# Multiple Sign-Changing Solutions for Kirchhoff-Type Equations

Authors: Xingping Li; Xiumei He
Journal: Discrete Dynamics in Nature and Society
 (2015)

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## Abstract

We study the following Kirchhoff-type equations-a+b∫Ω∇u2dxΔu+Vxu=fx,u, in Ω, u=0, in ∂Ω, where Ω is a bounded smooth domain of RN(N=1,2,3), a>0, b≥0, f∈C(Ω¯×R,R), and V∈C(Ω¯,R). Under some suitable conditions, we prove that the equation has three solutions of mountain pass type: one positive, one negative, and sign-changing. Furthermore, if f is odd with respect to its second variable, this problem has infinitely many sign-changing solutions.


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## Body

## 1. Introduction and Preliminaries

In this paper, we study the following Kirchhoff-type equations:(1)-a+b∫Ω∇u2dxΔu+Vxu=fx,u,inΩ,u=0,on∂Ω,where Ω is a bounded smooth domain of RN(N=1,2,3), a>0, b≥0, and f∈C(Ω¯×R,R) with F(x,t)=∫0t‍f(x,s)ds, x∈Ω¯, and V∈C(Ω¯,R).WhenV(x)=0, problem (1) is related to the stationary analogue of the Kirchhoff equation (2)utt-a+b∫Ω∇u2dxΔu=fx,u,proposed by Kirchhoff in [1] as an existence 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. Some interesting studies by variational methods can be found in [2–11]. In these papers, many achievements had been obtained on the existence of ground states, infinitely many radial solutions, soliton solutions, and high energy solution for (1) by using the Fountain Theorem, the mountain pass theorem, using the variational methods and the local minimum methods, and the invariant sets of descent flow. Particularly, in [12], the authors consider the following problem: (3)-a+b∫Ω∇u2dxΔu+ϕu=fx,u,inΩ-Δϕ=u2,inΩu=0on∂Ω,where Ω is a bounded smooth domain of RN (N=1,2 or 3), a>0, b≥0, and f:Ω¯×R→R is a continuous function which is 3-superlinear. The unbounded sequence of sign-changing solutions of (3) is obtained by using some variants of the mountain pass theorem. In [13], authors considered the following p-Laplacian equation coupled with the Dirichlet boundary condition:(4)-Δpu=λαxfuinΩ,u=0on∂Ω,where p>N, the parameter λ>0, α∈L1(Ω) is a nonzero potential, and f∈C([0,+∞),R) with f(0)=0. By using variational method, they proved that for every λ>1 problem (4) has at least two nonzero, nonnegative weak solutions, while there exists λ^>1 such that problem (4) has at least three nonzero, nonnegative weak solutions. In [14], Ricceri proved that there were at least three distinct weak solutions in H01(Ω) for the following equation: (5)-Δu=λfu+μguinΩu=0on∂Ωby using and improving the three critical points’ theorem, where f,g∈C(R,R); let Jμ be an open interval with Jμ⊂[0,+∞), λ∈Jμ.In this paper, we study the sign-changing solutions of problem (1). We need the following assumptions:(V)V∈C(Ω¯,R), V0≔infx∈Ω¯⁡V(x)>0.(f1)f∈C(Ω¯×R,R), and |f(x,t)|≤C(1+|t|p-1) for some 4<p<2∗, where C is a positive constant, 2∗=+∞ for N=1,2, and 2∗=6 for N=3.(f2)f(x,t)=o(|t|) uniformly in x∈Ω¯, as |t|→0.(f3) There exist μ>4 and M>0 such that (6)0<μFx,t≤fx,tt,∀x,t∈Ω¯×R,t≥M.(f4)f(x,-t)=-f(x,t), for all x∈Ω¯ and t∈R.We need the following several notations. LetX≔H01(Ω) with the inner produce and norm(7)u,v=∫Ω∇u·∇vdx,u=u,u1/2,u,v∈X.Recall that a functionu∈X is called a weak solution of problem (1) if (8)a+bu2∫Ω∇u∇φdx+∫ΩVxuφdx=∫Ωfx,uφdx,∀φ∈X.Seeking a weak solution of problem (1) is equivalent to finding a critical point of the C1-functional (9)Ju=a2∫Ω∇u2dx+b4∫Ω∇u2dx2+12∫ΩVxu2dx-∫ΩFx,udx,u∈X.SinceΩ is a bounded domain, it is well known that the embedding X↪Ls(Ω) is continuous for all s∈[1,2∗] and the embedding X↪Ls(Ω) is compact for all s∈[1,2∗). Furthermore, there is another norm (10)u0=∫Ω∇u2+Vxu2dx1/2,and we know that · and ·0 are equivalent on X; that is, there exist constants C_>0, C¯>0 such that(11)C_u≤u0≤C¯u,∀u∈X.By Lemma  1 in [9], we know that, under the conditions (V), (f1), and (f2), J∈C1(X,R) and for each u∈X, (12)J′u,φ=a+bu2∫Ω∇u∇φdx+∫ΩVxuφdx-∫Ωfx,uφdx,for all φ∈X.Our main result of this paper is the following.Theorem 1.
