Constant Sign and Nodal Solutions for Problems with the -Laplacian and a Nonsmooth Potential Using Variational Techniques

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Volume 2009, Article ID 820237,32pages doi:10.1155/2009/820237

Research Article

Constant Sign and Nodal Solutions for

Problems with the

p

-Laplacian and a Nonsmooth

Potential Using Variational Techniques

Ravi P. Agarwal,

1

_{Michael E. Filippakis,}

2

_{Donal O’Regan,}

3

and Nikolaos S. Papageorgiou

4

1_{Department of Mathematical Sciences, Florida Institute of Technology, Melbourne, FL 32901-6975, USA} 2_{Department of Mathematics, Hellenic Army Academy, Vari, 16673 Athens, Greece}

3_{Department of Mathematics, National University of Ireland, Galway, Ireland}

4_{Department of Mathematics, National Technical University, Zografou Campus, 15780 Athens, Greece}

Correspondence should be addressed to Ravi P. Agarwal,[email protected]

Received 10 December 2008; Revised 21 January 2009; Accepted 23 January 2009

Recommended by Juan J. Nieto

We consider a nonlinear elliptic equation driven by thep-Laplacian with a nonsmooth potential

hemivariational inequality and Dirichlet boundary condition. Using a variational approach based on nonsmooth critical point theory together with the method of upper and lower solutions, we prove the existence of at least three nontrivial smooth solutions: one positive, the second negative, and the third sign changing nodal solution. Our hypotheses on the nonsmooth potential incorporate in our framework of analysis the so-called asymptoticallyp-linear problems.

Copyrightq2009 Ravi P. Agarwal et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

1. Introduction

The aim of this work is to prove the existence of multiple solutions of constant sign and of nodal solutions sign changing solutionsfor nonlinear elliptic equations driven by thep -Laplacian and having a nonsmooth potentialhemivariational inequalities. So letZ ⊆ RN

be a bounded domain with a C2_-boundary _∂Z. _{The problem under consideration is the}

following:

−divDxzp−2_Dx_z_∈_∂j_{z, x}_z _a_._e_._on_Z,

x|∂Z 0 1< p <∞. 1.1

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Clarke 1. Problem 1.1 is a hemivariational inequality. Hemivariational inequalities are a new type of variational expressions, which arise in applications if one considers more realistic mechanical laws of multivalued and nonmonotone nature. Then the corresponding energyEulerfunctional is nonsmooth and nonconvex. Various engineering applications of hemivariational inequalities can be found in the book of Naniewicz-Panagiotopoulos2.

Multiple solutions of constant sign for problems monitored by thep-Laplacian and with aC1_{-potential were obtained by Ambrosetti et al.}₃_{, Garc´}_ı_{a Azorero-Peral Alonso}₄_,

and Garc´ıa Azorero et al.5. In all these works, the authors consider nonlinear eigenvalue problems and prove the existence of positive and negative solutions for certain values of the parameter λ ∈ R.The question of existence of nodal solutions was first addressed within the framework of semilinear problems i.e.,p 2. We mention the works of Dancer-Du

6and Zhang-Li7, which contain two diﬀerent approaches to the problem. In Dancer-Du

6, the authors use a combination of the variational methodcritical point theorywith the method of upper and lower solutions. In contrast Zhang-Li7use invariance properties of the negative gradient flow of the corresponding equation inC1

0Z.Recently these methods

were extended to “smooth” problems driven by thep-Laplacian diﬀerential operator. Carl-Perera8extended the work of Dancer-Du6, by assuming the existence of upper and lower solutions for the problem. Zhang-Li9and Zhang et al.10extended the semilinear work of7, by carefully constructing a pseudogradient vector field whose descent flow exhibits the necessary invariance properties. These works were extended recently by Filippakis-Papageorgiou 11. Recently the approach based on the invariance properties of descent flow was used by Zhang-Perera12to produce nodal solutions for certain Kirchhoﬀtype equations. Other recent works dealing withp-Laplacian equations are those by Ahmad-Nieto

13 monotone iterative technique, Kim et al.14 radial solutions, Lin et al. singular odes 15, and V¨ath16 degree theoretic approach.

In this paper using techniques from nonsmooth critical point theory in conjunction with the method of upper and lower solutions, we are able to extend the works of Dancer-Du 6 and Carl-Perera 8 to hemivariational inequalities. Helpful in this respect is the nonsmooth second deformation lemma of Corvellec17. Recently, sign-changing solutions for problems with discontinuous nonlinearities were obtained by Averna et al.18, but in contrast to our work they deal withp-superlinear problems.

2. Mathematical Background

In our analysis of problem1.1, we use the nonsmooth critical point theory which is based on the subdiﬀerential theory for locally Lipschitz functions and some basic facts about the spectrum of the negativep-Laplacian with Dirichlet boundary conditions. For easy reference, we recall some definitions and results from these areas, which will be used in the sequel.

We start with the subdiﬀerential theory for locally Lipschitz functions and the corresponding nonsmooth critical point theory. Details can be found in the books of Gasi ´nski -Papageorgiou19and Motreanu-Panagiotopoulos20. So letXbe a Banach space and let

X∗be its topological dual. By·,· we denote the duality brackets for the pairX, X∗. Given a locally Lipschitz functionϕ:X → R,the generalized directional derivativeϕ0_x_;_h_of_ϕ_at

x∈Xin the directionh∈Xis defined as follows:

ϕ0x;h lim sup

x→x λ↓0

ϕxλh−ϕx

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The functionh → ϕ0_x_;_h_{is sublinear continuous and so it is the support function of}

a nonempty, convex, andw∗-compact set∂ϕx⊆X∗defined by

∂ϕx x∗∈X∗:x∗, h≤ϕ0_x_;_h_∀h_∈_X_. _2.2

The multifunctionx → ∂ϕx is called the “generalized gradient” or generalized subdiﬀerential of ϕ. If ϕ : X → R is also convex, then ∂ϕx coincides with the subdiﬀerential in the sense of convex analysis∂cϕx,defined by

∂cϕx x∗∈X∗:x∗, y−x≤ϕy−ϕx∀y∈X. 2.3

Moreover ifϕ∈C1_X_,_then_ϕ_{is locally Lipschitz and}_∂ϕ_x _{ϕ_x_}.

