On Cond itions of the Nonexistence of Solu tions of Non linear Equationsw ith Data from Classes Close to L1

KOVALEVSKY A.A.and N ICOLOSIF.InstituteofApplied M athematicsand M echanics,National Academy ofSciences of Ukraine,R.Luxemburg St.74,83114Donetsk,Ukraine.

2DepartmentofM athematicsand Informatics,University ofCatania,VialeA.Doria 6,95125Catania,Italy.

On Cond itions of the Nonexistence of Solu tions of Non linear Equationsw ith Data from Classes Close to L1

KOVALEVSKY A.A.1,?and N ICOLOSIF.21InstituteofApplied M athematicsand M echanics,National Academy ofSciences of Ukraine,R.Luxemburg St.74,83114Donetsk,Ukraine.

2DepartmentofM athematicsand Informatics,University ofCatania,VialeA.Doria 6,95125Catania,Italy.

Received 14 June 2012;A ccep ted 2 January 2013

.We establish cond itionsof the nonexistence ofw eak solutionsof the Dirichlet p roblem for non linear ellip tic equations of arbitrary even order w ith som e righthand sides from Lmw here m＞1.The cond itions include the requirem entof a certain closeness of the param eter m to 1.

Non linear ellip tic equations in d ivergence form;Dirich let p roblem;w eak solu tion; existence and nonexistence ofw eak solutions.

1 In troduction

In the w ell-know n w ork[1],a theory of entropy solu tions for non linear ellip tic secondorder equationsw ith L1-dataw as developed.Accord ing to the resu lts of thisw ork,if?is a bounded open set of Rn(n≥2),1＜p＜n,and coefficients of the equations under consideration grow w ith respect to the grad ient of unknow n function u as|?u|p-1and satisfy natu ral coercivity and strictm onotonicity cond itions,then the Dirich let p roblem in?for these equations has a unique entropy solution for every L1-right-hand side.In add ition,if p＞2-1/n,the en tropy solu tion is aw eak solu tion.A t the sam e tim e in[1]it w as show n that if 1＜p≤2-1/n,the Dirichlet p roblem for the equation-Δpu+u=f in?doesnothavew eak solu tions for som e f∈L1(?).

In connection w ith the above,now w e note the follow ing tw o casesw here the Dirichlet p roblem for equations of the class under consideration has aw eak solu tion for every right-hand side in Lm(?)(see in[2,Theorem s 1.5.5 and 1.5.6]):

(a)p≥2-1/n,m＞1;

(b)p＜2-1/n,m≥n/(np-n+1).

We note that a cond ition of the nonexistence ofw eak solutions of the Dirichlet p roblem for high-order equationsw ith L1-dataw as established in the recentarticle[3].

As far as the solvability of non linear ellip tic high-order equationsw ith L1-right-hand sides is concerned,to ou r know ledge,there are no resu lts on this subject in the general case.Som e resu lts on the existence of entropy and w eak solu tions of the Dirich let p roblem for non linear ellip tic high-orderequationsw ith coefficientssatisfying a strengthened coercivity cond ition and L1-dataw ere obtained for instance in[4]and[5].In this connection see also[2,Chap ter 2]w here a theory of the existence and p roperties of entropy and w eak solu tions of the Dirich let p rob lem for non linear fou rth-order equations w ith strengthened coercivity and data from L1and classes close to L1is p resented.

2 Auxiliary assertions

Let n∈N,n≥2,and let?be a bounded open set of Rn.

Hence,taking in to accoun t the continuity of the functionals H f,Hg and H(αf+βg)along w ith the strong convergence of{?k}to?in Wl,p(?),w e deduce the equality〈H(αf+ βg),?〉=〈αH f+βHg,?〉.Then,due to the arbitrariness of?,w e get H(αf+βg)=αH f+ βHg.Therefore,the operator H is linear.

Now,w e pass to the imm ed iate p roofof the conclusion of the p roposition.

Ow ing to the linearity of the operator H,the functional F is linear.

and for every k∈N define the functional Fk:Lm(?)→R by

Due to the linearity of theoperator H,forevery k∈N the functional Fkis linear.M oreover, using(2.1),w e establish that if k∈N and f,g∈Lm(?),then

This im p lies that for every k∈N the functional Fkis con tinuous on Lm(?).Finally,it is obvious that forevery function f∈Lm(?)the sequenceof thenum bers〈Fk,f〉isbounded. Thegiven p ropertiesof the functionals Fkand the theorem on uniform boundedness(see, for instance[6,Charter 2])allow us to conclude that thereexists M＞0 such that for every k∈N and for every f∈Lm(?),

Hence,using the definition of the functionals Fkalong w ith(2.2),w e in fer that

Therefore,the functional F is con tinuous.

Thus,F∈(Lm(?))?.Then there existsa functionψ∈Lm/(m-1)(?)such that for every f∈Lm(?),

Thisand the definition of the functional F im p ly that

Letusshow that?=ψa.e.in?.Indeed,let f∈Lm(?)∩Lp/(p-1)(?).Since f∈Lm(?), by(2.1),for every k∈N w e have

M oreover,taking into account that f∈Lp/(p-1)(?)and using H¨older’s inequality,for every k∈N w e get

From(2.4),(2.5)and(2.2)it follow s that

On the other hand,by virtue of(2.2)and the continuity of the functional H f,w e obtain

From(2.3),(2.6)and(2.7)w e derive that

Hence,ow ing to the arbitrariness of the function f in Lm(?)∩Lp/(p-1)(?),w e get that ?=ψa.e.in?.Then,sinceψ∈Lm/(m-1)(?),w e have?∈Lm/(m-1)(?).

and

Next,let B be a closed ball in Rnw ith center y such that B??.We fix a function ?∈C∞0(?)such that0≤?≤1 in?and?=1 in B.