Suppose that(V) and (f1)–(f3) are satisfied. Then (1) has three solutions of mountain pass type: one positive, one negative, and one sign-changing. If moreover f is odd with respect to its second variable (i.e., (f4) holds), then problem (1) has infinitely many sign-changing solutions.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 (13)us≔∫Ωusdx1/s.The paper is organized as follows. In Section2, we introduce some notions and results of some critical theorem. In Section 3, we complete the proof of Theorem 1.


## 2. Some Critical Point Theorems

Let us begin by recalling some notions and results of some critical point theorems (see [15]).In the following,X will denote a Hilbert space endowed with the norm ·X,P⊂X, which is a closed convex cone.Forɛ>0, we denote by Vɛ(S) the ɛ-neighborhood of S⊂X; that is, (14)VɛS≔u∈X:dist⁡u,S≔infv∈Su-vX<ɛ.Define (15)+P≔u∈X:u≥0,-P≔u∈X:u≤0,Pɛ±≔Vɛ±P=u∈X:dist⁡u,±P<ɛ.LetJ∈C1(X,R). We denote by K the set of critical points of J and E=X∖K.Forε0>0, we consider the following situation.(
A
ε
0
): there exists a locally Lipschitz continuous vector field B:E→X (B odd if J is even) such that(i)
B(Pɛ±∩E)⊂Pɛ±, ∀ɛ∈(0,ε0);(ii)
there exists a constantα1>0 such that(16)J′u,u-Bu≥α1u-BuX2,∀u∈E;(iii)
forρ1<ρ2 and α<0, there exists β>0 such that u-B(u)X≥β if u∈X is such that J(u)∈[ρ1,ρ2] and J′(u)X∗≥α.Definition 2.
LetJ∈C1(X,R), c∈R. One says that J satisfies the (PS)c condition if each sequence {un}⊂X with J(un)→c and J′(un)→0 in X∗ has a convergent subsequence.Theorem 3 (see [15]).
LetJ∈C1(X,R) with J(0)=0. Assume there exists ε0>0 such that (Aε0) is satisfied. Assume also that there exist e±∈±P and r>0 such that (17)A1e±X>r,ρ≔infuX=ru∈X⁡Ju>δ≔max⁡J0,Je±.Then there exist sequences {u±n}⊂Pɛ±¯ such that (18)J′u±n⟶0inX∗,Ju±n⟶c±≔infγ∈Γ±⁡supu∈γ0,1⁡Ju≥ρ,∀ɛ∈0,ε0,where (19)Γ±≔γ∈C0,1,Pɛ±¯:γ0=0,γ1=e±.If in addition J satisfies the (PS)c condition for any c>0, then J has critical point u±∈±P∖{0}.Theorem 4 (see [15]).
