简介:<正>WeareconcernedwiththefollowingDirichletproblem:-△u(x)=f(x,u),x∈Ω.u∈H01(Ω).(P)wheref(x,t)∈C(Ω×R),f(x,t)/tisnondecreasingint∈RandtendstoanL∝-functionq(x)uniformlyinx∈Ωast→+∝(i.e.,f(x,t)isasymptoticallylinearintatinfinity).Inthiscase.anAmbrosetti-Rabinowitz-typecondition,thatis.forsomeθ>2.M>0,0<θF(x.s)≤f(x,s)s,forall|s|≥Mandx∈Ω,(AR)isnolongertrue,whereF(x,s)=integralfromn=0tosf(x,t)dt.Asiswellknown,(AR)isanimportanttechnicalconditioninapplyingMountainPassTheorem.Inthispaper,withoutassuming(AR)weprove,byusingavariantversionofMountainPassTheorem,thatproblem(P)hasapositivesolutionundersuitable,conditionsonf(x,t)andq(x).Ourmethodsalsoworkforthecasewheref(x,f)issuperlinearintatinfinity.i.e.,q(x)≡∞.
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简介:Ekeland’svariationalprincipleisafundamentaltheoreminnonconvesanalysis.Itsgeneralstatementisasthefollowing:Ekeland’sVariationalPrinciple'’a:.LetVbeacompletemetricspace,andF:F—*-RU{+°°}alowersemicontinuousfunction,notidentically+00andboundedfrom,below.Lets>0begiven,andapointu^VsuchthatF(u)
简介:Anewanalyticalmodelwasdevelopedtopredictthegravitywavedrag(GWD)inducedbyanisolated3-dimensionalmountain,overwhichastratified,non-rotatingnon-Boussinesqshearedflowisimpinged.Themodelisconfinedtosmallamplitudemotionandassumestheambientvelocityvaryingslowlywithheight.ThemodifiedTaylor-GoldsteinequationwithvariablecoefficientsissolvedwithaWentzel-Kramers-Brillouin(WKB)approximation,formallyvalidathighRichardsonnumbers.WiththisWKBsolution,genericformulaeofsecondorderaccuracy,fortheGWDandsurfacepressureperturbation(bothforhydrostaticandnon-hydrostaticflow)arepresented,enablingarigoroustreatmentontheeffectsbyverticalvariationsinwindprofiles.Inanidealtesttothecircularbell-shapedmountain,itwasfoundthatwhenthewindislinearlysheared,thattheGWDdecreasesastheRichardsonnumberdecreases.However,theGWDforaforwardshearedwind(windincreaseswithheight)decreasesalwaysfasterthanthatforthebackwardshearedwind(winddeceaseswithheight).Thisdifferenceisevidentwheneverthemodelishydrostaticornot.