Quasilinear and Hessian equations of Lane-Emden type

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1、QUASILINEARANDHESSIANEQUATIONSOFLANE–EMDENTYPENGUYENCONGPHUCANDIGORE.VERBITSKY∗Abstract.Theexistenceproblemissolved,andglobalpoint-wiseestimatesofsolutionsareobtainedforquasilinearandHessianequationsofLane–Emdentype,includingthefollowingtwomodelproblems:qq−∆pu=u+µ,Fk[−u]=u+µ,u≥0,onRn,oronabounded

2、domainΩ⊂Rn.Here∆isthep-pLaplaciandefinedby∆u=div(∇u

3、∇u

4、p−2),andF[u]isthepkk-Hessiandefinedasthesumofk×kprincipalminorsoftheHes-sianmatrixD2u(k=1,2,...,n);µisanonnegativemeasurablefunction(ormeasure)onΩ.Thesolvabilityoftheseclassesofequationsintherenormalized(entropy)orviscositysensehasbeenanopenpro

5、blemevenforgooddataµ∈Ls(Ω),s>1.Suchresultsarededucedfromourn(q−p+1)existencecriteriawiththesharpexponentss=forthepqn(q−k)firstequation,ands=forthesecondone.Furthermore,a2kqcompletecharacterizationofremovablesingularitiesisgiven.OurmethodsarebasedonsystematicuseofWolff’spotentials,dyadicmodels,andno

6、nlineartraceinequalities.WemakeuseofrecentadvancesinpotentialtheoryandPDEduetoKilpel¨ainenandMal´y,TrudingerandWang,andLabutin.Thisenablesustotreatsingularsolutions,nonlocaloperators,anddistributedsin-gularities,anddevelopthetheorysimultaneouslyforquasilinearequationsandequationsofMonge-Amp`erety

7、pe.arXiv:math/0501483v1[math.AP]26Jan20051.IntroductionWestudyaclassofquasilinearandfullynonlinearequationsandin-equalitieswithnonlinearsourceterms,whichappearinsuchdiversear-easasquasi-regularmappings,non-Newtonianfluids,reaction-diffusionproblems,andstochasticcontrol.Inparticular,thefollowingtwom

8、odelequationsareofsubstantialinterest:(1.1)−∆pu=f(x,u),Fk[−u]=f(x,u),∗SupportedinpartbyNSFGrantDMS-0070623.12NGUYENCONGPHUCANDIGORE.VERBITSKYonRn,oronaboundeddomainΩ⊂Rn,wheref(x,u)isanon-negativefunction,convexandnondecreasinginuforu≥0.Here∆u=div(∇u

9、∇u

10、p−2)isthep-Laplacian(p>1),andF[u]isthepkk-He

11、ssian(k=1,2,...,n)definedbyX(1.2)Fk[u]=λi1···λik,1≤i1<···