LetJ∈C1(X,R). Assume there exists ε0>0 such that (Aε0) is satisfied. Assume also that there exists a continuous map φ0:Δ→X such that, for any ɛ∈(0,ε0), the following conditions are satisfied: (1)
φ0(∂1Δ)⊂Pɛ+ and φ0(∂2Δ)⊂Pɛ-.(2)
φ0(∂0Δ)⋂‍Pɛ+⋂‍Pɛ-=Ø.(3)
c0≔supu∈φ0(∂0Δ)⁡J(u)<c∗≔infu∈∂Pɛ+⋂‍∂Pɛ-⁡J(u),where(20)Δ=s,t∈R2:s,t≥0,s+t≤1,∂1Δ=0×0,1,∂2Δ=0,1×0,∂0Δ=s,t∈Δ:s+t=1.Then there exists a sequence {un}⊂Vɛ/2(X∖(Pɛ+∪Pɛ-))¯ such that (21)J′un⟶0inX∗,Jun⟶c≔infφ∈Γ⁡supu∈φΔ∩X∖Pɛ+∪Pɛ-⁡Ju≥c0,∀ɛ∈0,ε0,where (22)Γ≔φ∈CΔ,X:φ∂1Δ⊂Pɛ+,φ∂2Δ⊂Pɛ-,φ∂0Δ=φ0.If in addition J satisfies the (PS)c condition for any c>0, then J has a sign-changing critical point.In the following, we assume thatX is of the form (23)X≔⨁j=1∞Xj¯,withdim⁡Xj<∞,and that there is another norm ·∗ on X such that (X,·X) embeds continuously into (X,·∗).We introduce the following notations:(24)Yk≔⨁j=1kXj,Zk≔⨁j=k∞Xj¯,fork≥2,Jα≔u∈X:Ju≤α,forα∈R.Notice that(25)X,·X↪X,·∗⟹∃C∗>0,s.t.u∗≤C∗uX,∀u∈X,dim⁡Yk<∞⟹∃θk>0,s.t.uX≤θku∗,∀u∈Yk.Assume there exist constants ρ>0 and q>2 and numbers ρk,dk>0 such that (26)ρk/θkqρk2+ρkρk/θkρk+C∗dkρk>ρ,and define (27)Bk≔u∈Yk:u≤ρk,Nk≔u∈Zk:u∗quX2+uX·u∗uX+dk·u∗=ρ.In the following, we introduce a sign-changing critical points theorem.Theorem 5 (see [15]).
LetJ∈C1(X,R) be an even functional. Assume that there exist ρ,ρk,dk>0 and q>2 such that (26) holds. Assume also that there exists ε0>0 such that (Aε0) and the following conditions are satisfied: (B1)ak≔supu∈∂Bk⁡J(u)≤0 and bk≔infu∈Nk∩Ja0⁡J(u)→+∞, as k→∞.(B2)Nk∩Ja0⊂X∖(Pɛ+∪Pɛ-), ∀ɛ∈(0,ε0), where a0≔maxu∈Bk⁡J(u).Then, for k large enough there exists a sequence {ukn}n⊂Vɛ/2(X∖(Pɛ+∪Pɛ-))¯ such that(28)J′ukn⟶0inX∗asn⟶∞,Jukn⟶ck≔infγ∈Γk⁡maxu∈γBk∩X∖Pɛ+∪Pɛ-⁡Ju≥bk,∀ɛ∈0,ε0,where(29)Γk≔γ∈CBk,X:γisodd,γ∂Bk=id,supu∈Bk⁡Jγu≤a0,γPɛ+∪Pɛ-⊂Pɛ+∪Pɛ-.If in addition J satisfies the (PS)c condition for any c>0, then it possesses a sequence {uk} of sign-changing critical points such that J(uk)→∞, as k→∞.


## 3. Proof of Theorem1

We divide the proof of Theorem1 into the following lemmas.Foru∈X fixed, we consider the functional (30)I~uv=12a+bu2∫Ω∇v2dx+12∫ΩVxv2dx-∫Ωfx,uvdx,v∈X.It is easy to prove that I~u is of class C1, coercive, bounded below, weakly lower semicontinuous, and strictly convex in X. Therefore, by Theorem  1.1 in [16], I~u admits a unique global minimizer in X which is the unique solution to the problem (31)-a+bu2Δv+Vxv=fx,u,u∈X.Here we introduce an auxiliary operatorA, which will be used to construct the descending flow for the functional J(·). We define an operator A:X→X: for u∈X, Au∈X is the unique solution of (31). Then the set of fixed points of A coincide with the set K of critical point of J.Furthermore, the operatorA has the following important properties.Lemma 6.
