R E S E A R C H
Open Access
Existence of solutions for nonlinear Robin
problems with the
p-Laplacian and
hemivariational inequality
Jing Zhang
**Correspondence: [email protected] Y.Y. Tseng Functional Analysis Research Center, Harbin Normal University, Harbin, 150025, P.R. China
Abstract
In this paper, we show the existence of at least three nontrivial solutions for a nonlinear elliptic equation driven by thep-Laplacian with a nonsmooth potential (hemivariational inequality) and Robin boundary condition. Two of these solutions are of constant sign (one is positive, the other negative). We mainly use a variational approach together with a sub-sup solution method.
Keywords: p-Laplacian; nonsmooth potential; hemivariational inequality; sub-sup solution method; second deformation theorem
1 Introduction
Consider the problem
–px+α|x|p–x∈∂j(z,x), z∈Z, |∇x|p–∂x
∂n+b(z)|x|p
–x= , z∈∂Z, (.)
whereZ⊂RN is a bounded domain withC-boundary∂Z,
px= div(|∇x|p–∇x) ( <p< ∞) is thep-Laplacian operator,α> ,b(z)∈L∞(∂Z),b(z)≥, andb(z)= on∂Z.j(z,x) is a measurable potential function onZ×R, which is locally Lipschitz in thex∈R,∂j(z,x) stands for the generalized subdifferential ofx→j(z,x). Also ∂∂xndenotes the outer normal derivative ofxwith respect to∂Z. The aim of this paper is to prove the existence of two constant sign solutions and furthermore prove the existence of at least three nontrivial solutions for problem (.).
A multiplicity of solutions for problems driven by thep-Laplacian has been obtained by Ambrosettiet al.[] and Garcia Azoreroet al.[]. In these works, the authors deal with a right-hand side nonlinearity of the form –px=λ|x|q–x+|x|r–xwithλ> being
a real parameter, <q<p<r<p∗(p∗=NNp–p ifp<N;p∗= +∞otherwise) and prove the existence of positive and negative solutions. The question of the existence of ap-Laplacian Robin problem –px+α|x|p–x=j(z,x) was also present in the work of Zhanget al.[] for
solutions. In Anello [] and Ricceri [], the main tool is an abstract variational principle of Ricceri and its use is made possible by the hypothesis thatp>N; by the fact that Sobolev spaceW,p(Z) is compactly embedded inC(Z), the authors obtain infinitely many weak solutions forp-Laplacian Neumann problem.
In all the aforementioned works, the nonlinearity is a Carathéodory function, a.e.j(z,x) is continuously differentiable in the variablex. In Barletta and Papageorgiou [], the au-thors consider a nonsmooth potential with an asymmetric behavior at +∞and at –∞ to get two nontrivial solutions using degree methods. Also, in Dancer and Du [], the au-thors use the critical point theory and a sub-sup solution method on smooth critical point theory.
In this paper, we use a combination of nonsmooth critical point theory with sub-sup solution methods. We also use the nonsmooth version of the second deformation theorem due to Corvellec []. Thus, we can extend the works of [, –] to a hemivariational inequality with the Robin boundary condition.
2 Preliminaries
Now we recall the subdifferential theory for locally Lipschitz functions and the corre-sponding nonsmooth critical point theory. LetXbe a Banach space and letX∗be its topo-logical dual. We denote by·,·the duality brackets for the pair (X,X∗). The generalized directional derivativeϕ(x;h) of a locally Lipschitz functionϕ:X→Ratx∈Xalong the directionh∈Xis defined as follows:
ϕ(x;h) =lim sup y→x,λ→
ϕ(y+λh) –ϕ(y)
λ .
It is well known that ϕ(x;·) is sublinear continuous and it is the support function of a
nonempty, convex, andw∗-compact set∂ϕ(x)⊆X∗defined by ∂ϕ(x) =x∗∈X∗:x∗,h≤ϕ(x;h),∀h∈X.
The function∂ϕ(x) is the ‘generalized subdifferential’ ofϕ. Ifϕ∈C(X), thenϕis locally
Lipschitz and∂ϕ(x) ={ϕ(x)}. Moreover, ifϕis also convex, then∂ϕ(x) coincides with the subdifferential in the sense of convex analysis,∂cϕ(x), which is defined by
∂cϕ(x) =
x∗∈X∗:x∗,h≤ϕ(x+h) –ϕ(x),∀h∈X.
If ∈∂ϕ(x), then we callx∈Xcritical point ofϕ. It is easy to see that ifx∈Xis a local minimum or a local maximum ofϕ, thenx∈Xis a critical point ofϕ.
