Positive Solu tions of Non linear Ellip tic Prob lem in a Non-Sm ooth Planar Dom ain

BEN BOUBAKERM oham ed Am ine

Institut Preparatoire aux Etudes D’Ing′enieur De Nabeul University of Carthage CampusUniversitaireM erezka,8000,Nabeul,Tunisia.

Positive Solu tions of Non linear Ellip tic Prob lem in a Non-Sm ooth Planar Dom ain

BEN BOUBAKERM oham ed Am ine?

Institut Preparatoire aux Etudes D’Ing′enieur De Nabeul University of Carthage CampusUniversitaireM erezka,8000,Nabeul,Tunisia.

Received 2Decem ber 2013;A ccep ted 28 June 2014

. Wef indandprove3Ginequal i t ies for theLaplacianGreenfunct ionwi th the D irich let boundary cond ition,w hich are app lied to show the existence of positive continuous solutions of the non linear equation w here V and g are Borelm easurable functions,required to satisfy suitable assum ptions related to a new functional class J.Our app roach uses the Schauder fixed point theorem.

Greenfunct ion;Schauder f ixedpoint theorem;superharmonicfunct ion.

1 In troduction

In this paper,w e w ork in the Euclidean space R2.By Sα,w e denote the dom ain defined by

The objective is to study the existence of con tinuous solu tions for the non linear ellip tic p roblem

w here V and g are Borelm easu rable functions,required to satisfy suitable assum p tions related to a new functional class J.The existence resu lts of p roblem(1.1)have been extensively stud ied w hen V=0 and the special non linearity g(x,t)=p(x)q(t),for both bounded and unbounded dom ain D in(n≥1),w ith sm ooth com pact boundary(see for exam p le[1–4]).On the other hand,in[5–7],the au thors considered the p roblem(1.1) w here there isno restrictionson the sign of g,and D isan unbounded dom ain in(n≥1) w ith a com pact Lipschitz boundary.Then,they p roved the existence of infinitely m any solu tionsp rovided that g is in a certain Kato class.Nam ely,they show ed that there exists a num ber b0＞0 such that for each b∈(0,b0],there existsa positive continuous solution u insatisfying

Theorem1.1.(3G-Theorem)There existsa constant C1=C1(α)＞0 such that forall x,y and z in Sα,wehave

whereδ(x)is the Euclidean distance from x to?Sα.

Thisenablesus to define and study,in Section 4 a new Kato class J of functionson Sα. Let

Definition1.1.Wesay that a Borelmeasurable function q on Sαbelongs to the Kato class Jifq satisfies thefollow ing conditions:

Note that ifα=1,then J is the Kato class K in troduced and stud ied in[8].For a Borel m easurable function q on Sα,let

w here hα(x)=|x|αcos(αarg x),for all x∈Sα.To study the p roblem(1.1)in section 5,w e assum e the follow ing cond ition on V and g:

(H1)The function g is a Borelm easu rable on Sα×(0,∞),continuousw ith respect to the second variable and satisfies

Ourm ain resu lt is the follow ing:

Theorem1.2.Assume(H1)-(H3).Then the problem(1.1)has infinitelymany continuous solutions.M oreprecisely,there existsε0＞0 such thatforeachε∈(0,ε0],thereexistsa solution u of

(1.1),continuouson Sαand satisfying

The follow ing notionsw illbe adop ted:

i)Let f and k be tw o positive functionson a set?.We say that f is com parable to k on?and w e denote fk,if there exists C＞1,such that?x∈?.

iii)B(x,r)denotes the Euclidean open ballof center x and rad ius r.

Throughout this paper,C denotes a generic positive constantw hichm ay vary from line to line and?cdenotes the com p lem entary of?in.