We say thatx ∈ X is a critical point of the locally Lipschitz functionϕ : X → R,if 0 ∈∂ϕx.It is easy to see that ifx∈Xis a local extremum ofϕi.e., a local minimum or a local maximum ofϕ, thenx∈Xis a critical point ofϕ.

A locally Lipschitz functionϕ : X → Rsatisfies the Palais-Smale condition at level

c ∈ RP Sc-condition for short, if every sequence{xn}_n_≥₁ ⊆ X such thatϕxn → cand

mxn inf{x∗:x∗∈∂ϕxn} → 0 asn → ∞has a strongly convergent subsequence. We say thatϕsatisfies theP S-condition, if it satisfies theP Sc-condition for everyc∈R.

The following topological notion is crucial in the minimax characterization of the critical values of a locally Lipschitz functionalϕ:X → R.

Definition 2.1. LetY be a Hausdorﬀtopological space andE0, E,andDare nonempty closed

subsets ofY withE0 ⊆E.We say that the pair{E0, E}is linking withDinY if and only if

aE0∩D∅;

bfor anyγ∈CE, Ysuch thatγ|E0id|E0,we haveγE∩D /∅.

Using this notion, we have the following general minimax principle for the critical values of a locally Lipschitz functionϕ:X → R.

Theorem 2.2. IfXis a reflexive Banach space,E0,E, andDare nonempty closed subsets ofXsuch

that {E0, E} is linking with D in X, ϕ : X → R is locally Lipschitz, sup_E₀ϕ < infDϕ, Γ {γ ∈ CE, X : γ|E0 id|E0}, c infγ∈Γsupv∈Eϕγv,andϕsatisfies theP Sc-condition, then c≥infDϕandcis a critical value ofϕ.

Remark 2.3. From this general minimax principle, by appropriate choices of the linking sets, one can produce nonsmooth versions of the mountain pass theorem, of the saddle point theorem, and of the generalized mountain pass theorem.

Definition 2.4. IfY is a subset of the Banach spaceX,a “deformation ofY” is a continuous maph:0,1×Y → Y such thath0,· idY. IfV ⊆ Y, then we can say thatV is a “weak deformation retract ofY”, if there exists a deformationh:0,1×Y → Ysuch thath1, Y⊆V

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Given a locally Lipschitz functionϕ:X → Randc∈R,we define

0

ϕc{x∈X:ϕ < c},

Kc{x∈x: 0∈∂ϕx, ϕx c}.

2.4

The next theorem is a partial extension to a nonsmooth setting of the so-called “second deformation theorem” see, e.g., Gasi ´nski -Papageorgiou21, page 628 and it is due to Corvellec17. In fact the result of Corvellec is formulated in the more general context of metric spaces, for continuous functions using the so-called weak slope. For our purposes, it suﬃces to use a particular form of the result which we state next.

Theorem 2.5. IfXis a Banach space,ϕ:X → Ris locally Lipschitz and satisfies theP S-condition, a∈R, b∈R∪ {∞}, ϕhas no critical points inϕ−1_{a, b}_,_and_Ka_{is discrete nonempty, then there}

exists a deformationh:0,1×

0

ϕb_→_ϕ0b_{such that}

aht,·|Ka idfor allt∈0,1;

bh1,

0

ϕb_⊆_ϕ0a_∪Ka_;

cϕht, x≤ϕxfor allt∈0,1and allx∈

0

ϕb _.

In particular the setϕ0a_∪Ka_{is a weak deformation retract of}_ϕ0b_.

Next let us recall some basic facts about the spectrum of the negativep-Laplacian with Dirichlet boundary conditions. So let Z ⊆ RN be a bounded domain with a C2_-boundary

∂Zandm∈L∞Z, m /0. We consider the following nonlinear weightedwith weightm

eigenvalue problem:

−divDxzp−2_Dx_z_λm _z_|x_z_|p−2_x_z _a_._e_._on_Z,

x|∂Z 0 1< p <∞. 2.5

The least number λ ∈ R for which problem 2.5 has a nontrivial solution is the first eigenvalue of−Δp, W₀1,pZ, mand it is denoted byλ1m. The first eigenvalueλ1m

is strictly positive i.e., λ1m > 0; it is isolated and it is simple i.e., the associated eigenspace is one dimensional. Moreover, using the Rayleigh quotient we have a variational characterization ofλ1m, namely,

λ1m min Dx

p p

Zm|x|pdz

:x∈W₀1,pZ, x /0

, 2.6

3. Solutions of Constant Sign

In this section, we produce two nontrivial solutions of1.1which have constant sign. The first is positive and the second is negative. To do this, we will need the following hypotheses on the nonsmooth potentialjz, x.

Hj₁:j :Z×R → Ris a function such thatjz,0 0 a.e. onZ, ∂jz,0 {0}a.e. onZ, and

ifor everyx∈R,z → jz, xis measurable;

iifor almost allz∈Z,x → jz, xis locally Lipschitz;

iiifor almost allz∈Z,allx∈R, and allu∈∂jz, x,we have

|u| ≤az c|x|p−1 witha∈L∞Z, c >0; 3.1

ivthere existsθ∈L∞Zsatisfyingθz≤λ1a.e. onZwith strict inequality on

a set of positive measure, such that

lim sup |x| → ∞

u

|x|p−2_x ≤θz 3.2

uniformly for almost allz∈Zand allu∈∂jz, x;

vthere existη,η∈L∞Zsatisfyingλ1 ≤ηz≤ηza.e. onZ,where the first

inequality is strict on a set of positive measure, such that

ηz≤lim inf

x→0

u

|x|p−2_x ≤lim sup

x→0

u

|x|p−2_x ≤ηz 3.3

uniformly for almost allz∈Zand allu∈∂jz, x;

vifor almost all z ∈ Z, allx ∈ R, and allu ∈ ∂jz, x,we haveux ≥ 0 sign condition.

