# Multiplicity of Nodal Solutions for a Class of Double-Phase Problems

Authors: Jie Yang; Haibo Chen; Senli Liu
Journal: Journal of Function Spaces
 (2020)

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## Abstract

We consider the following parametric double-phase problem:−div∇up−2∇u+ax∇uq−2∇u=λfx,uinΩ,u=0,on∂Ω.. We do not impose any global growth conditions to the nonlinearity fx,u, which refer solely to its behavior in a neighborhood of u=0. And we will show that they suffice for the multiplicity of signed and nodal solutions of the double-phase problem above when λ is large enough.


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## Body

## 1. Introduction and Statement of Results

In this paper, we deal with the existence and multiplicity of solutions for the following double-phase problem:(1)−div∇up−2∇u+ax∇uq−2∇u=λfx,u,inΩ,u=0,on∂Ω,Pλ,where λ>0 is a parameter, Ω⊂ℝNN≥2 is a bounded domain with smooth boundary, 1<p<q<minN,p∗,p∗=Np/N−p,
(2)qp<1+1N,a:Ω¯⟶0,+∞isLipschitzcontinuous,and we also assume that the nonlinearity f satisfies the following conditions:(f1) f:Ω×ℝ⟶ℝ is a C1 function, fx,0=0 for a.e.x∈Ω(f2) There exists γ∈q,p∗ such that
(3)limsupt→0+fx,ttγ−2t<+∞uniformly for a.e.x∈Ω(f3) There exists β∈p,p+pp∗−p/q such that
(4)liminft→0Fx,ttβ>0uniformly for a.e.x∈Ω, where Fx,t=∫0tfx,sds(f4) There exists a constant θ∈q,p∗,δ>0 such that
(5)0<θFx,t≤fx,ttfor a.e.x∈Ω and all 0<t≤δ(f5) For the δ in f4, fx,−u=−fx,u,∀x∈Ω,u≤δSimilar problems have been investigated, and it is well known they have a strong physical meaning because they appear in the models of strongly anisotropic materials (see [1, 2]). The energy functionals of the form
(6)u↦∫Ωℋx,∇uxdx,ℋx=tp+axtq,q>p>1,a⋅>0,where the integrand ℋ which switches between two different elliptic behaviors has been intensively studied since the late eighties (see [3–9]). Recently, Colombo and Mingione in [7] have obtained the regularity theory for minimizers of (6).The double-phase problem(7)−div∇up−2∇u+ax∇uq−2∇u=fx,u,inΩ,u=0,on∂Ω,has been studied extensively recently. The existence of a sign-changing ground state solution to problem (7) has been proven by Liu and Dai in [3], when f were assumed to satisfy the p-superlinear growth condition and Ambrosetti-Rabinowitz condition. In [4], by using Morse theory, Perera and Squassina obtained a nontrivial weak solution of problem (7), when fx,u=λup−2u+ur−2u+hx,u. In [5], by utilizing the Nehari method, Liu and Dai obtained three ground state solutions.Motivated by the above works, we intend to establish the multiplicity of both signed and nodal solutions of problem (Pλ), when λ>0 is large enough. Here, we note that the assumptions (f1)–(f5) that we make on the nonlinearity fx,u refer only to its behavior in a neighborhood of u=0.We present the main result of this paper as follows:Theorem 1.
Supposef1−f4 are satisfied. Then, there exists Λ>0, and if λ>Λ, problem (Pλ) has at least one positive solution, one negative solution, and a sign-changing solution.Theorem 2.
Supposef1−f5 are satisfied. Then, there exists Λ∗>0, and if λ>Λ∗, for any given k≥1, problem (Pλ) has k pairs of solutions ±ui,i=1,⋯,k, with ui∞≤δ. Moreover, ±ui,i=2,⋯,k, are k−1 pairs of sign-changing solutions.Remark 3.
Fromf1 and f2, it can be seen that γ≤β. We may think of fx,t=tβ−2t with θ<β, which clearly satisfy f1−f4.We remark that [3, 5] obtained only one sign-changing solution. However, in Theorem 2, since k is arbitrary, we get infinitely many sign-changing solutions. To the best of our knowledge, little has been done in the literature on the existence of multiple nodal solutions for the parametric Dirichlet problem with the minimal conditions on the nonlinearity fx,u.The proof will be done by variational techniques. Since we have no information on the behavior of the nonlinearityfx,u at the infinity, we adapt the argument introduced by Costa and Wang [10], which consists in making a suitable modification on f, solving a modified problem, and then checking that, for a large enough λ, the solutions of the modified problem are indeed solutions of the original one.The paper is organized as follows. In Section2, we modify the original problem and prove the Palais-Smale condition for the modified functional. We present some tools which are useful to establish a multiplicity result. In Section 3, we prove Theorem 1. Theorem 2 is proven in Section 4.