Now w e setλ=n/t and w=vλ?.We have

In fact,since t＞p,thenλp＜n.This along w ith the obvious estim ate wp≤vλpin?and(2.9)im p lies that w∈Lp(?).Clearly,wt=vnin B,and,by(2.10),v/∈Ln(?).Then w/∈Lt(?).Thus,inclusion(2.11)holds.

To this pu rpose for every j∈N w e define the function wj:?→R by

M oreover,wj→w in?{y}and for every j∈N,wj≤w in?{y}.Therefore,taking into account the inclusion w∈Lp(?),w e get

Using Leibniz’form u la of d ifferentiation of the p roduct of tw o functions,w e establish that there exists C＞0 such that for every j∈N and for every n-d im ensionalm u lti-index α,|α|≤l,

be an operator such that Z

Then

Proof.Suppose that inequality(2.14)is not valid.Then m/(m-1)＞np/(n-lp).Hence, by Proposition 2.2,

How ever,this contrad icts inequality(2.15).The contrad iction obtained p roves that inequality(2.14)is valid.

3 Existence and nonexistence of solu tions of second-order equations

Let 1＜p＜n,c1,c2＞0,g∈Lp/(p-1)(?),g≥0 in?,and let for every i∈{1,···,n},ai:?×Rn→R be a Carath′eodory function.We shall suppose that for alm ost every x∈?and for everyξ∈Rn,

M oreover,w e shallassum e that for alm ostevery x∈?and for everyξ,ξ′∈Rn,ξ6=ξ′,

For every f∈L1(?)by(Pf)w e denote the follow ing p roblem:

Definition3.1.

Let f∈L

1

(?).A weak solution ofproblem(P

f

)isafunction

such that:

(i)for every i∈{1,···,n},ai(x,?u)∈L1(?);

Letus recallsom e know n resu lts on the solvability of p roblem(Pf)in the casew here f∈Lm(?)w ith m＞1.

For everyλ∈[1,n)w e setλ?=nλ/(n-λ).

Now consider the casew here f∈Lm(?)w ith m lying in the interval(1,p?/(p?-1)).

This resu lt w as p roved in[8].In this connection w e observe that actually the conclusion of Theorem 3.1 holds if in the cond itions of the theorem w e assum e that the inequality p≥2-1/n is satisfied instead of the inequality p＞2-1/n(see[2,Theorem 1.5.5].

Theorem3.2.Let p＜2-1/n,n/(np-n+1)≤m＜p?/(p?-1),and let f∈Lm(?).Then

This resu ltw as established by the firstau thor in[2,Theorem 1.5.6].The sam e conclusion as in the given theorem under the cond itions p≤2-1/n and n/(np-n+1)＜m＜p?/(p?-1)has already been obtained in[9].

Them ain resu lt of this section given in the follow ing theorem show s that the cond ition on m in Theorem 3.2 cannotbew eakened.

Theorem3.3.Let p＜2-1/n,and let

Then there exists f∈Lm(?)such thatproblem(Pf)doesnothaveweak solutions.

Proof.Let us suppose that for every f∈Lm(?)there exists a w eak solu tion of p roblem

Then,app lying Proposition 2.3,w e get the inequality

Hence,by thedefinition of p1,w eobtain that m≥n/(np-n+1).How ever,thiscontrad icts (3.4).The con trad iction obtained p roves that the conclusion of the theorem is valid.

4 Nonexistence of solu tions of h igh-order equations

For every f∈L1(?)by(Pf)we denote the follow ing p roblem:

(i)for everyα∈Λ,Aα(x,?lu)∈L1(?);

Theorem4.1.Let p＜2-l/n,and let

Then there exists f∈Lm(?)such that problem(Pf)doesnothaveweak solutions.

Then,app lying Proposition 2.3,w e get the inequality

Hence,taking in to account the definition of p1,w e obtain that m≥n/(np-n+l).However,this contrad icts(4.2).The contrad iction obtained p roves that the conclusion of the theorem is valid.

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[5]Kovalevsky A.and N icolosi F.,Solvability of Dirichlet p roblem for a class of degenerate nonlinear high-order equationsw ith L1-data.NonlinearAnal.,47(2001),435-446.

[6]Yosida K.,Functional Analysis.Sp ringer-Verlag,Berlin,1965.

[7]Lions J.-L.,Quelques M′ethodes de R′esolu tion des Probl`em es aux Lim ites Non Lin′eaires. Dunod,Gau thier-Villars,Paris,1969.

[8]Boccardo L.and Gallou¨et T.,Nonlinear ellip tic equations w ith right hand sidem easures. Comm.PartialDifferential Equations,17(1992),641-655.

[9]KovalevskiiA.A.,Integrability of solutions of nonlinear ellip tic equationsw ith right-hand sides from classes close to L1.M ath.Notes,70(2001),337-346.

10.4208/jpde.v26.n1.4 M arch 2013

?Correspond ing au thor.Email addresses:alexkvl@iamm.ac.donetsk.ua(A.A.Kovalevsky),fnicolosi@ dmi.unict.i t(F.N icolosi)

AMSSubjectClassifications:35J25,35J40,35J60

ChineseLibraryClassifications:O175.8,O175.25,O175.29