Remark 3.1. HypothesesHj₁ivandvare nonuniform nonresonance conditions at zero and at ±∞, respectively. Moreover, as we move from 0 to±∞,the “slopes” u/|x|p−2_{x. u} _∈

∂jz, x cross the first eigenvalue λ1 > 0. So our framework incorporates the so-called

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The next lemma is an easy consequence of the strict positivity ofu1∈C1₀Zand of the

hypotheses onθ∈L∞ZseeHjiv. We omit the proof.

Lemma 3.2. Ifθ ∈ L∞Z satisfiesθz ≤ λ1 a.e. onZwith strict inequality on a set of positive

measure, then there existsξ0>0such that

Dxp p−

Z

θz|xz|pdz≥ξ0Dxpp ∀x∈W01,pZ. 3.4

Givenε > 0 andγε ∈ L∞Z, γε/0, we consider the following nonlinear Dirichlet problem:

−divDxzp−2_Dx_z _θ_z _ε_|x_z_|p−2_x_z _γε_z _a_._e_._on_Z,

x|∂Z0. 3.5

In the next proposition, we establish the solvability of3.5.

Proposition 3.3. Ifθ ∈L∞Zsatisfiesθ ≤λ1 a.e. onZwith strict inequality on a set of positive

measure, then for allε >0small problem3.5admits a solutionx∈intC1₀Z.

Proof. In what follows by ·,· we denote the duality brackets for the pair

W−1,p_Z_{, W}1,p

0 Z1/p1/p 1. We introduce the nonlinear operator A : W 1,p

0 Z →

W−1,p_Z_{defined by}

Ax, y

Z

Dxzp−2Dxz, Dyz_RNdz ∀x, y∈W₀1,pZ. 3.6

It is straightforward to check thatAis strictly monotone and demicontinuous, hence maximal monotone too. Also letNε:Lp_Z _→ _Lp_Z_{be the nonlinear, bounded, continuous}

map defined by

Nεx· θ· ε|x·|p−2x·. 3.7

Because of the compact embedding ofW₀1,pZintoLp_Z_,_Nε_{viewed as a map from} W₀1,pZ into Lp_Z _{is completely continuous. Therefore} _x _→ _Gε_x _A_x₋_Nε_x _is

pseudomonotone fromW₀1,pZintoW−1,pZ.Also for everyx∈W₀1,pZ,we have

Gεx, xDxpp−

Z

θz|xz|pdz−εxpp

≥

ξ0− ε

λ1

Dxp

psee Lemma 3.2 and2.6.

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Therefore, ifε < λ1ξ0,then by virtue of Poincare’s inequalityGε·is coercive. But a

pseudomonotone coercive operator is surjective. Hence we can find thatx ∈ W₀1,pZsuch that

Gεx Ax−Nεx γε, 3.9

⇒

−divDxzp−2_Dx_z _θ_z _ε_|x_z_|p−2

xz γεza.e.onZ, x|∂Z 0

. 3.10

Thusx∈W₀1,pZis a solution of3.5. We take duality brackets of3.9with the test function−x−−max{−x,0} ∈W₀1,pZ.We obtain

Dx−p p−

Z

θz|x−z|pdz≤εx−pp

sinceγε≥0,

⇒ξ0Dx−pp≤ ε λ1Dx

−p

p see Lemma 3.2 and2.6.

3.11

But recall thatε < λ1ξ0.So it follows thatDx−p 0,hencex− 0; that is,x ≥ 0.

Sinceγε/0,from3.10it follows that_{x /}0 andx∈C1

0Z nonlinear regularity theory. In

addition, from3.10, we see that

divDxzp−2Dxz≤0 a.e.onZ,

⇒x∈intC10Z see V´azquez23.

3.12

In fact the solutionx∈intC1₀Zof3.5is an upper solution for problem1.1.

Proposition 3.4. If hypotheses Hj₁(i)→(iv) hold and ε > 0 is small, then the solution x ∈ intC1

0Z of problem 3.5obtained in Proposition 3.3is a strict upper solution of problem 1.1

(strict means thatxis an upper solution of1.1which is not a solution).

Proof. Because of hypothesesHj₁iv, givenε > 0,we can findM1 M1ε >0 such that

for almost allz∈Z,allx≥M1, and allu∈∂jz, x,we have

u≤θz εxp−1. 3.13

Also due to hypothesisHj₁iii, we can findγε∈L∞Z, γε/0 such that for almost allz∈Z,allx∈0, M1, and allu∈∂jz, x,we have

u < γεz. 3.14

Therefore it follows that for almost allz∈Z,allx≥0, and allu∈∂jz, x,we have

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So for 0< ε < λ1ξ0andγεas above, we consider problem3.5. FromProposition 3.3,

we have a solution x ∈ intC1

0Z. Then due to 3.15, for all u ∈ Lp

Z with uz ∈

∂jz, xza.e. onZ,we have

uz<θz εxzp−1γεz a.e.onZ,

⇒x∈intC1₀Z is a strict upper solution for problem1.1.

3.16

Since∂jz,0 {0}a.e. onZ, x≡0 is a lower solution for problem1.1. We introduce the set

C{x∈W₀1,pZ: 0≤xz≤xza.e.onZ} 3.17

and the truncation functionτ:R → Rdefined by

τx

0 ifx≤0,

x ifx >0. 3.18

Then we setj·, x jz, τxand we consider the locally Lipschitz functionalϕ :

W₀1,pZ → Rdefined by

ϕx 1

pDx p p−

Z

jz, xzdz ∀x∈W01,pZ. 3.19

We will show that we can find a nontrivial solution of 1.1 in C, which is a local minimizer ofϕand ofϕ.To do this we will need the following simple result about ordered Banach spaces.

Lemma 3.5. IfXis an ordered Banach space,Kis the order cone ofX,int_{K /}∅, andx0∈intK,then

for everyy∈X,we can findtty>0such thattx0−y∈intK.