## 2. The Modified Problem

To prove our main results, we need to present the variational setting of our problem. Firstly, we introduce some notations and some necessary definitions. The Musielak-Orlicz spaceLℋΩ associated with the function
(8)ℋ:Ω×0,∞⟶0,∞,x,t↦tp+axtq,consists of all measurable functions u:Ω⟶ℝ with the ℋ-modular
(9)ρℋu≔∫Ωℋx,udx<∞.The Musielak-Orlicz spaceLℋΩ is defined by
(10)LℋΩ=u:Ω⟶ℝmeasurable:ρℋu<+∞,endowed with the norm
(11)uℋ=infλ>0:ρℋuλ≤1.The spaceLℋΩ is a separable, uniformly convex, and reflexive Banach space. We denote by ⋅p the norm in LpΩ and by LaqΩ the space of all measurable functions u:Ω⟶ℝ with the seminorm
(12)uq,a≔∫Ωaxuqdx1/q<∞.It is easy to check that the embeddings(13)LqΩ↪LHΩ↪LpΩ∩LapΩ,are continuous. Since ρℋu/uℋ=1 whenever u≠0, we have
(14)minuℋp,uℋq≤upp+uq,aq≤maxuℋp,uℋq,∀u∈LℋΩ.The related Sobolev spaceW1,ℋΩ is defined by
(15)W1,ℋΩ≔u∈LℋΩ:∇u∈LℋΩ,equipped with the norm
(16)u=uℋ+∇uℋ,where ∇uℋ=∇uℋ. The completion of C0∞Ω in W1,ℋΩ is denoted by W01,ℋΩ, and it can be equivalently renormed by
(17)u≔∇uℋ,via a Poincare-type inequality (cf [6], Proposition 2.18(iv)), under assumption (2). The spaces W1,ℋΩ and W01,ℋΩ are uniformly convex and hence are reflexive, Banach spaces. By Proposition 2.15 in [6], we know that the Sobolev embedding W01,HΩ↪LrΩ is compact since r<p∗. By (14), We have
(18)minup,uq≤∇upp+∇uq,aq≤maxup,uq,∀u∈W01,ℋΩ.From now on, we denote byX≔W01,ℋΩ for convenience in writing.We first mention thatf2−f3 imply that there exist c1,c2>0, such that
(19)fx,t≤c1tγ−1,(20)Fx,t≥c2tβ,for a.e.x∈Ω and all t≤δ.Now, define the even cutoff functionζt∈C2ℝ,0,1 as
(21)ζt=1ift≤δ,0ift>δ.Moreover, for allt∈ℝ, ζt satisfies that
(22)tζ′t≤0,tζ′t≤2δ.Let(23)F~x,t=ζtFx,t+1−ζtc1tγγ.Also, we set(24)f~x,t=F~t′x,t=∂F~∂tx,t.From hypothesesf1−f4, it is easy to check that f~ is a Carathéodory function and satisfies the following properties.Lemma 4.
If hypothesesf1,f2, and f4 hold, then the functions F~ and f~ satisfy the following properties:
(i)
There existc3>0 such that f~x,t≤c3tγ−1 for a.e.x∈Ω(ii)
There existsμ=minθ,γ such that for a.e.x∈Ω and t≠0(25)0<μF~x,t≤f~x,ttProof.
The proof is similar to that of Lemma 1.1 in [10].Now let us consider the modified problem of (Pλ):
(26)−div∇up−2∇u+ax∇uq−2∇u=λf~x,u,inΩ,u=0,on∂Ω,Pλ∗.The corresponding energy functional of (Pλ∗) is
(27)I~λu=∫Ω1p∇up+axq∇uq−λF~x,udx,∀u∈X.We easily get that the functionalI~λu∈C2X,ℝ, and its critical points are the solutions of (Pλ∗). We note that solutions of (Pλ∗) with u∞≤δ are also solutions of (Pλ). We shall search solutions of (Pλ∗) as critical points of I~λu.Let us now defineJ⋅:X⟶ℝ as
(28)Ju=∫Ω1p∇up+axq∇uqdx,and we denote the derivative operator by A, that is, A=J′:X⟶X∗ with
(29)Au,v=∫Ω∇up−2+ax∇uq−2∇u⋅∇vdx,u,v∈X.In the following lemma, we summarize some properties ofA, useful to study our problem.Lemma 5.
Under the condition (2), A is a mapping of type (S+); that is, if un⇀u in X and limsupn→+∞Aun−Au,un−u≤0, then un⟶u in X.Proof.
The proof is similar to that of Proposition 3.1(ii) in [3].
Firstly, we show the functionalI~λ satisfies the (PS) condition.Lemma 6.
If hypothesesf1–f2 and f4 hold, then I~λ satisfies the (PS) condition.Proof.
For everyc∈ℝ, let un⊂X be a PSc–sequence, that is,
