv2dx2≤C∫RN∇fvn2dx-∫RN∇fv2dx=C∫RNf′vn2∇vn2-∇v2dx+∫RNf′vn2-f′v2∇v2dx≤on1;thus(31)limn→∞∫RN∇fvn2dx2=∫RN∇fv2dx2.Consequently, I(vn)→I(v) and I is continuous in E. Similarly, it can be proved that 〈I′(v),φ〉 is continuous in v for each φ∈E∩L∞(RN). The proof of (24) is standard by using conditions (V), (g2), and (g3). It is obvious that 〈I′(v),φ〉 is linear in φ∈E∩L∞(RN). The proof is completed.

3. Nonsmooth Critical Framework

Let us begin by recalling some notions and results of nonsmooth critical point theory (see [21, 22]).

In the following, X will denote a metric space endowed with the metric d.

Definition 6.

Let f:X→R be a continuous function and let u∈X. One denotes by df(u) the supremum of the σ’s in [0,+∞) such that there exist δ>0 and a continuous map(32)H:Bu,δ×0,δ⟶Xsatisfying(33)dHv,t,v≤t,fHv,t≤fv-σtfor all (v,t)∈B(u,δ)×[0,δ], where B(u,δ) is the open ball of center u∈X and of radius δ. The extended real number df(u) is called the weak slope of f at u.

If X is a Finsler manifold of class C1 and f∈C1, it turns out that df(u)=f′(u).

The function df:X→[0,+∞] is lower semicontinuous.

Definition 7.

Let f:X→R be a continuous function. A point u∈X is called a critical point of f if df(u)=0. A real number c is called a critical value of f if there exists a u∈X such that f(u)=c and df(u)=0.

Definition 8.

Let f:X→R be a continuous function and let c∈R. One says that f satisfies the (PS)c condition if each sequence {un}⊂X with f(un)→c and df(un)→0 has a convergent subsequence.

We can state a generalized version of the symmetric Mountain Pass Theorem for the case of continuous functionals.

Theorem 9 (see [<xref ref-type="bibr" rid="B2">21</xref>]).

Let E be an infinite-dimensional Banach space and let f:E→R be continuous, even, and satisfying (PS)c condition for every c∈R. Assume the following:

There exist ρ>0, β>f(0), and a subspace V⊂E of finite codimension such that (34)∀u∈V:u=ρ⟹fu≥β.

For every finite-dimensional subspace W⊂E, there exists R>0 such that (35)∀u∈W:u≥R⟹fu≤f0.

Then there exists a sequence {cn} of critical values of f with cn→∞ as n→∞.

Now consider the functional I given in the previous section.

Lemma 10.

Consider dI(v)≥sup{〈I′(v),φ〉:φ∈E∩L∞(RN), φ=1}.

Proof.

We use a similar argument in the proof of Theorem 1.5 in [21]. Let v∈E and(36)Ψv=supI′v,φ:φ∈E∩L∞RN,φ=1.If Ψ(v)=0, the conclusion is obvious. Otherwise, take σ>0 such that Ψ(v)>σ. Then there exists a point φ∈E∩L∞(RN) with φ=1 and 〈I′(v),φ〉>σ. By Lemma 4, there exist δ0>0 such that 〈I′(u),φ〉>σ for every u∈B(v,δ0). Let δ=δ0/2 and H(u,t)=u-tφ. It is obvious that H:B(u,δ)×[0,δ]→E is continuous,(37)Hu,t-u≤t,IHu,t=IHu,0-t∫01I′Hu,st,φds≤Iu-σtfor (u,t)∈B(v,δ)×[0,δ]. Hence, by Definition 6, dI(v)≥σ and dI(v)≥Ψ(v) by the arbitrariness of σ. The proof is completed.

Lemma 11.

Under assumptions (g2), (g3), and (V), if v∈E satisfies 〈I′(v),φ〉=0, ∀φ∈E∩L∞(RN), then v∈L∞(RN).

Proof.

By (25), we have(38)a∫RN∇v∇φdx+b∫RN∇v21+2f2vdx∫RN∇v∇φ·1+2f2v-2∇v2fvf′vφ1+2f2v2dx=∫RNgx,fvf′vφdx-∫RNVxfvf′vφdxfor all φ∈E∩L∞(RN).

