Quasilinear Elliptic Problems with multiple regions of singularities and convexities for the p(x)-laplacian operator

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1 Uiversidade de Brasília Post-Graduatio Program, Mathematics Doctorate i Mathematics Quasiliear Elliptic Problems with multiple regios of sigularities ad covexities for the px-laplacia operator by Thiago Williams Siqueira Ramos Brasília - DF December/207

2 Resumo Nesta tese estabelecemos resultados de existêcia, uicidade, multiplicidade e regularidade de soluções para a seguite classe de problemas quasilieares que podem ser sigulares evolvedo expoetes variáveis px u = cxdx βx u αx + λfx, u i, u > 0 i ; u = 0 o. Na primeira parte, determiamos codições sucietes para existêcia de úica solução em W,px loc quado fx, t é subliear em t = 0 e t = + para todo x. Na seguda parte, obtemos multiplicidade de solução em W,px 0 quado fx, t é superliear em t = + em algum subdomíio de. Além disso, permitimos múltiplas regiões de sigularidades, tato o potecial quato a ão liearidade u > 0, equato que a seguda parte cosideramos β 0. Provamos também um pricípio de Comparação para sub e supersolução em W,px loc para problemas sublieares em t = 0 e em t = + evolvedo o operador pxlaplaciao. Etre as técicas utilizadas estão o Método de Galerki; Técica de regularização tipo Di Giorgi; Método de Sub-super solução e o Teorema do Passo da Motaha. Palavras-chave: pxlaplaciao, sigular com expoetes variáveis, Pricípio de Comparação, Regularidade de soluções ii

3 Abstract I this thesis we establish results of existece, uiqueess, multiplicity ad regularity of solutios for the followig class of quasiliear problems that may be sigular, ivolvig variable expoets px u = cxdx βx u αx + λfx, u i, u > 0 i ; u = 0 o. I the rst part, we determied suciet coditios for the existece of a uique solutio i W,px loc whe fx, t is subliear i t = 0 ad t = + throughout the domai. I the secod part, we obtai multiplicity of solutio i W,px 0 whe fx, t is superliear i t = + just i a subdomai of i some subdomai of. Besides this, we allow multiple regios of sigularity, both for the potetial ad for the o-liearity u > 0, while i the secod part we take β 0. I additio, we prove a Compariso priciple for sub ad supersolutio i W,px loc for subliear problems i t = 0 ad t = +, ivolvig the pxlaplacia operator. Amog the techiques used are the Galerki Method; the Di Giorgi regularizatio techique; the Sub-super solutio method; the Moutai Pass Theorem. Keywords: pxlaplacia, sigular variable expoet, Compariso Priciple, Regularity of Solutios iii

4 Cotets Itroductio Notatio ad Termiology Space of Fuctios About variable expoet spaces 20. Lebesgue spaces with variable expoets Sobolev spaces with variable expoets Regularity results i variable expoets spaces A Compariso Priciple for a kid of px subliear problems Itroductio Auxiliary results Proof of Theorem Completed Uiqueess of W,px loc -solutio for a oscillatig-sigular-cocave problem Itroductio A family of auxiliary problems Estimates i the variable expoets spaces Proof of mai results Proof of Theorem Completed Proof of Theorem Completed Proof of Theorem Completed Multiplicity of W,px 0 -solutios for local-sigular-covex problem 77

5 4. Itroductio Existece of a rst solutio Proof of Theorems 4..2 ad Completed Existece of a secod solutio Moutai Pass Geometry The Cerami Coditio Proof of Theorem Completed Ope problems ad future work 9 Appedix A Algebraic tools 93 A. Algebraic tools B Auxiliary Results 95 Referêcias 00 v

6 Itroductio This work presets a study of questios related to existece, uiqueess, multiplicity ad regularity of solutios for the followig class of quasiliear problems px u = axdx βx u αx + λfx, u i, u > 0 i, u = 0 o, ivolvig variable expoets ad powers, where R N is a bouded ope domai with smooth boudary, λ 0 is a real parameter, dx = if x y, for x, is the y stadard distace fuctio to the boudary of, p : R is a C -fuctio that satises < mi x px = p p + = max px < N, x ad px stads for the pxlaplacia operator, that is, px u = div u px2 u. Whe px p a costat, we have the well kow plaplacia operator. Because of the o-homogeeity of the pxlaplacia, these kids of problems are more delicate tha oes with plaplacia. For example, the rst eigevalue of pxlaplacia is zero i geeral, ad oly uder some special coditios the positivity holds see [32]. By quotig [28], the study of variable expoets spaces appeared i the literature for the rst time i 93, i a article by Orlicz [58], but the eld of variable expoet fuctio spaces has witessed a explosive growth i recet years. The developmets i sciece lead to a period of itese study of variable expoet spaces. Also observed were problems related to modelig of so-called electrorheological uids [63, 64], the study of thermorheological uids [67] ad image processig [9].