(1)A is continuous and maps bounded sets to bounded sets.
(2) For anyu∈X, one has (32)J′u,u-Au≥au-Au2,J′u≤a+C¯2+bu2u-Au,where C¯>0 is defined in (11).
(3) There existsε0>0 for enough small such that A(Pɛ±)⊂Pɛ±,∀ɛ∈(0,ε0).Proof.
(1) Let{un}⊂X such that un→u in X. For any w∈X, by the definition of A, we have (33)a+bun2∫Ω∇Aun∇wdx+∫ΩVxAunwdx=∫Ωfx,unwdx,(34)a+bun2∫Ω∇Au∇wdx+∫ΩVxAuwdx=∫Ωfx,uwdx.Let vn=Aun and v=Au. Taking w=vn-v∈X in (33) and (34), we obtain (35)a+bun2vn-v2+∫ΩVxvn-v2dx=∫Ωfx,un-fx,uvn-vdx+bu2-un2∫Ω∇v·∇vn-vdx.Using the Hölder inequality and the Sobolev embedding theorem, we have (36)Aun-Au≤C1fx,un-fx,up/p-1+baun2-u2·v,where C1>0 is a constant. By (f1) and Theorem  A.1 in [17], one has f(x,un)-f(x,u)→0 in Lp/(p-1)(Ω). Because un→u in X as n→∞, then un→u as n→∞. By (36), we obtain Aun-Au→0 in X, which implies that A is continuous on X.
On the other hand, for anyu∈X, taking w=Au∈X in (34), we obtain (37)a+bu2∫Ω∇Au2dx+∫ΩVxAu2dx=∫Ωfx,uAudx.Using the Hölder inequality, the Sobolev embedding theorem, (f1), and the fact b≥0, we obtain (38)Au≤C1+up-1,where C>0 is constant. This shows that Au is bounded in X whenever u is bounded in X.
(2) Takingw=u-Au∈X in (34), we have (39)a+bu2∫Ω∇Au·∇u-Audx+∫ΩVxAu·u-Audx=∫Ωfx,uu-Audx;thus (40)J′u,u-Au=a+bu2∫Ω∇u∇u-Audx+∫ΩVxuu-Audx-∫Ωfx,uu-Audx≥au-Au2,for all u∈X. Moreover, using again (34), we have (41)J′u,w=a+bu2∫Ω∇u∇wdx+∫ΩVxuwdx-∫Ωfx,uwdx=a+bu2∫Ω∇u-Au∇wdx+∫ΩVxu-Auwdx.By the Hölder inequality, we conclude that (42)J′u≤a+C¯2+bu2u-Au,where C¯>0 is defined in (11).
(3) From(f1) and (f2), for any ɛ>0, there exists Cɛ>0 such that(43)fx,t≤ɛt+Cɛtp-1,∀t∈R,(44)Fx,t≤ɛt2+Cɛtp,∀t∈R.Set u∈X and v=Au∈X. We denote w+=max⁡{0,w} and w-=min⁡{0,w}, for any w∈X. Taking w=v+ in (34) and using the Hölder inequality, we obtain (45)av+2≤ɛu+2v+2+Cɛu+pp-1v+p,which implies (46)v+2≤1aɛu+2v+2+Cɛu+pp-1v+p.Since z+s≤z-ws, for all z∈X, w∈-P, and 2≤s≤2∗, it follows from the Sobolev embedding theorem that there is a constant C1=C1(s)>0 such that u+s≤C1dist(u,-P). Moreover, one can easily verify that dist(v,-P)≤v+. Consequently, by (46) and the Sobolev embedding theorem, we have (47)dist⁡v,-Pv+≤v+2≤C2ɛdist⁡u,-P+Cɛdist⁡u,-Pp-1v+,where C2>0. Therefore, (48)dist⁡v,-P≤C2ɛdist⁡u,-P+Cɛdist⁡u,-Pp-1.Similarly, we can prove that (49)dist⁡v,+P≤C3ɛdist⁡u,+P+Cɛdist⁡u,+Pp-1,for some constant C3>0.