A locally Lipschitz functionϕsatisfies the Palais-Smale condition at levelc∈R, if every sequence{xn}n≥⊆Xsatisfyingϕ(xn)→candinf{x∗:x∗∈∂ϕ(xn)} → asn→ ∞has a strongly convergent subsequence. Ifϕsatisfies the Palais-Smale condition at levelc∈R
for allc∈R, then we say that it satisfies the Palais-Smale condition. For the details, we refer to [].
In the following study, denoteR(z,x) =|∇x|p–∂x
∂n+b(z)|x|p–x, and we will use the fol-lowing spaces:
Wn,p(Z) =x∈W,p(Z) :∃{xn} ⊂C∞(Z),xn→xinW,p(Z),R(z,xn) = ,∀z∈∂Z
Cn(Z) =x∈C(Z) :R(z,x) = ,∀z∈∂Z.
Both are ordered Banach spaces, and we denote
W+=x∈Wn,p(Z) :x(z)≥ a.e.z∈Z,
C+=x∈Cn(Z) :x(z)≥,∀z∈Z, intC+=x∈C+:x(z) > ,∀z∈Z.
It is well known that the principal eigenfunctione∈int(C+), soint(C+)=∅.
Furthermore, defineuas the normalized principal eigenfunction of (–p,Wn,p(Z)) (see []). It is well known thatu(z)≥, a.e.z∈Z, from the nonlinear regularityu∈Cn(Z) (see Di Benedetto [], [, Chapter IX]), furthermoreu∈int(C+) by virtue of the strong
maximum principle of Vazquez [].
We give the following minimax characterization (see []), suited for our purpose.
Proposition . Let S=Wn,p(Z)∩∂B and={γ ∈C([, ],S) :γ() = –u,γ() =u},
where∂B={x∈Lp(Z) :xp= }.Then the first eigenvalueλof(–p,Wn,p(Z))equals
λ=inf γ∈tmax∈[,]
Dγ(t) pp.
Next we recall the definitions of sub-sup solutions for problem (.). () A functionx∈W,p(Z)withR(z,x(z))≥is called a ‘sup solution’, if
Z
Dx(z)p–Dx(z),Dy(z)dz+α
Z
x(z)p–x(z),y(z)dz
+
∂Z
b(z)x(z)p–x(z),y(z)dz≥
Z
u(z)y(z)dz
for ally∈Wn,p(Z),y(z)≥a.e. onZand for someu∈Lq(Z),u(z)∈∂j(z,x(z))a.e. onZfor some <q<NNp–pifN>p,q= +∞ifN≤p.
() A functionx∈W,p(Z)withR(z,x(z))≤is called a ‘sub-solution’, if
Z
Dx(z)p–Dx(z),Dy(z)dz+α
Z
x(z)p–x(z),y(z)dz
+
∂Z
b(z)x(z)p–x(z),y(z)dz≤
Z
u(z)y(z)dz
for ally∈Wn,p(Z),y(z)≥a.e. onZand for someu∈Lq(Z),u(z)∈∂j(z,x(z))a.e. onZfor some <q<NNp–pifN>p,q= +∞ifN≤p.
Finally we recall the following topological notion which is crucial in critical point theory.
Definition .[] LetS,Qbe closed subsets of a Banach spaceX,Qwith relative bound-ary∂Q. We saySand∂Qlink if
() S∩∂Q=∅, and
From the definition, we give the following general minimax principle for the critical values of a locally Lipschitz functionϕ.
Proposition .[] Supposeϕis locally Lipschitz and satisfies the(PS)-condition. Con-sider closed subsets S,Q⊂X and Q with relative boundary∂Q.Suppose
() Sand∂Qlink, () infSϕ>sup∂Qϕ.
Let
=h∈C(X,X) :h|∂Q=id
.
Then the number
β=inf h∈supu∈Qϕ
h(u)
defines a critical valueβ≥infSϕofϕ.
Remark . From the above general minimax principle, a nonsmooth version of the mountain pass theorem, the saddle point theorem, and the generalized mountain pass theorem are available by choosing the link sets appropriately (see [, ]).
The following result is the so-called ‘second deformation theorem’ for a nonsmooth set-ting. In fact, this result is due to Corvellec []. We give the following sets:
K=x∈X: ∈∂ϕ(x),
Kc=
x∈X: ∈∂ϕ(x),ϕ(x) =c, ϕc=x∈X:ϕ(x) <c.
We know thatK,Kc, andϕcare the critical set ofϕ, the critical set at levelc∈Rofϕ, and the strict sublevel set ofϕatc, respectively.
Proposition . Let X be a Banach space,ϕ:X→Rbe locally Lipschitz satisfying the Palais-Smale condition.a,b∈Rwith a<b.Assume also that K∩ϕ–((a,b)) =∅and Kais
a finite set containing only local minimizers ofϕ.