2 Inequalities for the G reen function

Proposition2.1.([10])There existsa constant Cα＞0,such that forall x and y in Sα,wehave

Theorem2.1.([10])Letα∈]0,+∞[{1}.Then,forall x,y∈Sα

Lemma2.1.ThereexistsC=C(α)＞1,such thatforallx,y∈Sα,wehave thefollow ing properties:

Proof.i)Assum e|x|α-1δ(x)|y|α-1δ(y)≤|x-y|2(|x|∨|y|)2(α-1).Let Cαbe the constant defined in Proposition2.1 and x,y∈Sα.Let

Since|x|α-1δ(x)|y|α-1δ(y)≤|x-y|2(|x|∨|y|)2(α-1),it follow s,by Proposition2.1,that

Thus,w e deduce that

Using Proposition 2.1,w e get

By interchanging the role of x and y,w e obtain(i).

ii)Assum e|x-y|2(|x|∨|y|)2(α-1)≤|x|α-1δ(x)|y|α-1δ(y),then,by Proposition 2.1,w e get

In consequence,w e obtain

w hich gives

So,by using Proposition 2.1,w e deduce that

The p roof is com p lete.

Proposition2.2.Thereexists C=C(α)＞0 depending only on Sα,such that forall x,y in Sα

there exists C＞0 such that

Since|xα-yα|?|x-y|(|x|∨|y|)α-1andδ(x)≤|x|,there exists C＞0 such that

ii)The left hand side inequality in(2.7)follow s from(2.8).We p rove the righ t hand side inequality.There are tw o cases to d iscuss:

If|x|α-1δ(x)|y|α-1δ(y)≤|x-y|2(|x|∨|y|)2(α-1),then it follow s by Lemm a 2.1 that

If|x|α-1δ(x)|y|α-1δ(y)≥|x-y|2(|x|∨|y|)2(α-1),then it follow s by Lemm a 2.1 that

The p roof is com p lete.

ProofofTheorem1.1.Since the function z7-→zαis a con form alm app ing from Sαinto the half p lane S1.It follow s from 3G inequality p roved in[8]that,for all x,y and z in Sα

將需要反向計(jì)分的條目進(jìn)行轉(zhuǎn)換后，所有被試的數(shù)據(jù)根據(jù)量表總得分由高到低進(jìn)行排列，以總得分前27%的被試作為高分組，后27%的被試作為低分組，兩組被試每一條目得分的平均數(shù)差異均顯著，ps<0.01。被試每一條目得分與量表總得分之間均呈顯著正相關(guān)，相關(guān)系數(shù)在0.35～0.87之間。

Thus the resu lt follow s from Proposition 2.1.

3 Kato class

Proposition3.1.Let R＞0,then forall x,y∈Sα∩B(0,R),wehave

Proof.We can suppose that|x|≥|y|.Assum e12＜α＜1,then

Assum eα＞1,then

The p roof is com p lete.

Proposition3.2.Letq bea Borelmeasurablefunction on Sαsuch thatq∈J,then foreach R＞0,

Proof.By(2.7)w e have

This im p lies by Proposition 2.1,that for all x,y∈Sα

Since q∈J,then by(1.3)there exists r＞0 such that

The p roof is com p lete.

Proposition3.3.Ifq∈J,then k q k＜+∞.

Proof.Using Proposition 2.1,w e get

Let r＞0 and M＞0.Then,w e have

By(2.3)and Proposition 3.1,w e obtain

The resu lt follow s from(1.3),(1.4)and Proposition 3.2.

Proposition3.4.Let h beapositive superharmonic function in Sαand q in J.Then

Proof.Let h bea positive superharm onic function on Sα.By[11,Theorem 2.1],thereexists a sequence(fn)n∈Nof positivem easu rable function on Sαsuch that

Hence,w e need on ly to verify(3.3)-(3.5)for h(y)=Gα(y,z)uniform ly on Sα.

1.Let r＞0.Then by Theorem 1.1,w e have

Letηand M＞0.By using(2.6),w e get

We obtain(3.3)by using(1.3),(1.4)and Proposition 3.2.