For any T>0, let vT=v if v≤T and vT=sign(v)T if v≥T. For any r>0, take φ=vvT2r in the above equality; then it can be deduced from (38) that (39)a∫RN∇v·∇vvT2rdx+b∫RN∇v21+2f2vdx∫RN∇v·∇vvT2r1+2f2v-2∇v2fvf′v·vvT2r1+2f2v2dx=∫RNgx,fvf′v·vvT2rdx-∫RNVxfvf′v·vvT2rdx.By Lemma 2 and the Sobolev embedding theorem, (40)aâˆ«RNâˆ‡vÂ·âˆ‡vvT2rdx+bâˆ«RNâˆ‡v21+2f2vdxâˆ«RNâˆ‡vÂ·âˆ‡vvT2r1+2f2v-2âˆ‡v2fvfâ€²vÂ·vvT2r1+2f2v2dxâ‰¥aâˆ«RNâˆ‡vÂ·âˆ‡vvT2rdx=aâˆ«RNâˆ‡v2vT2r+2rv2râˆ‡vT2dxâ‰¥ar+2âˆ«RNâˆ‡v2vT2r+2r+r2v2râˆ‡vT2dx=ar+2âˆ«RNâˆ‡vvTr2dxâ‰¥C1r+1âˆ«RNvÂ·vTr2âˆ—dx2/2âˆ—,where C1 is positive constant.

For any 0<ɛ≤V0, by (27), (V), Lemma 2, Hölder inequality, and Sobolev inequality, for simplicity, taking q=p/2, we have (41)∫RNgx,fvf′v·vvT2rdx-∫RNVxfvf′v·vvT2rdx≤∫RNɛfv+Cɛfvp-1f′vv·vT2rdx-∫RNV0fvf′vv·vT2rdx≤C∫RNfvp·vT2rdx≤C∫RNvp/2·vT2rdx≤C∫RNv2vT2r2∗/2∗+2-qdx2∗+2-q/2∗∫RNv2∗dxq-2/2∗≤C2∫RNv2vT2r2∗/2∗+2-qdx2∗+2-q/2∗,where C and C2 are positive constants.

Set d=2∗/(2∗+2-q). Because 2<q=p/2<2∗, 1<d<2∗/2. Moreover, by (39), (40), and (41), we have (42)∫RNv2∗vT2∗rdx1/2∗r+1≤Cr+11/2r+1∫RNv2dvT2drdx1/2dr+1.Taking the limit T→∞ in (42), (43)∫RNv2∗r+1dx1/2∗r+1≤Cr+11/2r+1∫RNv2dr+1dx1/2dr+1.Set r0=0 and 2d(rk+1+1)=2∗(rk+1), ∀k∈N. Then rk→∞ as k→∞ and (44)∫RNv2drk+1dx1/2drk+1≤∏i=1kCri+11/2ri+1∫RNv2∗dx1/2∗.Because(45)∏i=1kCri+11/2ri+1=exp∑i=1klnC2∗/2d-1i22∗/2d-1i=exp∑i=1klnC22∗/2d-1i+iln2∗/2d-122∗/2d-1iis convergent as k→∞, let Ck=∏i=1k[C(ri+1)]1/2ri+1; then Ck→C∞>0 as k→∞. Hence (46)vL2drk+1RN≤CkvL2∗RN.Let k→∞; then, we have (47)vL∞RN≤C∞vL2∗RN≤C,where the positive constant C is independent of k. The proof is completed.

Corollary 12.

If v∈E is a critical point of I, then v∈L∞(RN).

Proof.

The proof is completed by Lemmas 10 and 11.

Lemma 13.

∀c∈R, under assumptions (g2), (g3), and (g4), the functional I satisfies (PS)c condition.

Proof.

Let {vn}⊂E be a sequence with I(vn)→c and dI(vn)→0. Then, it follows from Lemma 10 that(48)δn=supI′vn,φ:φ∈E∩L∞RN,φ=1⟶0.Moreover,(49)I′vn,φ≤δnφ,∀φ∈E∩L∞RN.

Note that(50)I′vn,φ=a∫RN∇vn∇φdx+∫RNVxfvnf′vnφdx-∫RNgx,fvnf′vnφdx+b∫RN∇vn21+2f2vndx∫RN∇vn∇φ·1+2f2vn-2∇vn2fvnf′vnφ1+2f2vn2dx.Letting φ=wn=f(vn)/f′(vn)=1+2f2(vn)f(vn) and(51)∇wn=1+2f2vn1+2f2vn∇vn≤2∇vn,we deduce that wn≤Cvn for some positive constant C. From (49) and (50), we have(52)CÎ´nvnâ‰¥aâˆ«RN1+2f2vn1+2f2vnâˆ‡vn2dx+bâˆ«RNâˆ‡vn21+2f2vndx2+âˆ«RNVxf2vndx-âˆ«RNgx,fvnfvndxfor each n. Since I(vn)→c, by (g4) and Lemma 2, ∀λ>0, we have (53)c+on1+Cδnvn≥Ivn-1μI′vn,wn=a∫RN12-1μ1+2f2vn1+2f2vn∇vn2dx+14-1μb∫RNf′vn2∇vn2dx2+12-1μ∫RNVxf2vndx+∫RN1μgx,fvnfvn-Gx,fvndx≥a12-2μ∫RN∇vn2dx+12-1μ∫RNVxf2vndx≥C1∫RN∇vn2dx+1+∫RNVxf2vndx-C1≥C1∫RN∇vn2dx+infλ>01λ1+∫RNVxf2λvdx-C1,where C1>0. Their estimates imply that vn is bounded in E.