Then there exists a continuous deformation: [, ]×ϕb→ϕbsuch that
() (t,x) =xfor allt∈[, ],x∈Ka, () (,ϕb)⊆ϕa∪Ka,
() ϕ((t,x))≤ϕ(x)for allt∈[, ],x∈ϕb.
Definition .[] LetXbe a topological space andAa subspace ofX. A weak deforma-tion retracdeforma-tion fromXtoAis a homologyF:X×I→Xsuch that for allx∈Xanda∈A, we haveF(x, ) =x,F(a, ) =a, andF(x, )∈A.
In particular, the setϕa∪K
ais a weak deformation retract ofϕb.
We now recall another notion, which will be useful in the following. SupposeW is a Banach space and A:W →W∗ is a mapping, we say thatAis a type (S)+ if for every
Considering the nonlinear mappingA:Wn,p(Z)→Wn,p(Z)∗defined for allx∈Wn,p(Z) by
A(x),y=
Z
Dx(z)p–Dx(z)·Dy(z)dz. (.)
We have the following result (see [, Proposition .]).
Proposition . The mapping(.)is continuous and of the type(S)+.
Definition .[] Given a functionalϕ:Wn,p(Z)→R,x∈Wn,p(Z) is called aW-local minimizer ofϕ if there existsr> satisfying for ally∈Wn,p(Z) withyW,p
n (Z)≤r, we
have
ϕ(x)≤ϕ(x+y).
Definition .[] x∈Wn,p(Z) is called aC-local minimizer ofϕ if there existsr> satisfying for ally∈C
n(Z) withyCn(Z)≤r, we have
ϕ(x)≤ϕ(x+y).
As the study of problems like (.) is reduced to seeking the critical points of correspond-ing energy functional onWn,p(Z) or onCn(Z), in this section we introduce the notations used along the paper together with the main abstract results that we will use later on for aC-local minimizer to be aW-local minimizer. Such a result forp> was first proved in []. Then it has been extended to the Neumann boundary condition and a nonsmooth potential by [].
We denoteψ:Wn,p(Z)→Rfor allx∈Wn,p(Z)
ψ(x) =
pDx
p p+
p
∂Z
b(z)|x|pds–
Z
jz,x(z)dz.
From Clarke [, pp.-], we know thatψ is locally Lipschitz. By [], we know that if we letx∈Wn,p(Z) be aC-local minimizer ofψ, then x∈Cn(Z) and it is aW-local minimizer ofψ.
3 Solutions of constant sign
In this section, by using a sub-sup solution method, we get two solutions of (.) with constant sign, one positive and the other negative.
Our general assumptions on the nonsmooth potentialj(z,x) are the following:
A(j) (i) z→j(z,x)is measurable for allx∈R; (ii) x→j(z,x)is locally Lipschitz for a.e.z∈Z;
(iii) |u| ≤γ(z) +C|x|p–for a.e.z∈Z, allx∈R,u∈∂j(z,x), withγ∈L∞(Z)
+and
C> ;
(iv) lim sup|x|→∞ u
|x|p–x≤ω(z)for a.e.z∈Z, allu∈∂j(z,x), withω∈L
∞(Z)
+
(v) η(z) +α≤lim infx→|x|pu–x for a.e.z∈Z, allu∈∂j(z,x), withη∈L∞(Z)+
satisfyingλ≤η(z)a.e. inZandλ<η(z)in some set of positive measure,
λis the first eigenvalue of–pwith Robin boundary condition; (vi) ux≥for a.e.z∈Z, allx∈R,u∈∂j(z,x).
Theorem . Assume that A(j)(i)-(vi)hold.Problem(.)has at least two solutions x∈
int(C+)and x∗∈–int(C+).
Example The following potential functionj satisfies assumptionsA(j) (for the sake of simplicity we drop thez-dependence):
j(x) =
η+α
p |x|
p, |x| ≤,
ω
p|x|
p+Cln|x|p+η+α–ω
p , |x|> ,
where <ω<α,η>λ, andC> . Note that, ifC=η+α–ωp , thenj∈C(R).
Note that
∂j(x) =
⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩
[–ω–pC, –η–α], x= –, (η+α)|x|p–x, |x| ≤,
, x= ,
ω|x|p–x+pC x
|x|, |x|> , [η+α,ω+pC], x= .
It is easy to see thatjsatisfiesA(j)(i)-(iii), (vi). For allu∈∂j(x), we have lim sup
|x|→∞
u
|x|p–x≤ω
and
η+α≤lim inf x→
u
|x|p–x.
Then the potential functionjsatisfies assumptionsA(j).
Remark . In fact, problem (.) has the trivial solution ∈∂j(z, ) for a.e.z∈Z ac-cording to assumptionA(j)(vi) and the upper semicontinuity of the subdifferential∂j(z,·) (see Clarke []). What we are interesting in is whether it has nontrivial solutions.