(30)I~λun⟶c,andI~λ′un⟶0,asn⟶∞.We claim thatun is bounded in X. Indeed, if un≤1, we have done. If un>1, by Lemma 4(ii), then we have that
(31)I~λun−1μI~λ′un,un=∫Ω1p∇up+axq∇uqdx−λ∫ΩF~x,un−1μ∫Ω∇up+ax∇uq−λf~x,unundx≥1q−1μunp,where μ=minθ,γ>q. Hence, un is bounded. Therefore, there is a subsequence (which we still denote by un) that converges weakly to some u∈X and strongly in LγΩ. It is easy to check from Lemma 4(i) and Hölder’s inequality that
(32)∫Ωf~x,un−f~x,uun−udx≤c4un−uγ⟶0.Then,(33)Aun−Au,un−u=I~′un−I~′u,un−u+λ∫Ωf~x,un−f~x,uun−udx⟶0.Soun⟶u follows from Lemma 5.Next, we will show the functionalI~λ satisfies the Mountain Pass Geometry [11].Lemma 7.
Assume that hypothesesf1–f4 are satisfied. Then, the I~λ satisfies the following conditions:
(i)
For everyλ>0, there exist αλ,ρλ>0, such that I~λu≥αλ>0 with u=ρλ>0(ii)
There existsλ0>0 such that e∞≤δ/2,e≥ρλ and I~λe≤0<αλ, for all λ>λ0Proof.
(i) It follows from Lemma4(i) that there exists c5>0 such that
(34)F~x,u≤c5uγ,fora.e.x∈Ω.Then, for allu∈X,u≤1, we have
(35)I~λu≥1quq−λc5uγγ≥1quq−λc6uγ=1q−λc6uγ−quq.Forγ>q, we can choose ρλ=min1/2qλc61/γ−q,1, then
(36)I~λu≥12qρλq>0,forallu∈Xwithu=ρλ.(ii) Lete∈C01Ω\0, such that
(37)ex≤δ2,forallx∈Ω.Firstly, we note thatρλ⟶0,asλ⟶+∞. So there exists λ0>0 such that for all λ>λ0,
(38)e>ρλ,1p∫Ω∇ep+ax∇eqdx≤λc2eββ.Then, by (20), (37), and (38), we get
(39)I~λe=∫Ω1p∇ep+axq∇eq−λF~x,edx≤1p∫Ω∇ep+ax∇eqdx−λc2eββ≤0<αλ.This completes the proof of Lemma7.Lemma 8.
Assume that hypothesesf1–f2 and f4 are satisfied, λ>0 and u∈KI~λ≔u∈XI~λ′u=0,I~λu=c, then there exists d>0 such that minuq,up≤dI~λu.Proof.
Letu∈KI~λ. We observe from Lemma 4(ii) that
(40)μI~λu=μI~λu−I~λ′u,u=μ∫Ω1p∇up+axq∇uq−λF~x,udx−∫Ω∇up+ax∇uq−λf~x,uu≥μq−1minuq,up.Sinceμ=minθ,γ>q, there exists d=μ/μ/q−1>0 such that
(41)minuq,up≤dI~λu.From Theorem 8.4 in [12], we say that the solutions of (Pλ∗) enjoy the L∞ estimates given in the next lemma.Lemma 9.
Letu~λ∈W01,ℋΩλ>0 be the weak solution to (Pλ∗) under the assumptions f1–f4. Then, the following estimate holds:
(42)u~λ∞≤c7λp∗−γ−1u~p∗−p/p∗−γ,where c7=c7γ,N,Ω.Now we are ready to show the existence of a solution to (Pλ∗) for a large λ.Lemma 10.
Assume that conditionsf1–f4 hold. Then, for any λ>λ0, there exists a nontrivial solution u~λ∈X of (Pλ∗), such that
(43)u~λ∞≤c9λκ,where κ=pp∗−p−β−pq/p∗−γβ−pq>0.Proof.
By Lemma6 and Lemma 7, we conclude that I~λu satisfies the (PS) condition and the Mountain Pass Geometry. Consequently, by the Mountain Pass Theorem (Theorem 2.2 in [13]), there exists a uλ∈X such that
(44)I~λ′uλ=0,αλ≤I~λuλ=c~λ=infτ∈Γmax0≤t≤1I~λτt,where Γ=τ∈C0,1,X,τ0=0,τ1=e.From (38), we easily obtain that the function
(45)hλt=tpp∫Ω∇ep+ax∇eqdx−λc2tβeββt∈0,1,λ>λ0,is continuous and hλ0=0,hλ1<0. It follows from 1<p<β, for t∈0,1 which is small enough, we get
(46)hλt>0.Hence, there existst0∈0,1 such that
(47)hλt0=max0≤t≤1hλt,where t0=∫Ω∇ep+ax∇eqdx/c2βλeββ1/β−p. Then, we obtain
(48)hλt0=c8λp/β−p,where
(49)c8=1p−1β∫Ω∇ep+ax∇eqdxβ/β−pc2βeββp/β−p>0.Letτ0=te. Obviously, τ0∈Γ, and then we get from (44) that
(50)I~λuλ=c~λ≤max0≤t≤1I~λte≤max0≤t≤1hλt=c8λp/β−p.Therefore, there existsλ1≥λ0 such that dc8/λp/β−p<1. Together with Lemma 8, we obtain
(51)uq≤dc8λp/β−p.It follows from Lemma9, we get that
(52)u~λ∞≤c9λκ,where c9>0,κ=pp∗−p−β−pq/p∗−γβ−pq>0, recalling that β∈p,p+pp∗−p/q.