7 By goig back to the problem, we otice that it exhibits a sigular behavior at the origi whe αx > 0, that is, s αx s for all x {αx > 0}. Moreover, the weight dx βx also presets a sigular behavior ear the boudary whe βx > 0, that is, dx βx dx 0 + for all x {βx > 0}. The study of sigular problems relies maily of their applicatio to physical models such as o-newtoia uids [2], boudary layer pheomea for viscous uids [], chemical heterogeous [0] ad e theory of heat coductio i electrically coductig materials [22]. Our objective i this work is exploit the variable expoet to study the problem i two dieret ways. The rst beig whe fx, t is subliear i t = 0 ad t = + throughout the domai ad αx, βx allowig to chage the sigal. I secod part, fx, t is superliear i t = + oly i a subdomai of, β 0 ad αx allowig the sigal to chage. Our work is divided ito three chapters ad two appedix. I Chapter, we are goig to remember some deitios ad results ivolvig the Lebesgue ad Sobolev spaces with variable expoets which will be used throughout this thesis. I Chapter 2 we preset a Compariso priciple to the problem px u = gx, u i, u > 0 i, u = 0 o, where R N is a smooth domai, p : R is a C -fuctio that satises < mi x ad gx, t fullls the followig coditios: px = p p + = max px < N, x g g : 0, [0, + is a fuctio such that t gx, t is a cotiuous g 2 t fuctio a.e. x ad for each t > 0 the fuctio x gx, t is mesurable, gx, t t p is strictly decreasig o 0, for a.e. x, g 3 the fuctioal I h : W,px 0 R, deed by u px I h u = dx px G h x, udx, 2 2

8 is coercive ad weakly lower semicotiuous o {w W,px 0 / 0 w u} with respect to W,px 0 -orm, where for each h > 0 give. G h x, s := s 0 g h x, t + hdt ad g h x, t := gx, t + h Brezis ad Oswald [8] studied the semiliear case to the problem 2, that is, u = gx, u i, 3 u = 0 o, with gx, t satisfyig: BO t gx, t is a cotiuous fuctio a.e. x ad for each t 0 the fuctio x gx, t belogs to L a.e. x, BO 2 t gx, t t is strictly decreasig o 0, for a.e. x, BO 3 there is a costat C > 0 such that gx, t C + t for a.e. x ad t 0. Besides this, they itroduce the exteded fuctios a 0 x = lim t 0 + gx, t t ad a x = lim t gx, t t for x ad the quatity λa = { } if v 2 a v 2 dx v H, v 2 = {v 0} for ay exteded fuctio a : R {, + } give. With this set of hypotheses, they showed that the problem 3 has at most oe weak solutio. Moreover, a weak solutio of 3 exists if ad oly if λa 0 < 0 < λa. Later, Diaz ad Saá [27] exteded the result of Brezis ad Oswald for plaplacia operator with p > ad similar hypotheses. The fudametal tool to prove the uiqueess of solutio is the followig iequality w p p2 w p w w 2 w p p w p 2 p2 w p which became kow as Diaz-Saá Iequality [27]. 2 w w 2 p dx 0, 4 p w 2 3