We introduce a useful extension of the notion of maximal monotonicity (see [, p.]).
Definition . LetXbe a reflexive Banach space andA:X→X∗ an operator. We say thatAis pseudomonotone if
() Ahas nonempty, bounded and convex values;
() Ais upper semicontinuous for every finite dimensional subspace ofXintoX∗; () ifxnxinX,xn∗∈A(xn), andlim supn→+∞x∗n,xn–xX≤, then for everyy∈X,
there existsu∗(y)∈A(x), such that
u∗(y),x–yX≤lim inf n→+∞
x∗n,xn–y
Definition .[] A is said to be demicontinuous on Xif {xn} ⊂X andxn→x∈X together implyA(xn)A(x).
It is well known that (.) is the Euler-Lagrange equation of the functionalϕ:Wn,p(Z)→ R,
ϕ(x) =
pDx
p p+
α
px
p p+
p
∂Z
b(z)|x|pds–
Z
jz,x(z)dz, ∀x∈Wn,p(Z).
We introduce the truncation functionν+:R→R+by
ν+(x) =
x, x> , , x≤;
then define the locally Lipschitz functionalϕ+:Wn,p(Z)→Rby
ϕ+(x) =
pDx
p p+
α
px
p p+
p
∂Z
b(z)|x|pds–
Z
j+
z,x(z)dz, ∀x∈Wn,p(Z),
wherej+(z,x) =j(z,ν+(x)) for allz∈R,x∈R, which is locally Lipschitz.
We consider the nonlinear Robin problem for givenε> andδε(z)∈L∞(Z)+,δε= :
–px+α|x|p–x= (ω(z) +ε)|x|p–x+δε(z), z∈Z,
|∇x|p–∂x
∂n+b(z)|x|
p–x= , z∈∂Z. (.)
Define the mappingI:Wn,p(Z)→Wn,p(Z)∗for allx,y∈Wn,p(Z) by
I(x),y=
Z
Dx(z)p–Dx(z)·Dy(z)dz+α
Z
x(z)p–x(z)·y(z)dz
+
∂Z
b(z)x(z)p–x(z)·y(z)ds.
It is well known thatIis strictly monotone and demicontinuous, furthermore, maximal monotone (see []). We denoteKε:Lp(Z)→Lp
(Z) (p+p = ) and we have
Kε(x)(·) =
ω(·) +εx(·)p–x(·),
which is bounded and continuous. Then the mapping I(x) –Kε(x) is pseudomonotone
fromWn,p(Z) into W–,p
n (Z), in fact,Wn,p(Z)→Lp(Z) is compact embedding andKε:
Wn,p(Z)→Lp
(Z) is completely continuous.
Next, we will show that (.) has a solutionx∈int(C+).
Lemma . Letω∈L∞(Z)+satisfyω(z)≤αa.e.in Z andω(z) <αin some set of positive
measure.Then(.)has a solution x∈int(C+)forε> small enough.
Proof First, we claim that there existsξ> such that
J(x) =Dxpp+αxpp+
∂Z
b(z)|x|pds–
Z
In fact, from assumptionb≥, we know thatJ(x)≥, for allx∈Wn,p(Z). Suppose the conclusion is false, we havexn∈Wn,p(Z),J(xn) <nxnpp,xn= . If we setxn =xxnn, then
J(xn) < n (Jisp-homogeneous). We may assumexnxinWn,p(Z),xn →xinLp(Z) by passing to a subsequence if necessary. Then
xpp= lim n→∞ x
n p
p, Dx
p
p≤lim infn→∞ Dxn p p,
∂Z
b(z)|x|pds–
Z
ω(z)x(z)pdz= lim n→∞
∂Z
b(z)xnpds–
Z
ω(z)xn(z)pdz
.
So, by passing to the limit ofJ, we have Dxpp+αxpp+
∂Z
b(z)|x|pds–
Z
ω(z)x(z)pdz≤.
This implies
Dxpp≤
Z
ω(z) –αx(z)pdz–
∂Z
b(z)|x|pds≤.
Hence, we havex(z) =Cfor a.e.z∈ZwhereC∈R. In fact,C= , if not, from the above inequality,
≤ |C|p
Z
ω(z) –αdz–
∂Z
b(z)ds
< .
It produces a contradiction. On the other hand,
Dxn pp=Jxn–α xn pp–
∂Z
b(z)xnpds–
Z
ω(z)xn(z)pdz.
We have Dxnp →, together with xn → x in Lp(Z), so xn → in Wn,p(Z), but xnW,p
n (Z)= ,n∈Z. So the assumption is false, we have the conclusion.