## 3. Proof of Theorem 1

In this section, we prove our main result. We will show (Pλ∗) has a positive solution, a negative solution, and a nodal solution. And the solutions obtained satisfy the estimate u∞≤δ. This fact implies that these solutions are indeed solutions of the original problem (Pλ).Proof of Theorem 1.
Consider the following problem:(53)−div∇up−2∇u+ax∇uq−2∇u=λf~+x,u,inΩ,u=0,on∂Ω,where
(54)f~+x,t=f~x,t,t>0,0,t≤0.Define the corresponding functional(55)I~λ+u=∫Ω1p∇up+axq∇uq−F~+x,udx,∀u∈X,where F~+x,u=∫0uf~+x,sds. Obviously, I~λ+∈C2X,R and f~+ satisfy all the conditions of Theorem 1. Let uλ+ be a nontrivial critical point of I~λ+, which implies that uλ+ is a weak solution of (53). It is known by Lemma 4.1 of [5] that uλ+≥0 a.e. in ℝℕ. From Theorem 3.3 in [5], we conclude that uλ+>0. Therefore, by Lemma 10,
(56)uλ+∞≤c9λκ,where κ>0, so there exists λ2≥λ1 such that
(57)c9λκ≤δ,for all λ>λ2. Thus, uλ+>0 is also a nontrivial solution of original problem (Pλ) for all λ≥λ2.Similarly, we can define(58)f~−x,t=f~x,t,t<0,0,t≥0,I~λ−u=∫Ω1p∇up+1qax∇uq−F~−x,udx,∀u∈X,where F~−x,u=∫0uf~−x,sds. We also get a negative solution uλ−<0 of our original problem (Pλ) for all λ≥λ3≥λ2.We next show that there is a sign-changing solution forλ large enough. We can apply the method introduced by Li and Wang [14] to our case. Since X is a real, reflexive, and separable Banach space, there are ej⊂X and ej∗⊂X∗ such that
(59)X=spanej,j=1,2,⋯¯,X∗=spanej∗,j=1,2,⋯¯,ej,ej∗=1,i=j,0,i≠j.Fork=1,2,⋯, we denote
(60)Yk≔spane1,e2,⋯,ek,Zk≔spanek,ek+1,⋯¯.OnX, we define a closed convex cone
(61)PX=u∈Xux≥0,a.e.inΩ.AndE=C01Ω is a Banach space densely embedded in X. Assume that P=PX∩E has interior points in P. As in Section 3 of [14], we may define a partial order relation in E: u,v∈E,u>v⇔u−v∈P\0;u≫v⇔u−v∈P∘. And we define W=P∪−P. In order to apply the method introduced by Li and Wang in [14] (Example 3.2 and Corollary 3.2), we take Q=u=s1e1+s2e2:s1≤R,0≤s2≤R,u≤R on Y2, and for 0<r<R, T=u∈Z2u=r. Hence, we obtain T and ∂Q link.It follows from Lemma4(ii) that there exist c10,c11>0 such that
(62)F~x,u≥c10uμ−c11.Then, for eachu∈Y2,u≥1, we obtain that
(63)I~λu≤1puq−λ∫ΩF~x,udx,≤1puq−λc12uμ+λc11Ω,for all norms on Y2 are equivalent. Since μ>q, we may get R≥1 which is large enough such that I~λ≤0 for all u∈Y2,u=R and for all λ>0.Using (34) and the Sobolev embedding W01,HΩ↪LrΩ for γ∈q,p∗, we can choose some α>0,0<ρ<1 such that
(64)I~λu≥1quq−λc13uγ>α,for u=ρ. Let us define
(65)cλ=infh∈ΓsuphQ\WI~λu,where Γ=h∈CQ,EhQ1∈W,hu=u,foru∈Q2. Therefore, by [14], we see that cλ>0 is a critical value of I~λ and I~λ has a sign-changing critical point u~λ at this critical value. Therefore, by Lemma 10,
(66)u~λ∞≤c9λκ,where κ>0, so there exists Λ≥λ3 such that
(67)c9λκ≤δ,for all λ>Λ. Thus, u~λ is also a nontrivial sign-changing solution of original problem (Pλ) for all λ≥Λ.