9 Goig back to our problem, the hypotheses g g 3 do ot imply ay growth restrictio o g with respect to the variable t ad allow gx, t to have sigular behavior at 0. Furthermore, the techical hypothesis g 3 is ot a strage assumptio. I fact, with the hypothesis λa > 0 cosidered i Brezis-Oswald [8] Laplacia operator or Diaz-Saa [27] plaplacia operator together with BO BO 3 ad px p, lead us to show that fuctioal I h is coercive ad sequetially weakly lower semicotiuous o {w W,px 0 / 0 w u}. O the other side, i a Sobolev variable expoet space, the amout { } if v px a x v px dx v W,px, v px = {v 0} px may ot be positive, see for istace [32]. We also poit out that if we take gx, t = ax/t αx, for some αx > p ad a L rx with a suitable choice of rx, ad, i particular, px has a strictly local miimum or maximum i, the the ima i 5 is ull, but the hypothesis g g 3 are still satised. Before euciatig our rst result, we eed of the ext deitio. Deitio 0.0. We say that u W,px loc is a subsolutio of 2 if u 0, gx, u L loc ad u px2 u φdx gx, uφdx 0, φ C0, φ 0. Aalogously, u W,px loc is a supersolutio of 2 if u 0, gx, u L loc ad u px2 u φdx gx, uφdx 0, φ C0, φ 0. Now, we have. Theorem Assume that g g 3 holds true for each h > 0 give. If u, u are subsolutio ad supersolutio for 2, respectively, such that u L loc with u ɛ + W,px 0 for each ɛ > 0 give ad ess if ux > 0 for each U, x U the u u a.e. i. The importace of our rst result is pricipally because it may be applied to subsolutios ad supersolutios just i W,px loc. The proof is quite techical, because we have to keep away from the boudary of to avoid the possible sigularity of gx, t at t = 0. The rst part of our proof is ispired o ideas i [6] that show the compariso 5 4

10 betwee a sub ad a supersolutio for a olocal ad sigular problem by trucatig the sigularity i a suitable way. The secod part is ispired o ideas of [43], to take advatage to the covexity of the fuctioal J = J K : L loc, ] be give by u p px dx, u 0, u p,px W loc, Ju = K px +, otherwise, for each K give to derive a Diaz-Saá type iequality p w px2 p w w w 2 p p w 2 px2 w K p w where w, w 2 L loc {u L p loc / u 0, u p 2 w w 2 dx 0, w p p 2 W,px loc } with w i /w j L loc, i j. To our kowledge, this result is ew eve for Laplacia operator. I the Chapter 3, we study issues about existece, regularities ad uiqueess of solutios to the problem. To euciate these results, let us deote the δ-strip aroud to the boudary of by δ := {x / dx < δ} ad cosider the umbers px βx max θ = x δ px + αx if βx + αx > i δ, if βx + αx i δ, px βx θ 2 = mi x δ px + αx, for each δ > 0 give. So, let us assume that there exists a δ > 0 such that: H α : R is a C 0, -fuctio that satises αx mi x αx := α > p, H 2 f : [0, [0, is a Carathéodory fuctio such that fx, t bx + t qx for all x holds true, for some fuctios q C ad 0 b L sx L δ with < q q + p ad sx > N/p for x, H 3 i β : R is a C 0, -fuctio that satises β + < p, ii 0 < c L rx L δ for some r C with rx +, iii cx/ αx L rx L δ, 5

11 H 4 fx, t t p is strictly decreasig o 0, for a.e. x. Uder the coditio H, the problem may be sigular at u = 0 i multiple regios of the domai. For example, if = B R 0 is the ball cetered at origi of R N with radius R = 0π, the the problem oscillates from sigular i the rigs B 2k+π 0\B 2kπ 0 to o-sigular oe i B 2kπ 0\B 2kπ 0 for k =,, 5. Beside this, we allow the sigal of αx oscillates from a sub-liearity p < αx 0 passig through a weak sigularity 0 < αx, to reach a strog sigularity αx both i the domai ad its boudary. Before sharig our pricipal results, we here briey recall the literature about related sigular problems. Cradall, Rabiowitz ad Tartar [24] have cosidered a class of sigular problems which icluded, as special model, u = ax u α i, u > 0 i, u = 0 o, 6 for some 0 < a C ad α > 0 beig a real umber, showig ot oly existece of classical ad weak solutios but also some boudary regularity. A broad literature o problems like 6 is available to this date. Sice the, may authors have cosidered the above problem with other operators. I a famous paper, Lazer ad McKea [5] studied the problem u = ax u α i, u > 0 i, u = 0 o, 7 where a C with a > 0 i, ad α > 0 is a real costat. They proved that 7 has a solutio i H 0 if ad oly if 0 < α < 3, while for α > the solutios are ot i C. A extesio of the Lazer ad McKea's obstructio was proved by Zhag ad Cheg [72] whe ax is like dx β with β R i.e., c, C > 0 s.t. cdx β ax Cdx β o, where they showed that 7 has a solutio still i H 0 if, ad oly if, α 2β < 3. Boccardo ad Orsia [6], by combiig the techique of trucatio with some ecessary apriori estimates o the solutios of the correspodig approximatio problem, showed existece ad regularity results for div Mx u = ax u α i, u > 0 i, u = 0 o, 6