Proof. Sincex0∈intK,we can findδ >0 such that

Bδx0 x∈X:x−x0 ≤δ

⊆intK. 3.20

Lety ∈ X, y /0 ify 0,then clearly the lemma holds for all t > 0. We have the following:

x0−δ_yy ∈intK,

⇒ y

δ x0−y∈intK.

3.21

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Using this lemma, we can prove the following result.

Proposition 3.6. If hypothesesHj₁hold, then there existsx0∈Cwhich is a local minimizer ofϕ and ofϕ.

Proof. From3.15, we know that givenε >0,we can findγε∈L∞Z, γε/0 such that

u <θz εxp−1γεz for a.a. z∈Z, allx≥0,

and all u∈∂jz, x ∂jz, x.

3.22

Because of hypotheses Hj₁i, ii, for almost all z ∈ Z, x → jz, x is almost everywhere diﬀerentiable on R Rademacher’s theorem and at every point of diﬀerentiability we have

d

dxjz, x∈∂jz, x,

⇒ d

dxjz, x<θz εx

p−1_γε_z _{for a}_._a_{. z}_∈_Z, _all_x_≥₀ _see₃_.₂₂_.

3.23

Integrating this inequality and sincejz, x|R−0 for almost allz∈Z,we obtain

jz, x< 1

pθz ε|x|

p_γε_z_|x| _{for a}_._a_{. z}_∈_Z, _all _x_∈_R_. _3.24

Then for everyx∈W₀1,pZ,we have

ϕx 1_pDxpp−

Z

jz, xzdz

> 1 pDx

p p−

1

p

Z

θz|xz|pdz− ε px

p

p−c1Dxp

for somec1>0 see3.24

≥ 1 p

ξ0− ε

λ1

Dxp

p−c1Dxp see Lemma 3.2.

3.25

Choosingε < λ1ξ0,becausep >1,from3.25and Poincare’s inequality, we infer that

ϕis coercive. Also it is easy to see thatϕis weakly lower semicontinuous onW01,pZ.Hence

by virtue of the theorem of Weierstrass, we can findx0∈Csuch that

ϕx0 inf

C ϕ. 3.26

First we show thatx0/0.To this end, note that hypothesisHj1vimplies that given

ε >0,we can findδδε>0 such that

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As before, integrating3.27, we obtain

jz, x≥ 1

pηz−εx

p _{for a}_._a_{. z}_∈_Z, _all_x_∈₀_{, δ}_. _3.28

We know thatx ∈ intC1

0Z seeProposition 3.3. So usingLemma 3.5, we can find

μ >0 small such that

μu1z≤min{xz, δ} ∀z∈Z. 3.29

Then, because of3.28, we have

ϕμu1

μp

p Du1 p p−

Z

jz, μu1zdz

≤ μp p Du1

p p−

μp p

Z

ηzu1zpdzμ p_ε

p u1 p p

μp

p Z

λ1−ηz

u1zpdzεu1p_p

.

3.30

Letσ _Zλ1−ηzu1zpdz.Using the hypothesis onηseeHj1vand the fact

thatu1z>0 for allz∈Z,we see thatσ <0.So, if we chooseε <−σ/u1pp,we have

ϕμu1

<0 ∀μ >0 small. 3.31

Note that forμ >0 small,μu1∈C.Hence

ϕx0

inf

C ϕ≤ϕ

μu1

<0ϕ0 see3.31,

⇒x0/0, x0∈C.

3.32

Given anyy ∈C,we definek0t ϕty 1−tx0, t ∈ 0,1.Thenk is Lipschitz continuous, hence diﬀerentiable almost everywhere and k00 ≤ k0t for all t ∈ 0,1.

From Chang27, page 106, we know that we can findu ∈ Lp_Z_{, u}_z _∈ _∂j

z, x0z

∂jz, x0za.e. onZ,such that

0≤Ax0

, y−x0

−

Z

uzy−x0

zdz, ∀y∈C. 3.33

For anyv∈W₀1,pZandε >0,we define

yz

⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩

0 if z∈x0εv≤0

,

x0z εvz if z∈0< x0εv < x

, xz if z∈x≤x0εv

.

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Clearlyy∈C.We use thisy∈Cin3.33. Hence we obtain

0≤ε

{0<x0εv<x}

Dx0p−2Dx0, DvRNdz−

{0<x0εv<x}

uεvdz

−

{x0εv≤0}

Dx0pdz

{x0εv≤0} ux0dz

{x0εv≥x}

Dx0p−2Dx0, Dx−x0_RNdz−

{x0εv≥x}

ux−x0dz

ε

Z

Dx0p−2Dx0, DvRNdz−ε

Z uv dz

−

{x0εv≥x}

Dxp−2_{Dx, D}_x

0εv−xRNdz

{x0εv≥x}

ux0εv−xdz

u∈LpZ, uz∈∂jz, xza.e.and clearly from the definition ofu, we can always assumeuua.e.onxx0

{x0εv≤0}

ux0εv

dz

{x0εv≥x}

u−ux−x0−εv

dz

−

{x0εv≤0}

_Dx₀p dz−ε

{x0εv≤0}

_Dx₀p−2

Dx0, Dv

RNdz

{x0εv≥x}

_Dxp−2

Dx−Dx0p−2Dx0, D

x0−x

RNdz

ε

{x0εv≥x}

_Dxp−2

Dx−Dx0p−2Dx0, Dv

RNdz.

3.35

Usingh x0εv−x∈W01,pZas a test function, from the definition of an upper

solution for problem1.1, we have

−

{x0εv≥x}

Dxp−2_{Dx, D}_x

0εv−x

RNdz

{x0εv≥x}

ux0εv−x

dz≤0. 3.36

Also from thestrictmonotonicity of the operatorA,we have

{x0εv≥x}

_Dxp−2

Dx−Dx0p−2Dx0, D

x0−x

RNdz≤0. 3.37

From hypothesisHj₁vi, it follows that

{x0εv≤0}

ux0εv

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Sincex0∈C1₀Z,we have

{x0εv≥x}

u−ux−x0−εv

dz

{x0εv≥x>x0}

u−ux−x0−εv

dz

recall that uua.e.onxx0

, x0≤xsincex0∈C

≤c2

{x0εv≥x>x0}

x0εv−x

dzfor somec2>0

see hypothesisHj₁iii

≤εc2

{x0εv≥x>x0}

v dzsincex0≤x

.

