Existenceof Non trivialWeak Solu tions to Quasi-Linear Ellip tic Equationsw ith Exponen tial G row th

WANG Chong

DepartmentofM athematics,theGeorgeWashington University,

Washington DC 20052,USA.

Existenceof Non trivialWeak Solu tions to Quasi-Linear Ellip tic Equationsw ith Exponen tial G row th

WANG Chong?

DepartmentofM athematics,theGeorgeWashington University,

Washington DC 20052,USA.

Received 13 June 2012;A ccep ted 1 Decem ber 2012

.In this paper,w e study the existence of nontrivial w eak solu tions to the follow ing qu asi-linear ellip tic equations

AMSSubjectClassifications:35J20,35J60

ChineseLibraryClassifications:O175

Trud inger-M oser inequality;exponentialgrow th.

1 In troduction

Consider non linear ellip tic equationsof the form

w here?isa sm ooth bounded dom ain in Rn,and-△pu=-d iv(|?u|p-2?u).Br′ezis[1], Br′ezis-N irenberg[2]and Bartsh-W illem[3]stud ied this p roblem under the assum p tions that p=2 and|f(x,u)|≤c(|u|+|u|q-1).Garcia-A lonso[4]stud ied this p roblem under the assum p tions that p≤n and p2≤n.W hen?=Rnand p=2,Kryszew ski-Szu lkin[5], A lam a-Li[6],Ding-N i[7]and Jean jean[8]stud ied the follow ing equations in stead of (1.1):

In thispaperw e consider quasi-linear ellip tic equations in thew hole Euclidean space

D.Cao[9]and Cao-Zhang[10]stud ied p roblem(1.2)in the case n=2 andβ=0. Panda[11],do′O etal.[12,13]and A levs-Figueiredo[14]stud ied p roblem(1.2)in general d im ension andβ=0.Whenβ6=0,(1.2)w as stud ied by Ad im urthi-Yang[15],do′O et al.[16],Yang[17],Zhao[18],and others.Sim ilar p roblem s in R4or com p letenoncom pact Riem annianm anifoldsw ere also stud ied by Yang[19,20].

We define a function space

w ith the norm

We say that u∈E is aw eak solu tion of p roblem(1.2)if for all?∈E w e have

Ifaw eak solution u satisfies u(x)≥0 for alm ostevery x∈Rn,w e say u is positive.

Throughou t thispaperw eassum e the follow ing tw o cond itionson the poten tial V(x):

(V1)V(x)≥V0＞0;

We also assum e that the non linearity f(x,s)satisfies the follow ing:

(H1)There exist constan tsα0,b1,b2＞0 such that for all(x,s)∈Rn×R+,

(H2)There existsμ＞n,such that for all x∈Rnand s＞0,

(H3)There exist constan ts R0,M0＞0,such that for all x∈Rnand s＞R0,

F(x,s)≤M0f(x,s);

(H4)

uniform ly w ith respect to x∈Rn,w here

(H5)There exist constan ts p＞n and Cpsuch that

for all s≥0 and all x∈Rn,w here

Ou rm ain resu lt is the follow ing theorem:

Theorem1.1.Assume that V(x)isacontinuousfunction satisfying(V1)and(V2).f:Rn×R→R isa continuous function and the hypotheses(H1)-(H6)hold.Then Eq.(1.2)has a nontrivial positiveweak solution.

Here the assum p tion(H5)is d ifferent from that of[17].(H5)w as also used in[16] and[18].An exam p le of f satisfying(H1)-(H6)reads

w here l≥N is an integer,Cpis as in(H5),χ:[0,∞)→R is a sm ooth function such that 0≤χ≤1,χ≡0 on[0,A],χ≡1 on[2A,∞),and|χ′|≤2/A,w here A is a large constan t,say A＞4n-1.For detailsw e refer the reader to in[20,Proposition 2.9].Other exam p lesw ere also given in[16]and[18]respectively.

2 Com pactness analysis

We w ill give som e p relim inary resu lts before p roving Theorem 1.1.Define a function ζ:N×R→R by

Let s≥0,p≥1 be realnum bers and n≥2 be an integer,then thereholds(see[17])

Problem(1.2)is closely related to a singu lar Trud inger-M oser type inequality[15]. That is,for allα＞0,0≤β＜n,and u∈W1,n(Rn)(n≥2),there holds

In thispaper,w e also need the follow ing resu ltw hich is taken from Lemm a 2.4 in[17]. That is,if V:Rn→R is continuous and(V1),(V2)are satisfied,then for any q≥1,there holds

Define a functional J:E→R by

w hereζ(n,s)is defined by(2.1).Thus J isw ell defined thanks to(2.3).

Lemma2.1.Assume V(x)≥V0in Rn,(H1),(H2)and(H3)hold.Then for any nonnegative, compactly supported function u∈W1,n(Rn){0},thereholds J(tu)→-∞as t→+∞.

Proof.We follow the line of[15].(H2)and(H3)im p ly that there exists R0＞0 such that for all s＞R0,

therefore,

It follow s that

Sinceμ＞n,this im p lies J(tu)→-∞as t→+∞.

Lemma2.2.Assume that V(x)≥V0in Rn,(H1)and(H4)are satisfied.Then there exist δ＞0 and r＞0 such that J(u)≥δforall‖u‖=r.