Hence(50)dist⁡v,±P≤C4ɛdist⁡u,±P+Cɛdist⁡u,±Pp-1,where C4=max⁡{C2,C3}. We can choose ε0>0 small enough so that, for all ɛ∈(0,ε0), (51)dist⁡v,±P≤12dist⁡u,±Pwhenever  distu,±P<ɛ.It then follows that A(Pɛ±)⊂Pɛ±, ∀ɛ∈(0,ε0).Notice that the vector fieldA is not locally Lipschitz. However, it can be used as [18] to construct a locally Lipschitz vector field which will satisfy condition (Aε0). More precisely, we have the following result.Lemma 7 (see [19, Lemma  3.4]).
There exists a locally Lipschitz continuous operatorB:E≜X∖K→X (B odd when J is even) such that (1)
J′u,u-Bu≥(1/2)u-Au2, for any u∈E.(2)
(1/2)u-Bu≤u-Au≤2u-Bu, for any u∈E.(3)
B(Pɛ±∩E)⊂Pɛ± for any ɛ∈(0,ε0), where ε0 is obtained in Lemma 6(3).Remark 8.
Lemmas6 and 7 imply that (52)J′u,u-Bu≥18u-Bu2,J′u≤2a+C¯2+bu2u-Bu.Lemma 9.
Letρ1<ρ2 and α>0. Then there exists β>0 such that u-Bu≥β if u∈X is such that J(u)∈[ρ1,ρ2] and J′(u)≥α.Proof.
By the definition of the operatorA, we have (53)a+bu2∫Ω∇u·∇Audx+∫ΩVxuAudx=∫Ωfx,uudx,∀u∈X.It follows that (54)Ju-1μa+bu2∫Ω∇u·∇u-Audx-1μ∫ΩVxuu-Audx=a12-1μ+b14-1μu2·u2+12-1μ∫ΩVxu2dx+∫Ω1μfx,uu-Fx,udx≥a12-1μu2+b14-1μu4.
Ifb>0, using Lemma 7(2) and the Hölder inequality, one has (55)a12-1μu2+b14-1μu4≤Ju+1μa+C¯2+bu2u·u-Au≤Ju+2μa+C¯2+bu2u·u-Bu,for C¯>0 (see (11)). Suppose that there exists a sequence {un}⊂X such that J(un)∈[ρ1,ρ2], J′(un)≥α and un-Bun→0. By (55), we see that {un} is bounded. It follows from Remark 8 above that J′(un)→0, which is a contradiction.
Ifb=0, the conclusion is concluded by Remark 8.Lemma 10.
The functionalJ satisfies the (PS) condition at any level c∈R.Proof.
In view of (9) and (12), (f3), for any u∈X, one has (56)Ju-1μJ′u,u=a12-1μu2+b14-1μu4+12-1μ∫ΩVxu2dx+∫Ω1μfx,uu-Fx,udx≥a12-1μu2.Let {un}⊂X be a sequence such that supn⁡|J(un)|<∞ and J′(un)→0 in X∗ as n→∞. Inequality (56) implies that {un} is bounded in X. Then, up to a subsequence, we have un⇀u in X and un→uLs(Ω) for 2≤s<2∗. Using a standard argument, one has J′(u)=0. Notice that (57)on1=J′un-J′u,un-u=a+bu2∫Ω∇un-u2dx+bun2-u2∫Ω∇u·∇un-udx+∫ΩVxun-u2dx-∫Ωfx,un-fx,uun-udx≥aun-u2+bun2-u2∫Ω∇u·∇un-udx-∫Ωfx,un-fx,uun-udx.Consequently, by (43), the Hölder inequality, and the boundness of {un} in X, we know that, for any ɛ>0, there is a constant Cɛ>0 such that (58)aun-u2≤ɛun2·un-u2+Cɛunpp-1·un-up+ɛu2·un-u2+Cɛupp-1·un-up+on1.By the arbitrariness of ɛ and X↪Ls(Ω) which is compact for s∈[2,2∗), we have un→u in X.Lemma 11.
Fors∈[1,2∗], there exists k>0 such that for any ɛ>0(59)us≤kɛ,∀u∈Pɛ+∩Pɛ-.Proof.