3.39

Returning to3.35and using3.36→3.39, we obtain

0≤ε

Z

_Dx₀p−2

Dx0, Dv

RNdz−ε

Z uv dz

εc2

{x0εv≥x>x0}

v dz−ε

{x0εv≤0}

_Dx₀p−2

Dx0, Dv

RNdz

ε

{x0εv≥x}

_Dxp−2

Dx−Dx0p−2Dx0, Dv

RNdz.

3.40

We denote by| · |Nthe Lebesgue measure onRN.Then

_x₀_εv_≥_{x > x}₀

N↓0 as ε↓0. 3.41

Moreover, from Stampacchia’s theorem, we know that

Dx0z 0 a.e.onx00

, Dx0z Dxz a.e.onx0x

. 3.42

If we divide3.40byε >0 and then we pass to the limit asε↓0,because of3.41and

3.42, we obtain

0≤Ax0

, v−

Z

uv dzAx0

−u, v. 3.43

Recall thatv∈W₀1,pZwas arbitrary. So from3.43, it follows that

Ax0 u,

⇒x0 ∈W₀1,pZis a solution of problem1.1.

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The nonlinear regularity theory implies thatx0 ∈C₀1Zand then sincex0/0, x0 ≥0

from the nonlinear strong maximum principle of V´azquez23, we havex0 ∈intC10Z.

From3.22, we know that

uz<θz εxzp−1γεz a.e.onZrecall that x0 ≤x

. 3.45

Then Proposition 2.2 of Guedda-V´eron28implies that

x0z< xz ∀z∈Z, ∂x ∂nz<

∂x0

∂nz ∀z∈∂Z,

⇒x−x0∈intC1₀Z.

3.46

Recall also thatx0∈intC10Z.Thus we can findδ >0 such that

BC10Z δ

x−x0

y∈C1₀Z:y−x−x0_C1 0Z< δ

⊆intC1₀Z,

BC10Z δ

x0

y∈C1₀Z:y−x0_C1 0Z< δ

⊆intC1₀Z.

3.47

These inclusions imply that

x−x0BC

1 0Z δ

⊆intC1₀Z, x0BC

1 0Z δ ⊆intC

1

0Z. 3.48

The solutionx0 was obtained as a minimizer ofϕonC.Then3.48implies thatx0

is also a local minimizer of ϕ and of ϕon C10Z.But then from Motreanu-Papageorgiou 29 see also Gasi ´nski -Papageorgiou21, pages 655-656, it follows thatx0 is also a local

W₀1,pZ-minimizer ofϕand ofϕtoo.

So far we have worked on the positive semiaxis. Next we repeat the same analysis on the negative semiaxis. More precisely, givenε > 0 andγε ∈L∞Z, γε/0,we consider the following auxiliary problem:

−divDvzp−2Dvzθz εvzp−2vz−γεz a.e.onZ,

v|∂Z0.

3.49

Then as in the proof ofProposition 3.3, through the surjectivity of the pseudomonotone coercive operatorv → Gεv Av−Nεv,we obtain a solutionu∈ −intC1

0Zof3.49.

We can check thatvis a strict lower solution of1.1, while clearlyv≡0 is an upper solution

in fact a solutionof1.1. This time we consider the set

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and the truncation functionτ−:R → R−defined by

τ−x

x ifx <0,

0 ifx≥0. 3.51

We setj−z, x jz, τ−x and then introduce the locally Lipschitz functionalϕ− : W₀1,pZ → Rdefined by

ϕ−x 1 pDx

p p−

Z

j−z, xzdz ∀x∈W01,pZ. 3.52

We consider the minimization problem

inf

D ϕ−. 3.53

Arguing as inProposition 3.3, we obtain the following.

Proposition 3.7. If hypothesesHj₁hold, then there existsv0∈Dwhich is a local minimizer ofϕ− and ofϕ.

Now combining Propositions3.6and3.7, we obtain a multiplicity result for problem

1.1with solutions of constant sign.

Theorem 3.8. If hypotheses Hj₁ hold, then the problem1.1 has at least two solutions x0 ∈

intC1

0Zandv0∈ −intC01Z.

Remark 3.9. From Propositions3.6and3.7, we know that bothx0andv0are local minimizers

ofϕ.So we must have a third critical point ofϕ,distinct fromx0, v0.However, at this point

we cannot guarantee that it is nontrivial. In the next section by strengthening our hypothesis onjz,·near the originseeHj₁v, we will be able to show that this third critical point is nontrivial and in fact is a nodal solution.

4. Existence of Nodal Solution

Recall that every eigenfunction of2.5corresponding to an eigenvalue_{λ /}λ1 must change

sign. So we expect that in general the sign changing solutions of1.1must be more than the solutions of constant sign. Nevertheless to produce a sign-changing solutionalso known as nodal solutionfor1.1is a rather involved process.

Here we follow an approach first employed by Dancer-Du6for semilinear problems

i.e.,p2and recently extended to problems with thep-Laplacian and a smooth potential by Carl-Perera8. Roughly speaking, the strategy is as follows. Continuing with the argument used in Section 3, we produce the smallest positive solution y and the largest negative solutiony−.Then we form order intervaly−, y.Using variational techniquesin particular Theorem 2.2we produce a solutiony0of1.1iny−, ydiﬀerent fromy−andy.Evidently

ify0/0, theny0must be sign changing. To show thaty0is nontrivial, we employTheorem 2.5

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method can also be found in the works of Ambrosetti-Garcia Azorero-Peral Alonso3and Jin30. A diﬀerent approach based on the construction of a pseudogradient vector field with appropriate invariance properties can be found in Zhang-Li7,9, Zhang et al.10 see also Li-Wang31.