For allx∈Wn,p(Z), from the above discussion, we get
I(x) –Kε(x),x
=Dxpp+αxpp+
∂Z
b(z)|x|pds–
Z
ω(z)x(z)pdz–εxpp
≥(ε–ε)xpp.
So ifε<εsmall enough, we haveI(·) –Kε(·) is coercive. But a pseudomonotone coercive
operator is surjective (see [, Theorem .]), forδε, we can findx∈Wn,p(Z) such that
I(x) –Kε(x) =δε.
That is,
–px+α|x|p–x= (ω(z) +ε)|x|p–x+δε(z), z∈Z,
|∇x|p–∂x
∂n+b(z)|x|p–x= , z∈∂Z.
(.)
Next we showx∈int(C+). Take –x–= –max{–x, } ∈Wn,p(Z) forδε≥, then
I(x) –Kε(x), –x–
= –Dx–pp–αx–pp–
∂Z
b(z)|x–|pds
+
Z
ω(z)x–(z)
p
dz+εx–pp≥.
So
εx–pp≤ Dx–pp+αx–pp+
∂Z
b(z)|x–|pds–
Z
ω(z)x–(z)
p
dz≤εx–pp.
Butε<ε, we havex–= , that is,x≥. Sinceδε> , from (.), we havex= and
x∈Cn(Z) (nonlinear regularity theorem, see []), furthermore, px≤ onZ, sox∈
int(C+).
Now we prove that the solutionxof (.) is a strict sup solution of (.) forε> small enough.
Lemma . Let assumptions A(j)(i)-(iv)hold.Then the solution x of(.)is a strict sup solution of(.)forε> small enough.
Proof FromA(j)(iv), for givenε> , we can findM> , such that for allz∈Z,x≥M,
u∈∂j(z,x), we have
u
xp–≤ω(z) +ε.
FromA(j)(iii), we can findδε∈L∞(Z)+,δε= , such that for all z∈Z, x∈[,M],u∈
∂j(z,x), we have
u<δε(z).
So for allz∈Z,x≥,u∈∂j(z,x), we have
u<ω(z) +εxp–+δε(z).
From Lemma ., we see that (.) has a solutionx∈int(C+), so whenε<εsmall enough,
for allz∈Z,x∈Lp(Z)
+,u∈∂j(z,x(z)), we have
u<ω(z) +εxp–+δε(z),
that is,
u< –px+α|x|p–x,
Remark . We have found a sup solution of (.) and∂j(z, ) ={}a.e. onZ, we also find
x≡ is a sub-solution of (.). Define the set
W=x∈Wn,p(Z) : ≤x(z)≤x(z), a.e. onZ.
Next, we will find a nontrivial solution of (.) inW.
Proof of Theorem.
Step : Claim: We can findx∈Wwhich is a local minimizer ofϕ+and ofϕ.
From the discussion of Lemma ., for a.e.z∈Z, allx≥,u∈∂j+(z,x) =∂j(z,x), we
have
u<ω(z) +εxp–+δε(z).
Furthermore, for a.e.z∈Z, allx≥, from assumptionsA(j)(i), (ii),
d
dxj+(z,x) <
ω(z) +εxp–+δε(z)
then for a.e.z∈Z, allx≥, we have
j+(z,x) <
p
ω(z) +ε|x|p+δε(z)|x|.
So, for someC> , we have
ϕ+(x) =
pDx
p p+
α
px
p p+
p
∂Z
b(z)|x|pds–
Z
j+
z,x(z)dz
>
pDx
p p+
α
px
p p+
p
∂Z
b(z)|x|pds–
p
Z
ω(z)x(z)pdz–ε
px
p
p–Cxp ≥
p(ε–ε)x
p
p–Cxp.
Because ofε<ε,p> , we see thatϕ+ is coercive, and together withϕ+ weakly lower
semicontinuous onW. Thus by the Weierstrass theorem, we can findx∈W, satisfying
ϕ+(x) =inf
Wϕ+.
We claim thatx= . In fact, from assumptionA(j)(v), we see that, for givenε> , we can
find someδ> , for a.e.z∈Z, allx∈[,δ],u∈∂j+(z,x) =∂j(z,x),
u
xp–≥η(z) +α–ε;
then for a.e.z∈Zand allx∈[,δ], we get
j+(z,x)≥
p
Furthermore, letebe the first eigenfunction of Robin problem of –p(see []), then for
x∈int(C+), we can findθ> , such that
θe(z)≤min
x(z),δ, ∀z∈Z.
Thenθe∈int(C+), and
ϕ+(θe) =
θp
pDe
p p+
αθp
p e
p p+
θp
p
∂Z
b(z)|e|pds–
Z
j+
z,θe(z)
dz
≤θp
pDe
p p–
θp
p
Z
η(z)|e|pdz+
θp
p
∂Z
b(z)|e|pds+
εθp
p e
p p
= θ p
p
Z
λ–η(z)
|e|pdz+εepp
.