On the other hand,

2.(3.5)follow s imm ed iately by using Theorem 1.1.

Corollary3.1.Letq bea Borelmeasurablefunction in J.Then,

Proof.By using(3.5)w ith h=1 and Proposition 3.3,w e obtain(3.7).Let x0∈Sα.By(2.6) and(3.7)w e get

Finally,(3.9)follow s from(3.8).

Proposition3.5.Letη,μ∈R such thatη＜2＜μandα∈]12,1[.Then thefunctionσdefined on Sαby

is in J.

Proof.Letη,μ∈R such thatη＜2＜μand r＞0.Then w e have

Let us recall that ifα∈]12,1[,then|x-y|(|x|∨|y|)α-1≤21-α|x-y|α.This im p lies by using Lemm a 2.1,that

On the other hand,by Lemm a2.1 there exists C＞1 such that for all y∈?2,xw e have

Since ln(1+t2)≤C tγ,?t≥0,?γ∈](0∨η),2[,then

Now let M＞0,

and show ing that limM-→+∞IM=0.Letε＞0,then there exists r＞0 such that

Then

w here r0=21-αrα.

Fixing this r0＞0 and letting M＞1,w e get

Since the function z7-→zαis a conform alm apping,by using(2.2)w e obtain

The p roof is com p lete.

Lemma3.1.Let hα(x)=|x|αcos(αarg x),x∈Sα.Let M＞0 and r＞0.Then there exists a constant C＞0,such that forall x,y∈Sαsatisfying|x-y|≥r and|y|≤M

4 Proof of Theorem 1.2

Let C(Sα)denotes the space of allbounded con tinuous functions in Sαendow ed w ith the uniform norm k k∞and let

w here Cαis the constant defined in Proposition 2.1.For v∈C1(Sα),w e define

In the follow ing,for sim p licity,w ew rite q(y)=|V(y)|+ψ(y).

Proof.Let x0∈and r＞0.Let M＞|x0|+r,and x,It follow s from(H1) and v≤1/Cαthat

Hence Lv is continuous in Sαuniform ly for v∈C1(Sα).

We next consider x0=∞.Let M＞0 and x∈Sαsuch that|x|≥M+1,then

By(1.4),the second term of the righ t hand side is bounded byε,w henever M is su fficiently large.Note that from Proposition 3.1 and Lemm a 3.1,there exists C＞0 such that

It follow s from Proposition 3.2 and Lebesgue’s convergence theorem that

M oreover,Lε(Gε)isrelatively compact in C(Sα∪{∞}).

Let0＜λ＜1 and define

so it follow s from(H1)that for x∈Sα,w e have

Lemma4.3.Let0＜ε＜ε0.Then Lεis continuouson Gε.

Proof.Let(vk)kbe a sequence in Gεw hich convergesuniform ly to v∈Gε.It follow s from (H1)-(H3)and Lebesgue’s theorem that

Since Lε(Gε)is a relatively com pact fam ily in C(Sα∪{∞}),the pointw ise convergence im p lies the uniform convergence.Therefore k Lεvk-Lεv k∞-→0 as k-→+∞.Thus Lεis continuouson Gε.

ProofofTheorem1.2.Let 0＜ε≤ε0,w hereε0is the positive constant defined in Lemm a

Let u(x)=v(x)hα(x).Then u＞0 in Sα,and u=0 on?Sαsince hαsatisfies these cond itions. A lso,w e have for x∈Sα,

The p roof is com p lete.

Example4.1.Letν＞0,α∈]12,1[andη＜2＜μ.Let h be a Borelm easu rable function on Sαsuch that

Then,there existsε0＞0 such that for eachε∈(0,ε0],the p roblem

has a solution w continuouson Sαand satisfying

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10.4208/jpde.v27.n3.2 Sep tem ber 2014

?Correspond ing au thor.Emailaddress:MohamedAmine.BenBoubaker@ipein.rnu.tn(M.A.Ben Boubaker)

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