We start executing the solution strategy outlined above by proving first a lemma which establishes that the set of upper solutions for problem1.1is downward directed.

Lemma 4.1. Ify1, y2 ∈W1,pZare two upper solutions for problem1.1andy min{y1, y2} ∈

W1,pZ,thenyis also an upper solution for problem1.1.

Proof. Givenε >0, we consider the truncation functionξε:R → Rdefined by

ξεs

⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩

ε ifs≥ε, s ifs∈−ε, ε, −ε ifs≤ −ε.

4.1

Clearlyξεis Lipschitz continuous. So from Marcus-Mizel32, we have

ξεy1−y2

−_∈

W1,pZ,

Dξεy1−y2

− ξε

y1−y2

−

Dy1−y2

− .

4.2

Consider a test functionψ ∈C1

cZwithψ≥0.Then

ξεy1−y2

−

ψ∈W1,p_Z_∩_L∞_Z_,

Dξεy1−y2

−

ψψDξεy1−y2

−

ξεy1−y2

− Dψ.

4.3

Becausey1, y2∈W1,pZare upper solutions for problem1.1, we have

Ay1

, ξεy1−y2

−

ψ≥u1, ξε

y1−y2

− ψ,

Ay2

,ε−ξεy1−y2−

ψ≥u2,

ε−ξεy1−y2

− ψ

4.4

for someuk ∈Lp_Z_with_uk_z_∈_∂j_{z, yk}_z_{a.e. on}_{Z, k} ₁_,₂_._{Adding these inequalities,}

we obtain

Ay1

, ξεy1−y2

₋

ψ Ay2

,ε−ξεy1−y2

₋

ψ

≥u1, ξε

y1−y2

₋

ψ u2,

ε−ξεy1−y2

₋

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Note that

Ay1

, ξεy1−y2

− ψ

Z

_Dy₁p−2

Dy1, D

y1−y2

₋

RNξε

y1−y2

₋

ψ dz

Z

_Dy₁p−2

Dy1, Dψ

RNξε

y1−y2

− dz

−

{−ε≤y1−y2≤0}

_Dy₁p−2

Dy1, D

y1−y2

RNψ dz

Z

_Dy₁p−2

Dy1, Dψ

RNξε

y1−y2

− dz,

Ay2

,ε−ξεy1−y2

₋

ψ

{−ε≤y1−y2≤0}

_Dy₂p−2

Dy2, D

y1−y2

RNψ dz

Z

_Dy₂p−2

Dy2, Dψ

RN

ε−ξεy1−y2

− dz.

4.6

Adding4.6and recalling thatψ≥0,we obtain

Ay1

, ξεy1−y2

−

ψAy2

,ε−ξεy1−y2

− ψ

{−ε≤y1−y2≤0}

_Dy₂p−2

Dy2−Dy1p−2Dy1, D

y1−y2

RNψ dz

Z

_Dy₁p−2

Dy1, Dψ

RNξε

y1−y2

− dz

Z

_Dy₂p−2

Dy2, Dψ

RNε−ξε

y1−y2

− dz

≤

Z

_Dy₁p−2

Dy1, Dψ

RNξε

y1−y2

− dz

Z

_Dy₂p−2

Dy2, Dψ

RN

ε−ξεy1−y2

₋

dz.

4.7

Returning to4.5, using4.7, and dividing byε >0,we get

Z

_Dy₁p−2

Dy1, Dψ

RN 1

εξε

y1−y2

− dz

Z

_Dy₂p−2

Dy2, Dψ

1−1

εξε

y1−y2

− dz ≥ u,1 εξε

y1−y2

− ψ

u2,

1−1

εξε

y1−y2

− ψ

.

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We observe that

1

εξε

y1−y2

₋

z−→χ{y1<y2}z a.e.onZasε↓0,

χ{y1≥y2}1−χ{y1<y2}.

4.9

Therefore, if we pass to the limit asε↓0 in4.8, we obtain

{y1<y2}

_Dy₁p−2

Dy1, Dψ

RNdz

{y1≥y1}

_Dy₂p−2

Dy2, Dψ

RNdz

≥

{y1<y2}

u1ψ dz

{y1≥y2} u2ψ dz.

4.10

Sinceymin{y1, y2} ∈W1,pZ,we have

Dyz

⎧ ⎨ ⎩

Dy1z for a.a. z∈y1< y2

, Dy2z for a.a. z∈y1≥y2

. 4.11

Also ifu χ{y1<y2}u1 χ{y1≥y2}u2,then u ∈ Lp

Zanduz ∈ ∂jz, yza.e. onZ.

Therefore

Z

Dyp−2_{Dy, Dψ} RNdz≥

Z

uψdz 4.12

for someu ∈ Lp_Z_with_u_z _∈ _∂j_{z, y}_z_{a.e. on}_Z._Since_ψ _∈ _C1

cZ was arbitrary and

C1

cZ is dense inW01,pZ,from4.12we conclude thaty min{y1, y2} ∈ W1,pZis an

upper solution for problem1.1.

Arguing similarly, we can also show that the set of lower solutions for problem1.1 is upward directed. Namely, we have the following.

Lemma 4.2. If v1, v2 ∈ W1,pZ are lower solutions for problem 1.1 and v max{v1, v2} ∈

W1,p_Z_,_then_v_{is also a lower solution for problem}_1.1_.

Now that we have established that the sets of upper solutions and of lower solutions are directed, we will show that problem1.1admits the smallest positive solution and the largest negative solution. To this end, we need to strengthen the hypotheses onjz, x.

Hj₂:j :Z×R → Ris a function such thatjz,0 0 a.e. onZ, ∂jz,0 {0}a.e. onZ; it satisfies hypothesesHj₁i→ivandviand

vthere existsη∈L∞Zsuch that

λ1<lim inf

x→0

u

|x|p−2_x ≤lim sup

x→0

u

|x|p−2_x ≤ηz 4.13

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Remark 4.3. Note that in the new hypotheses, we have strengthened the condition concerning the behavior of ∂jz,·near the origin. This has as a consequence that we can replace the origin as a lower solution in the positive axis and as an upper solution in the negative axis

seeSection 3, by functions which are strictly positive and strictly negative, respectively. This is done in the next lemma.