From assumptionA(j)(v) ande> , we have
Z
λ–η(z)
|e|pdz< .
If we chooseεsmall enough, we can getϕ+(θe) < for allθ> small enough. So, we have
ϕ+(x) =inf
Wϕ+≤ϕ+(θe) < =ϕ+(), then we havex= ,x∈W.
Step : The local minimizer ofϕ+,x∈Wn,p(Z) is a nontrivial solution of (.).
Firstly, we claim thatxis also a localWn,p(Z)-minimizer ofϕ+. In fact, the nonlinear
regularity theory (see for example []) assures thatx∈C(Z). Hence, as the boundary
relation is understood in a pointwise sense and we getx∈Cn(Z), also, byx= ,x≥,
and the nonlinear strong maximum principle of Vazquez,x∈int(C+),x–x∈int(C+). So
we can findδ> satisfying
Bδ(x) =
x∈Cn(Z) :x–xCn(Z)<δ
⊆int(C+),
Bδ(x–x) =
x∈Cn(Z) : x– (x–x) C
n(Z)<δ
⊆int(C+).
Then
x+Bδ⊆int(C+), x–x+Bδ⊆int(C+).
So,xis also a local minimizer ofϕ+onCn(Z); also from [],xis also a localWn,p(Z )-minimizer ofϕ+and ofϕtoo.
Also, from [], there existsω(z)∈∂j+(z,x(z)) =∂j(z,x(z)),u∈Lp
(Z) satisfying
≤I(x),y–x
–
Z
Using
y(z) =
⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩
x(z), z∈ {x≤x+εv}=A,
x(z) +εv(z), z∈ { <x+εv<x}=B,
, z∈ {x+εv≤}=C.
We havey∈Wfor allv∈Wn,p(Z),ε> , then we have
≤
A
|Dx|p–
Dx,D(x–x)
dz+α
A
|x|p–x,x–xdz–
A
u(x–x)dz
+ε
B
|Dx|p–Dx,Dvdz+α
B
|x|p–x,εvdz–
B
u(εv)dz
+
∂B
b(z)|x|p–x,εvds
–
C
|Dx|pdz–α
C
|x|pdz+
C
uxdz–
∂C
b(z)|x|pds
+
∂A
b(z)|x|p–x,x–xds
=ε
Z|
Dx|p–Dx,Dvdz+εα
Z|
x|p–x,vdz–ε Z uv dz +ε ∂Z
b(z)|x|p–x,vds
–
C
|Dx|pdz–α
C
|x|pdz–ε
C
|Dx|p–Dx,Dvdz–
∂C
b(z)|x|pds
–
A
|Dx|p–Dx,D(x+εv–x)
dz–α
A
|x|p–x,x+εv–xdz
+
A
u(x+εv–x)dz
–
∂A
b(z)|x|p–x,x+εv–xds+
C
u(x+εv)dz+
A
(u–u)(x–x–εv)dz
+
A
|Dx|p–Dx–|Dx|p–Dx,D(x–x)
dz+α
A
|x|p–x–|x|p–x,x–x
dz
+
∂A
b(z)|x|p–x–|x|p–x,x–x
ds+ε
C
|Dx|p–Dx–|Dx|p–Dx,Dv
dz
–ε
∂A
b(z)|x|p–x,vds–ε
∂C
b(z)|x|p–x,vds+ε
∂A
b(z)|x|p–x,vds
–εα
A
|x|p–x,vdx–εα
C
|x|p–x,vdx+εα
A
|x|p–x,vdx.
From the monotonicity ofI, we have
A
|Dx|p–Dx–|Dx|p–Dx,D(x–x)
dz+α
A
|x|p–x–|x|p–x,x–x
dz
+
∂A
b(z)|x|p–x–|x|p–x,x–x
From the definition of a sup solution of (.), we have
–
A
|Dx|p–Dx,D(x+εv–x)
dz–α
A
|x|p–x,x+εv–xdz+
A
u(x+εv–x)dz
–
∂A
b(z)|x|p–x,x+εv–xds≤.
FromA(j)(vi), we haveCu(x+εv)dz≤. Furthermore,
C
(u–u)(x–x–εv)dz≤εc
{x+εv≥x>x}
v dz.
Also,
m{x+εv≥x>x} →, asε→,
and
Dx(z) = a.e. on{x= },
Dx(z) =Dx(z) a.e. on{x=x}.
So, we have
≤ε
Z
|Dx|p–Dx,Dvdz+εα
Z
|x|p–x,vdz
+ε
∂Z
b(z)|x|p–x,vds–ε
Z
uv dz
–ε
C
|Dx|p–Dx,Dvdz+εc
{x+εv≥x>x}
v dz
+ε
C
|Dx|p–Dx–|Dx|p–Dx,Dv
dz.