Proof.Accord ing to(H4),there existτ,δ＞0 such that if|s|≤δ,

Therefore for all x∈Rn,|s|≤δ,w e have

On the other hand,accord ing to(H1),w e can obtain that for any|s|≥δ,

w here

Com bining(2.6)and(2.7),w e have for all(x,s)∈Rn×Rn,

w here C=Cδ.Herew e also use the inequality

w hich is taken from Lemm a 4.2 in[15].Accord ing to the definition ofλβ,w e get

Thanks to(2.8),(2.9),and(2.10),w e obtain

For su fficien tly sm all r＞0,w e have

w hich is due toτ＞0.Therefore,accord ing to(2.11)and(2.12),for all‖u‖=r,

Lemma2.3.Criticalpoints of Jareweak solutionsof(1.2).

Proof.Though the p roof is standard,w e w rite it for com p leteness.Define a function g(t)=J(u+t?),nam ely

By a sim p le calcu lation,

Let f1(t)=|?(u+t?)|n,f2(t)=|u+t?|n,and

Clearly w e have

Com bining the above,w e have for all?∈E,

Therefore,J′(u)·?=0 is equivalent to

Hencew e get the desired resu lt.

Lemma2.4.Assume(H5)issatisfied,then thereexistsafunction up∈Ewhich satisfies‖up‖= Sp,and for t∈[0,+∞),wedefine

Thereholds

Proof.Sim ilar to[18],assum e{uk}isabounded positivesequenceof functions in E w hich satisfies

Mean while we assum e that uk?upin E,uk→upin Lq(Rn)for all q≥1,uk(x)→up(x) alm ost everyw here.Using the Ho¨lder inequality and the M ean Value Theorem,w e can easily p rove that for any ＞k＞K,

Therefore,

Nextw ew ill p rove

Since uk?upw eak ly in E,w e know?uk??upw eak ly in Ln(Rn).Accord ing to the definition ofw eak convergence and the H¨older inequality,w e get

Sim ilarly to the p roof of(2.15),w e know

Thanks to(2.17)and(2.18),(2.16)holds.M eanw hile,by the definition of Sp,w e know Sp≤‖up‖.Therefore,w e know‖up‖=Sp.Accord ing to(H5),w e have

Due to the definition of J(tup)and(2.19),w e have

Let

and by calcu lation w e know for any realnum ber t,

Thism eans

Ifw e set

then w e have

In view of(H5),w e get(2.14)imm ed iately.

Lemma2.5.Assume that(V1),(V2),(H1),(H2)and(H3)hold and{uk}?E bean arbitrary Palais-Smale sequenceof J,i.e.,

in E?as k→∞,where E?denotes the dual space ofE.Then there exists a subsequence of{uk} (still denoted by{uk})and u∈E such that uk?u weakly in E,uk→u strongly in Lq(Rn)for allq≥1,and

Furthermore,u isaweak solution of(1.2).

Proof.Assum e{uk}is a Palais-Sm ale sequence of J.Since J(uk)→c,w e obtain

Accord ing to(2.13),w e know

for all?∈E,w hereτk=‖J′(uk)‖,andτk→0 as k→∞.Taking?=ukin(2.21),w e have

By(H2),w e obtain

A ccord ing to(2.24),it’s easy to p rove that‖uk‖is bounded.Due to(2.20)and(2.22),w e get

w here C is a constantw hich dependson ly onμand n.A ccord ing to(2.5)w e obtain that for som e u∈E and any q≥1,up to a subsequence,uk→u strongly in Lq(Rn).Then w e know uk→u alm osteveryw here in Rn.Nextw ew ill p rove thatup to a subsequence

Due to f(x,·)≥0,it is su fficien t for us to p rove thatup to a subsequence

Due to

w e know

For anyδ＞0,there exists M＞Csuch that

w here C is the constant in(2.25).Accord ing to(2.25),w e know

For all x∈{x∈Rn:|uk|＜M},by ou r assum p tion(H1),w e can deduce that

Since

accord ing to the generalized Lebesgue’sdom inated convergence theorem,w e know

Accord ing to(2.28),(2.29)and(2.31),w e can p rove that(2.27)holds.Therefore w e get (2.26).By(H3)and(H1),w e obtain that

3 Proof of Theorem 1.1

Next w e w ill p rove Theorem 1.1.By Lemm as 2.1 and 2.2,w e know J satisfies all the hypothesesof the M oun tain-pass Theorem w ithou t the Palais-Sm ale cond ition:

Accord ing to the M ountain-pass Theorem excep t for the Palais-Sm ale Cond ition[21], there existsa sequence{uk}?E such that

in E?,w here

A ccord ing to Lemm a 2.5,w e know thatup to a subsequence

Nextw ew ill p rove that the solution u w hich w e get in the above is nontrivial.Suppose u≡0.Due to F(x,u)≡0 for all x∈Rn,w e get

A ccord ing to(2.20),w e have

Com bining(3.1)and(3.2),w e can obtain that

By Lemm a 2.4,w e get

Accord ing to(3.3)and(3.4),w e know there exists som eη0＞0 and K＞0,such that

for all k＞K.A ccord ing to(3.5),w e can choose q＞1 su fficien tly close to 1 such that

for all k＞K.By(H1)and(2.1),w e have

It follow s that

Letting 1/q′+1/q=1,and accord ing to(3.6),(3.7),the H¨older Inequality,(2.2),and(2.4), w e have

Accord ing to(2.22)and(3.8),w e have

This is in con trad iction w ith(3.3).Thus the solu tion u of(1.2)is nontrivial.

Testing Eq.(1.2)w ith u-,the negative part of u,w e conclude that u-≡0.Hence u≥0.

Acknow ledgm en ts

The au thor is supported by the China Scholarship Council.

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10.4208/jpde.v26.n1.3 M arch 2013

?Correspond ing au thor.Emailaddress:chongwang@gwu.edu(C.Wang)