## 4. Proof of Theorem 2

In this section, we present a multiplicity result for the modified problem. We shall use arguments in [13] to get solutions for Pλ∗ first. Next, we use proofs in [14] to get the nodal property of the solutions.Proof of Theorem 2.
By (63), we can choose R≥0 such that I~λ≤0 for all u∈Y2,u=R and for all λ>0. Set D=BR∩Yk. Let G=h∈CD,Xhisoddandh=idon∂BR∩Yk. Let Σ denote the family of sets A⊂X\0 such that A is closed in X and symmetric with respect to 0. For A∈Σ, iA defines the genus of A. Then, for j=1,⋯k, we can define
(68)cj,λ=infA∈Γjsupu∈AI~λu,where Γj=hD\Y¯h∈G,k≥j,Y∈Σ,iY≤k−j. Now we can apply Proposition 9.30 in [13] to functional I~λ, we get that 0<c1,λ≤c2,λ≤⋯≤ck,λ are all critical values of I~λ. And Pλ∗ possesses at least k pairs of distinct nontrivial solutions. Then, by Lemma 10, there exists Λ∗>0, such that λ>Λ∗; these k pairs solutions of Pλ∗ are also solutions of the original problem Pλ.Secondly, we will obtain the nodal property of the solutions by Theorem 2.3 in [14]. We need a procedure similar to the one we used earlier to rule this out. Let
(69)ΣE=A⊂EAiscompact,A=−A.LetG=h∈CD,Ehisoddandh=idon∂BR∩Yk. For A∈ΣE, iA also denotes the genus of A. By (63), we can also choose R≥0 such that I~λ≤0 for all u∈Y2,u=R and for all λ>0. By (64), we can also choose α>0,0<r<R such that I~λ≥α for all u∈X,u=r and for all λ>0. By Theorem 2.3 in [14], I~λ has at least k−1 pairs of critical points ±uj in E\P∪−P, with critical values
(70)I~λuj=cj,λ=infA∈ΓjsupA\WI~λu,where Γj=hD\Y¯h∈G,k≥j,Y∈Σ,iY≤k−j,j=2,⋯k. Moreover, 0<c2,λ≤c3,λ≤⋯≤ck,λ. Finally, by Lemma 10,
(71)uj∞≤c9λκ,where κ>0, so there exists Λ∗>0 such that
(72)c9λκ≤δ,for all λ>Λ∗. Thus, ±uj,j=2,⋯,k, are also nontrivial sing-changing solutions of the original problem (Pλ) for all λ≥Λ∗.



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