Asε→, for allv∈Wn,p(Z), we obtain
≤I(x),v
–
Z
uv dz=I(x) –u,v
.
That is,
I(x) =u.
Thenx∈Wn,p(Z) is a solution of (.).
Step : In a similar way, we introduce another truncation functionν–:R→R–by
ν–(x) =
then define the locally Lipschitz functionalϕ–:Wn,p(Z)→Rby
ϕ–(x) =
pDx
p p+
α
px
p p+
p
∂Z
b(z)|x|pds–
Z
j–
z,x(z)dz, ∀x∈Wn,p(Z),
where j–(z,x) =j(z,ν–(x)) for allz∈R, x∈Rwhich is locally Lipschitz. Then we have
another nontrivial solutionx∗∈Wn,p(Z) which is a local minimum ofϕ– and ofϕ too.
4 Existence of the third nontrivial solution
In this section, we prove the existence of the third solution. Then we give the new assump-tions which differ slightly fromA(j)(v):
A(j) (i) z→j(z,x)is measurable for allx∈R; (ii) x→j(z,x)is locally Lipschitz for a.e.z∈Z;
(iii) |u| ≤γ(z) +C|x|p–for a.e.z∈Z, allx∈R,u∈∂j(z,x), withγ ∈L∞(Z) +and
C> ;
(iv) lim sup|x|→∞| u
x|p–x≤ω(z)for a.e.z∈Z, allu∈∂j(z,x), withω∈L∞(Z)+
satisfyingω(z)≤αa.e. inZandω(z) <αin some set of positive measure; (v) η(z)≤lim infx→|x|pu–x for a.e.z∈Z, allu∈∂j(z,x), withη∈L∞(Z)+
satisfyingλ+α≤η(z)a.e. inZandλ+α<η(z)in some set of positive
measure,λis the first eigenvalue of–pwith the Robin boundary condition; (vi) ux≥for a.e.z∈Z, allx∈R,u∈∂j(z,x).
Theorem . Let assumptions A(j)(i)-(vi)hold.Then we can find three nontrivial solu-tions x∈int(C+),x∗∈–int(C+),and y∈Cn(Z)of(.).
Proof From Theorem ., we have two constant sign solutions x∈int(C+) and x∗ ∈
–int(C+) which are the local minimizers ofϕ+and ofϕ–, also ofϕ. We may assume thatx
is the only nontrivial critical point ofϕ+andx∗is the only nontrivial critical point ofϕ–.
In fact, if there exists another nontrivial critical pointxofϕ+,x=x. Then, by a similar
discussion,x∈int(C+) and it solves (.). Thus we have a third nontrivial solution, a.e.
y=x.
Moreover, as forϕ, we see thatϕis coercive and so we can easily prove the Palais-Smale condition. In fact, as in the proof of Theorem ., for a.e.z∈Z, allx∈R, we have
j(z,x) <
p
ω(z) +ε|x|p+δε(z)|x|,
whereωsatisfies (iii),δε∈L∞(Z)+,δε= .
Then, using Lemma ., we have
ϕ(x)≥
pDx
p p+
α
px
p p+
p
∂Z
b(z)|x|pds–
p
Z
ω(z)x(z)pdz–ε
px
p
p–Cxp ≥
p(ε–ε)x
p
We setS=∂Bδ(x) ={x∈Wn,p(Z) :x–xW,p
n (Z)=δ},Q= [x∗,x], with relative
bound-ary ∂Q={x∗,x}. If we choose <δ <x∗–xW,p
n (Z). Then S and ∂Q link. In fact,
S∩∂Q=∅, and for any maph∈C(Q,W,p
n (Z)) such thath|∂Q=Id, we can choose some
t∈(, ) satisfying
htx∗+ ( –t)x
–x W,p n (Z)=δ,
soh(Q)∩S=∅,Sand∂Qlink.
When we chooseδ, we can also assumeδsatisfyinfx∈Sϕ>ϕ(x) andinfx∈Sϕ>ϕ(x∗) (x,
x∗are local minimizers ofϕ), we may assume thatϕ(x∗) <ϕ(x). Therefore, we can apply
Proposition .; let={h∈C(Q,W,p
n (Z)) :h|∂Q=Id}, producey∈Wn,p(Z), a critical point ofϕ, such that
∈∂ϕ(y),
ϕ(x∗) <ϕ(x) <inf
x∈Sϕ≤ϕ(y) =hinf∈supx∈Qϕ
h(x).
From the above inequality, we havey=x,y=x∗.