Lemma 4.4. If hypothesesHj₂hold, then problem1.1has a strict lower solutionx∈intC1 0Z

and a strict upper solutionv∈ −intC1 0Z.

Proof. By virtue of hypothesisHj₂v, we can findc > λ 1andδ >0 such that for almost all

z∈Z,allx∈0, δ, and allu∈∂jz, x,we have

cxp−1≤u. 4.14

We know that foru1 the principal eigenfunction of−Δp, W₀1,pZ i.e.,m ≡ 1, we

have thatu1∈intC10Z.Thus we can findμ∈0,1small enough such that 0< μu1z≤δ

for all z ∈ Z. Letx ∈ intC1

0Z be the strict upper solution for problem1.1obtained in

Proposition 3.4. InvokingLemma 3.5, we can findt >1 such thattx−μu1∈intC₀1Z.We set

x μ

tu1∈intC

1

0Z. 4.15

Note that sincet >1, xz∈0, δfor allz∈Z.Hence

−divDxzp−2Dxz λ1|xz|p−2xz

<c|x z|p−2xz ≤uz a.e.onZ

4.16

for everyu∈Lp_Z_with_u_z_∈_∂j_{z, x}_z_{a.e. on}_Z_see_4.14_{. Hence for all}_ψ_∈_W1,p_Z

, we have

Z

_Dxp−2

Dx, Dψ_RNdz <

Z uψ dz,

⇒x∈intC1

0Z is a strict lower solution for problem1.1.

4.17

Note that from the definition ofx,we havex−x∈intC1₀Z.

A similar reasoning applied on the negative semiaxis produces an upper solutionv

μ/t−u1for some 0 ≤ μ <1 < t.Thenv∈ −intC₀1Z and as forx,we will havev−v ∈

intC1 0Z.

Using{x, x}and{v, v},we introduce the following order intervals inW₀1,pZ:

x, x x∈W₀1,pZ:xz≤xz≤xza.e.onZ,

v, v v∈W₀1,pZ:vz≤vz≤vz a.e.onZ.

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In the next proposition, we establish the existence of the smallest solution of1.1in

x, xand of the greatest solution of1.1inv, v.

Proposition 4.5. If hypothesesHj₂ hold, then problem1.1admits the smallest solution in the order intervalx, xand the greatest solution in the order intervalv, v.

Proof. We will show that the existence of the smallest solution inx, xand the proof of the greatest solution inv, vis similar.

LetS be the set of solutions of1.1belonging in the order intervalE x, x.We will show thatSis downward directed. So letx1, x2∈S.In particular bothx1, x2are upper

solutions for problem1.1. ThenLemma 4.1implies thatx min{x1, x2} ∈ W01,pZis also

an upper solution for problem1.1. We set

E x,x x∈W01,pZ:xz≤xz≤xz

. 4.19

Using standard truncation and penalization techniques, we can obtain x0 ∈ E a solution of 1.1 see Carl-Heikkil¨a 33 and Gasi ´nski -Papageorgiou 19. Nonlinear regularity theory implies thatx0∈C₀1Zand we have

x≤x0≤min

x1, x2

,

⇒S is downward directed.

4.20

Consider a chainΓ⊆Si.e., a totally ordered subset ofS. By virtue of34, Corollary 7, page 336by Dunford-Schwartz, we can find{xn}_n_≥₁⊆Γsuch that

inf

n≥1xninfΓ. 4.21

Because of4.20, we may assume that{xn}_n_≥₁is decreasing. Also since thexn’s are solutions of1.1inE, we see that there existsc3>0 such that

_Dxn

p≤c3 ∀n≥1. 4.22

Therefore{xn}_n_≥₁⊆W₀1,pZis bounded and so we may assume that

xn−→w y inW₀1,pZ, xn−→yinLpZ asn−→ ∞. 4.23

We can findun∈Lp_Z_with_un_z_∈_∂j_{z, xn}_z_{a.e. on}_Z_{such that}

Axnun ∀n≥1. 4.24

Because of hypothesisHj₂iii,{un}n≥1⊆Lp

Zis bounded. So we may assume that

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we haveuz∈ ∂jz,yza.e. onZrecall that the multifunctionx → ∂jz, xhas closed graph; see Clarke1, page 29. From4.24, we have

Axn, xn−y

Z

unxn−ydz−→0 asn−→ ∞. 4.25

SinceAis maximal monotone, we havesee Gasi ´nski -Papageorgiou19, page 84

Axn, xn−→Ay,y,

⇒Dxn_p−→Dy_p. 4.26

Recalling thatDxn →w DyinLp_Z,_RN_,_{we infer that}_Dxn _→ _D_y_in_Lp_Z,_RN

Kadec-Klee propertyand so we conclude thatxn → yinW₀1,pZasn → ∞.So, if we pass to the limit asn → ∞in4.24and sinceAis demicontinuous, it follows that

Ay u 4.27

withu∈Lp_Z_, _u_z_∈_∂j_z,_y_z_{a.e. on}_Z._Therefore_y_∈_S

andyinfΓ.BecauseΓwas an arbitrary chain, invoking Zorn’s lemma, we obtainx∗∈Sa minimal element. Then from

4.20, we conclude thatx∗is the smallest solution of1.1inE.

We will use this proposition to produce the smallest positive and the greatest negative solutions for problem1.1.

Proposition 4.6. If hypothesesHj₂hold, then problem1.1has the smallest nontrivial solution y∈intC10Zand the greatest nontrivial negative solutiony−∈ −intC10Z.