For anyu∈X, this follows from the fact that (60)u±s=infw∈∓P⁡u-ws≤Csinfw∈∓P⁡u-w=Csdist⁡u,∓P,where Cs>0 is the Sobolev constant in the continuous embedding X↪Ls(Ω) for all s∈[1,2∗].Lemma 12.
Forɛ>0 small enough, one has (61)Ju≥a8ε2,∀u∈∂Pɛ+∩∂Pɛ-.Proof.
Letu∈∂Pɛ+∩∂Pɛ-. It is clear that (62)u±≥dist⁡u,∓P=ɛ,∀ɛ>0.Using Lemma 11, we obtain (63)Ju≥a4ε2-Cεp,where C>0.Let us denote by0<λ1<λ2<λ3≤⋯ the distinct eigenvalues of the problem(64)-Δu=λuinΩ,u=0,on⁡∂Ω.It is well known that each λj(j≥2) has finite multiplicity, the principle eigenvalue λ1 is simple with positive eigenfunction e1, and the eigenfunctions ej corresponding to λj(j≥2) are sign-changing. Let Xj be the eigenspace associated with λj. We set k≥2(65)Yk≔⨁j=1kXj,Zk≔⨁j=k∞Xj¯.Note that any element of Zk∖{0} is sign-changing.We define(66)Nk≔u∈Zk:uppu2+u·upu+βk-σ·up=ρ,where (67)ρ≔a8Cɛ,Cɛis  obtained  in  (43),(68)βk≔supu=1u∈Zk⁡up,σ≔2pp-2.Lemma 13.
One has for any fixedα>0(69)infu∈Nk∩Jα⁡Ju⟶+∞,k⟶∞.Proof.
By the definitionNk, we have (70)uppu2≤ρ,∀u∈Nk.For each fixed u∈Nk, for ɛ>0 small enough, by (44) and (67), we have (71)Ju≥a4u2-Cɛupp=u2a4-Cɛuppu2≥a8u2.Notice that (72)up≤ρ1+Cpβk-σ,∀u∈Nk,where Cp>0 is the Sobolev constant in the embedding X↪Lp(Ω).
For anyu∈Nk∩Jα, one has (73)a4u2≤α+Cupp.This implies that (74)u≤C∗≔Cα,p,ρ1+βk-σ/2p,∀u∈Nk∩Jα,where C(α,p,ρ)>0.
Furthermore,∀u∈Nk∩Jα, one has (75)ρ=uppu2+u·upu+βk-σup≤uppu2+u·up2u·βk-σup1/2=upu2upp-2+12u·βkσ·up1/2≤βk2·Cpp-2·up-2+Cp2·βkσ/2·u=βk2·Cpp-2·C∗p-2·1C∗up-2+Cp2·βkσ/2·C∗·1C∗u≤βk2Cpp-2C∗p-2+Cp2βkσ/2C∗max⁡1C∗u,1C∗up-2,where Cp>0 is the Sobolev constant in the embedding X↪Lp(Ω). By (74), one has (1/C∗)u≤1, for u∈Nk∩Jα. Hence, (76)max⁡1C∗u,1C∗up-2=1C∗u,ρ≤βk2Cpp-2C∗p-3+Cp2βkσ/2u,∀u∈Nk∩Jα.For any u∈Nk∩Jα, we deduce (77)ρ≤C~α,p,ρβk2+βkσ/2+βk2-σ/2pp-2u,where C~(α,p,ρ)>0 is a constant. From (71), one has(78)Ju≥a8ρ2·C~α,p,ρ-2·βk2+βkσ/2+βk2-σ/2pp-2-2,∀u∈Nk∩Jα.By (68), one has 2-(σ/2)p(p-2)=1>0. From Lemma  3.8 in [17], we know that βk→0, as k→∞. Set k→∞ in (78); we have (79)infu∈Nk∩Jα⁡Ju⟶+∞,ask⟶∞.Lemma 14.
For anyα>0, one has(80)δ0α≔dist⁡Nk∩Jα,P>0.Proof.
The proof is similar to the proof of Lemma  5.4 in [20].Proof of Theorem1.
Step
1 (the existence of a positive and a negative solution). By (44) and the Sobolev embedding theorem, ∀0<ɛ<(1/4)a, there exists constant C>0 such that (81)Ju≥a4u2-Cup.Consequently, there exists r>0 (small enough) such that (82)infu=r⁡Ju≥18r2>0.