From ∈∂ϕ(y), we know that
–py(z) +α|y(z)|p–y(z)∈∂j(z,y(z)), z∈Z,
|∇y(z)|p–∂y∂n(z)+b(z)|y(z)|p–y(z) = , z∈∂Z,
(.)
and from the regularity theory (see []), we havey∈Cn(Z), hence (.) holds in allz∈Z, we gety∈Cn(Z).
Finally, we prove thaty= . It is equivalent to proving that there is a pathh∈ such
that for allx∈Q,
ϕh(x)< =ϕ().
From Proposition ., recall thatS=Wn,p(Z)∩∂B,∂B={x∈Lp(Z) :xp= }endowed with the Wn,p(Z)-topology. Furthermore, setSc =S∩Cn(Z) equipped with theCn(Z )-topology. Then we can findh∈Sc by virtue of the density ofSc in Sin the Wn,p(Z )-topology, soC(Q,Sc) is dense inC(Q,S), and
max
Dxpp+
∂Z
b(z)|x|pds,x∈h(Q)
≤λ+δ. (.)
From assumptionA(j)(v), we can findδ> , such that for a.e.z∈Z, all <|x|<δ,u∈
∂j(z,x), we get
η(z)≤ u |x|p–x.
So for a.e.z∈Z, all <|x|<δ,
η
p|x|
Sinceh(Q)∈Sc, for theδ, we can findε> , such that for a.e.z∈Z,x∈h(Q), we have
εx(z)≤δ. (.)
Then letδ> be such thatλ+α+δ<η, from (.), (.), (.), andxp= , we have
ϕ(εx) = ε p
pDx
p p+
αεp
p x
p p+
εp
p
∂Z
b(z)|x|pds–
Z
jz,εx(z)dz
≤ εp
pDx
p p+
αεp
p x
p p+
εp
p
∂Z
b(z)|x|pds–ηε p
p x
p p
= ε p
p
Dxpp+
∂Z
b(z)|x|pds
+α–η
p ε
p
≤ λ+δ+α–η
p ε
p< .
We consider the continuous pathhε=εh, then for allx∈Q,
ϕhε(x)
< , ∀x∈Q.
Next recall thatϕis coercive and satisfies the Palais-Smale condition. From the discussion, we seta=ϕ(x) =infϕ< ,b= ,ϕhas no critical points inϕ–(a,b),Ka={x}. Then with
the help of Proposition ., there exists a deformation: [, ]×ϕb→ϕbsuch that
(t,·)|Ka=Id, ∀t∈[, ],
,ϕb⊆ϕa∪Ka, (.)
ϕ(t,x)≤ϕ(x), ∀(t,x)∈[, ]×ϕb.
In fact, the continuous path can be seen as ={h∈C([, ],Wn,p(Z)) :h() =
x∗,h() =x}. Then we defineh: [, ]→ϕbby
h(t) =(t,εu), ∀t∈[, ].
Then it is a continuous path, so from (.), we have
h() =(,εu) =εu,
h() =(,εu) =x
ϕa=∅,Ka={x}
,
ϕh(t)
=ϕ(t,εu)
≤ϕ(εu) < , ∀t∈[, ] (ϕ|hε< ).
Thus, we construct a continuous pathhjoiningεuandxsuch that
ϕ|h< .
Similarly, we construct a continuous pathhjoining –εuandx∗such that
Then we joinh,hε,h, and we construct a continuous pathh∈such that
ϕ|h< .
It follows thatϕ(y) < =ϕ() and soy= .
Therefore, we find the third nontrivial solution of (.).
5 Open related questions
Consider the problem
–px+α(z)|x|p–x∈∂j(z,x), z∈Z,
|∇x|p–∂∂xn+b(z)|x|p–x= , z∈∂Z, (.)
whereZ⊂RNis a bounded domain withC-boundary∂Z,
px= div(|∇x|p–∇x) ( <p< ∞) is thep-Laplacian operator,α(z),b(z)∈L∞(∂Z),b(z)≥, andb(z)= on∂Z.j(z,x) is a measurable potential function onZ×R, which is locally Lipschitz in thex∈R,∂j(z,x) stands for the generalized subdifferential ofx→j(z,x). Also ∂∂nxdenotes the outer normal derivative ofxwith respect to∂Z.
Whether problem (.) has more solutions and whether it has oscillating solutions, we will discuss in the future.
Competing interests
The author declares that she has no competing interests.
Author’s contributions
The author wrote, read, and approved the final manuscript.
Acknowledgements
The author was supported by Tianyuan Foundation for Mathematics under Grant No. 11326098 of National Natural Science Foundation and Doctor Scientific Research Startup Foundation of Harbin Normal University No. XKB 201311.
Received: 5 January 2014 Accepted: 1 December 2014 References
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doi:10.1186/s13661-014-0257-5