Proof. Letx_n εnu1 withεn ↓ 0 and letEn xn, x.FromProposition 4.5, we know that

problem1.1admits the smallest solutionxn

∗ in the order intervalEn.We know that{xn∗}n≥1⊆

W₀1,pZis bounded and so we may assume that

xn∗ −→w y inW01,pZ, x∗n−→y inLpZ asn−→ ∞. 4.28

We know that

Axn∗un∗ ∀n≥1, 4.29

withun

∗ ∈ LpZ, un∗z ∈ ∂jz, x∗nza.e. onZ. Taking duality brackets withxn∗ −y and arguing as before , we can check that

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Suppose thaty0.Thenx∗n → 0 asn → ∞.We setwnxn∗/xn∗, n≥1.Then by passing to a suitable subsequence if necessary, we may assume that

wn−→w w inW₀1,pZ, wn−→w inLpZn−→ ∞. 4.31

From4.29, we have

Awn u n ∗ xn

∗p−1 ∀n≥1 4.32

and so taking duality brackets withwn−w, again we obtain

lim

n→ ∞

Awn, wn−w0 4.33

from which it follows that

wn−→w inW₀1,pZ 4.34

and so we havew1_{, w /}0.By virtue of hypothesisHj₂v, we can findδ >0 such that for almost allz∈Z,all 0<|x| ≤δ, and allu∈∂jz, x,we have

β≤ u

|x|p−2_x ≤ηz 1 withβ > λ1. 4.35

In addition, due to hypothesis Hj₂iii, for almost all z ∈ Z, all |x| ≥ δ, and all

u∈∂jz, x,we have

|u| ≤αz c|x|p−1≤

αz δp−1 c

|x|p−1_. _4.36

So finally from4.35and4.36, we infer that

|u| ≤c3|x|p−1 for a.a. z∈Z, allx∈R, and all u∈∂jz, x. 4.37

From4.37it follows that

un ∗ xn

∗p−1

n≥1

⊆Lp_Z_{is bounded}_. _4.38

Therefore we may assume that

hn u n ∗ xn

∗p−1 w

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Forε >0 andn≥1 we introduce the sets

Σn

z∈Z:xnz>0, β−ε≤ u n ∗z xn

∗zp−1

≤ηz ε

,

Σn −

z∈Z:xnz<0, β−ε≤ u n ∗z xn∗zp−1

≤ηz ε

.

4.40

Sincexn

∗ → 0 inW01,pZ,we may assumeat least for a subsequencethatxn∗z → 0

a.e. onZ.Thenxn

∗z → 0a.e. on{w >0}andxn∗z → 0−a.e. on{w <0}and so because of hypothesisHj₂v, we have

χΣn

z−→1 a.e.on{w >0}, χΣ−nz−→1 a.e.on{w <0}. 4.41

Then we have

χΣn

un

∗

_xn ∗p−1

w

−→h inLp_{{w >}₀_}_, _χ

Σn −

un ∗

_xn ∗p−1

w

−→h inLp_{{w <}₀_}_. _4.42

From the definition of the setΣn

,we have

χΣn

zβ−εwnz p−1_≤_χ

Σn

z un

∗z x∗nzp−1

wnzp−1χΣn

zhnz

≤χΣn

zηz εwnz p−1 _a

.e.onZ.

4.43

Taking weak limits inLp_{{w >}₀_}_,_{via Mazur’s lemma, and since}_{ε >}_{0 was arbitrary,}

we obtain

βwzp−1≤hz≤ηzwzp−1 a.e.on{w >0}. 4.44

Similarly working onΣn₋,we obtain

ηz|wz|p−2wz≤hz≤β|wz|p−2wz a.e.on{w <0}. 4.45

Moreover, from4.36we see that

hz 0 a.e.on{w0}. 4.46

So from4.44,4.45, and4.46it follows that

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withξ1∈L∞Zandλ1< ξ1z≤ηza.e. onZ.Therefore, if we pass to the limit asn → ∞

in4.32, we obtain

Aw ξ1|w|p−2w,

⇒

−divDwzp−2Dwzξ1z|wz|p−2wza.e.onZ, w|∂Z 0 _{w /}0

.

4.48

Note thatλ1ξ1 < λλ1 1.So from4.48, it follows thatwmust change sign. But

wnxn

∗/x∗n ≥0 for alln≥1 and sow≥0,a contradiction. This proves that we cannot have y0,hencey/0 and of coursey≥0.Moreover, as before we can check that

xn_∗ −→y inW01,pZ, Ay u see4.29, 4.49

withu∈LpZ, uz∈∂jz, yza.e. onZ.It follows that

−divDyzp−2Dyzuz a.e.onZ,

y|∂Z 0.

4.50

From nonlinear regularity theory we have y ∈ C01Z, y/0. Moreover, from

hypothesisHj₂v the sign condition, we haveuz≥ 0 a.e. onZ. So via the nonlinear strong maximum principle of V´azquez23, we obtain thaty∈intC10Z.

We claim thaty is the smallest nontrivial positive solution of1.1. Indeed letybe another nontrivial positive solution of1.1and assume thaty ≤x.As above we can verify thaty∈intC1

0Z.UsingLemma 3.5, we can findε > 0 such thatεu 1≤y. Then forn≥1 large

we haveεnu1 ≤εu 1≤y≤x.So forn≥1 large, working on the order intervalεnu1,y,we can

obtain a solutiony0of1.1in the interval. Thenxn∗ ≤y0forn≥1 large and soy ≤y0 ≤y.

This proves the claim.

In a similar fashion working on En− v, vn with vn εn−u1, we obtain y− ∈ intC1

0Z,the largest nontrivial negative solution of1.1.

Now we are ready to conclude our plan and produce a nontrivial nodal solution. More precisely, using variational methods andTheorem 2.5, we will obtain a nontrivial solutiony0

of1.1in the order intervaly−, ydistinct fromy− andy.Evidently y0 must be a sign

changingnodalsolution. We will achieve this by exploiting the variational characterization ofλ2provided in2.10. This requires a further strengthening of hypothesesHj₁.Suppose

f :Z×R → Ris a measurable function with the following property, for everyr > 0 there existsαr ∈L∞Zsuch that|fz, x| ≤αrzfor a.a.z∈Zand all|x| ≤r.Then we define

f1z, x lim inf