Hence, up to a subsequence, vn⇀v in E. Because δn→0, from (49) and (50), (54)on1+∫RNgx,fvnfvndx=a∫RN1+2f2vn1+2f2vn∇vn2dx+b∫RNf′vn2∇vn2dx2+∫RNVxf2vndx.Because 〈I′(v),φ〉=0 for each φ∈E∩L∞(RN), set w=f(v)/f′(v); it follows from Lemma 11 that 〈I′(v),w〉=0. Consequently, (55)∫RNgx,fvfvdx=a∫RN1+2f2v1+2f2v∇v2dx+b∫RNf′v2∇v2dx2+∫RNVxf2vdx.By Proposition 3, we get vn→v in Ls(RN) for 2≤s<2∗. Using (27), Lemma 2, and Lebesgue’s theorem, we obtain(56)∫RNgx,fvnfvndx⟶∫RNgx,fvfvdx.

By assumption (V), Lemma 2, the Fatou Lemma, and the Lebesgue dominated convergence theorem, one has(57)liminfn→∞∫RN∇vn2dx≥∫RN∇v2dx,liminfn→∞∫RN2f2vn1+2f2vn∇vn2dx≥∫RN2f2v1+2f2v∇v2dx,liminfn→∞∫RNf′vn2∇vn2dx≥∫RNf′v2∇v2dx,liminfn→∞∫RNVxf2vndx≥∫RNVxf2vdx.These imply that(58)∫RN∇vn2dx⟶∫RN∇v2dx,(59)∫RNVxf2vndx⟶∫RNVxf2vdx.Notice that ∇vn⇀∇v in L2(RN); then, we conclude that ∇vn→∇v in L2(RN). By Proposition 3 and (59), we have (60)infλ>01λ1+∫RNVxf2λvn-vdx⟶0.We obtain vn→v in E; that is, the functional I satisfies (PS)C condition.

4. Proof of the Main Result

The following lemma implies that I possesses the symmetric Mountain Pass Geometry.

Lemma 14.

Suppose that (V), (g2), and (g3) are satisfied. Then the functional I satisfies condition (i) of Theorem 9.

Proof.

Let S(ρ)={v∈E:v=ρ}, 0<ρ<4. If v≥(1/2)ρ, by Lemma 2, we have (61)12ρ≤infλ>01λ1+∫RNVxf2λvdx≤14ρ+14ρ∫RNVxf24ρvdx≤14ρ+4ρ∫RNVxf2vdx,which implies that(62)∫RN∇v2+Vxf2vdx≥116ρ2for v≥(1/2)ρ. Note that ∇v2≥(1/2)ρ if v∈S(ρ) with v<(1/2)ρ. Hence (63)infv∈Sρ∫RN∇v2+Vxf2vdx≥116ρ2.By (V), (27), Lemma 2, and Proposition 3, let 0<ɛ≤(1/4)V0; then, we have (64)Iv≥a2∫RN∇v2dx+12∫RNVxf2vdx-∫RNɛf2v+Cɛvpdx≥a2∫RN∇v2dx+14∫RNVxf2vdx-C∫RNvp/2dx≥116mina2,14ρ2-Cρp/2for v∈S(ρ) and ρ<1. Hence condition (i) in Theorem 9 holds for small ρ>0.

Lemma 15.

Assume that (V), (g2), and (g4) hold; then the functional I satisfies condition (ii) of Theorem 9.

Proof.

By Lemma 2, for m>4,∃C>0, such that (65)tm/2≤t2+Cftm,∀t∈R.By (27) and (g4), ∃C1>0 and C2>0; one has (66)Gx,s≥C1sμ-C2s2,∀x,s∈RN×R.By Lemma 2, we have(67)Gx,fs≥C1fsμ-C2f2s≥C3sμ/2-C4s2∀(x,s)∈RN×R, C3>0, C4>0. Let W be any finite-dimensional subspace of E, ∀v∈W; by (65), (67), and Lemma 2, (68)Iv≤C∫RN∇v2dx+∫RNVxv2dx+C∫RNv2dx-∫RNGx,fvdx≤C∫RN∇v2+Vxv2dx+C∫RNv2dx-C3∫RNvμ/2dx+C4∫RNv2dx.Since all norms are equivalent in a finite-dimensional space W, I(v)→-∞ as v→∞. So the conclusion follows.

Proof of Theorem <xref ref-type="statement" rid="thm1.1">1</xref>.

The functional I is evidently even. Therefore, Theorem 9 implies that I has a critical sequence {vn}⊂E such that I(vn)→∞ as n→∞. Note that I(vn)=J(f(vn)); the proof is completed.

Conflict of Interests

The author declares that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

This work is supported by Scientific Research Foundation of Yunnan Province Education Department (2015C075Y) and the Scholarship Award for Excellent Doctoral Student Granted by Yunnan Province.

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