From(f3) and (44), we have (83)Fx,t≥C1tμ-C2t2,∀x,t∈Ω¯×R,for some positive constants C1 and C2. Thus, by (83) and the Sobolev embedding theorem, one has (84)Ju≤C3u2+b4u4-C4uμμ,where C3>0, C4>0. Fixing e∈X∖{0}, it is easy to prove that(85)Jte⟶-∞ast⟶+∞.Therefore, we can find e±∈±P such that (86)e±>r,Je±<0.This shows that condition (A1) of Theorem 3 is satisfied. By Lemmas 6, 7, and 9, condition (Aε0) is satisfied for ε0>0 small enough. By Lemma 10, J satisfies the (PS) condition at any positive level c. Hence, the existence of a positive and a negative solution follows from Theorem 3.
In the following proof, we adopt the notations of Theorem4.
Step
2 (the existence of a sign-changing solution). Using the the main idea of [21], we will verify the assumptions of Theorem 4.
Letv1,v2∈C0∞(Ω)∖{0} be such that supp⁡(v1)∩supp⁡v2=Ø, v1≤0 and v2≥0. We define the continuous map φ0:Δ→X by φ0(s,t)=R(sv1+tv2) for all (s,t)∈Δ, where R>0 is a constant to be determined later. Obviously, φ0(0,t)∈Pɛ+ and φ0(s,0)∈Pɛ- for all ɛ>0. This implies that φ0(∂1Δ)⊂Pɛ+ and φ0(∂2Δ)⊂Pɛ-; that is, Theorem 4(1) holds. Now a simple computation is as follows: (87)δ≔min⁡1-tv1+tv22:t∈0,1>0.Then u2≥δR for u∈φ0(∂0Δ) and it follows from Lemma 11 that φ0(∂0Δ)∩Pɛ+∩Pɛ-=Ø, for R large enough and for any ɛ>0.
By (84), there exist constants C3>0 and C4>0 such that(88)Ju≤C3u2+b4u4-C4uμμ.Combining with Lemma 12, for R large enough and ɛ>0 small enough, we obtain (89)c0=supu∈φ0∂0Δ⁡Ju<0<c∗≔infu∈∂Pɛ+∩∂Pɛ-⁡Ju.
By Theorem4, J has a sign-changing critical point.
Step
3 (the existence of infinitely many high energy solutions). Because dim⁡Yk<+∞, there exists θk>0 such that u≤θkuμ for any u∈Yk. Combining with (83), one has (90)Ju≤C1u2+b4u4-C2uμ,∀u∈Yk,where C1>0, C2>0. Hence, we have J(u)→-∞ on Yk as u→∞.
We can then chooseρk>0 large enough so that (91)ρk/θkpρk2+ρkρk/θkρk+Cpβk-σρk>ρ,ak≔maxu=ρku∈Yk⁡Ju<0,where σ is given by (67) and Cp is the Sobolev constant. Combining with Lemma 13, condition (B1) of Theorem 5 is satisfied. By Lemma 14, (92)δ0a0≔dist⁡Nk∩Ja0,P>0,where a0≔maxu∈Bk⁡J(u)>0.
For anyu∈Nk∩Ja0, v∈P, and w∈Pɛ+∩Pɛ-, one has (93)0<δ0a0=infv∈Pu∈Nk⋂Ja0⁡u-v≤u-w+w+-v++w--v-,where v+=max⁡{v,0}, v-=min⁡{v,0}, w+∈Pɛ+, and w-∈Pɛ-. Hence (94)0<δ0a0=dist⁡u,Pɛ+∪Pɛ-+2ɛ.Set ε0∈(0,(1/2)δ0(a0)); one has (95)dist⁡u,Pɛ+∪Pɛ-≥δ0a0-2ɛ>0,∀ɛ∈0,ε0.This implies that condition (B2) of Theorem 5 holds. Thus, by Theorem 5, we obtain that J possess a sequence {uk} of sign-changing critical point such that J(uk)→∞ as k→∞.



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*Source: 985986-2015-12-13.xml*