12 where α > 0 is a real costat, ad 0 < a L r. I particular, they showed that if α ad r = 2 /2 + α, the their solutio u H0, while u Hloc if α > ad r =. Followig these ideas, Chu, Gao ad Gao [20] have geeralized the mai result i [6] for the case whe α > 0 is a variable power, by cosiderig three cases: 0 < α < α + <, α < < α + ad < α < α + i. Carmoa ad Aparicio [7] also cosidered α > 0 as a variable power that may have a regio iside with αx ad aother oe with αx >. They proved the existece of solutio i H 0 whe αx i a strip aroud the boudary ad belogs to the Hloc with zero o the boudary i a geeral sese for the other cases. Most of these results was geeralized for dieret operators. We would like to metio [7, 20, 26, 57] ad their refereces. Results for pxlaplace equatios with pure sigular o-liearity have bee recetly explored. I [7], Zhag has studied the problem px u = λ u αx i, u > 0 i, u = 0 o, with αx > 0. By usig the sub-solutio method, he has obtaied the existece result of solutios i W,px loc C ad has preseted a asymptotic behavior of these positive solutios whe λ > 0 is large eough. The same author i [7] has improved his above existece result by cosiderig the problem px u = λkxfx, u + βu γx i, u > 0 i, u = 0 o, where γ C with γx < p, fx, t C 0,, 0, is a decreasig ad sigular fuctio at the origi, 0 K L qx for some qx > N, ad λ, β > 0 are real parameters. After Lazer ad Mackea [5], it is well kow that our problem may ot have solutios with zero-boudary value i the sese of the trace fuctio. Alog this chapter, we are goig to cosider the ext oe. Deitio Let u W,px loc. We say that u 0 o if uɛ + W,px 0 for every ɛ > 0 give. Furthermore, we also say that u 0 o if u 0 o, ad u = 0 o if u 0 ad u 0 o, simultaeously. It is readily see that if u W,px 0, the u = 0 o i the sese of above 7

13 deitio. Moreover, for each small δ > 0 give, the fuctio σdx θ if dx < δ, dx 2 2δ t ux = σδ θ + σθδ θ p dt if δ dx < 2δ, δ δ 2δ 2 2δ t σδ θ + σθδ θ p dt if 2δ dx, δ δ does ot belog to W,px 0 if θ > /p +, but u ɛ + W,px 0 for each ɛ > 0 give. Ispired by the ideas i [7], for each Γ smooth eough, ad h C give, let us deote by W,hx Γ = { u W,hx U / u Γ = 0 i the trace sese } for all ope set U such that U = Γ. I particular, we otice that W,hx Γ = W,hx loc if Γ =, W,hx 0 if Γ =. We otice that the trace over Γ is well deed if, for example, is Lipschitz cotiuous see [28, Chapter 2] Deitio A positive fuctio u W,px Γ is a solutio to problem if u = 0 o i the sese of Deitio 0.0.3, ad i cxdx βx ux αx L loc, ii ess if x K iii u px2 u φdx = ux > 0 for all K, cxdx βx u αx φdx+λ fx, uφdx, φ C0. From the results i [6, 7, 26, 5, 72], it is reasoable to expect that admits a solutio that fullls the boudary datum i the sese of the trace fuctio whe the trio cx, αx, βx satises some compatibility coditio. For this reaso, let us cosider the C 0, -maifold ad the umber Γ t = {x / [βx + t αx] /rx + > 0}, σ = max { p + β + /θ 2 + α +, p } + α +. p p Our rst result is related to existece of solutios ad it is formulated as follows. 8

14 Theorem Assume H H 4. If σp p σ+αx if βx + αx > > 0 i δ, rx = p x α if βx + αx > = 0 i δ, the there exists a 0 < λ such that the problem admits a solutio u = u λ W,px Γ Γ θ2 with ux Cdx, x for each 0 λ < λ give. I additio: i if q + < p i H 2, the λ =, ii if cx c δ i δ for some c δ > 0, the ux Cdx θ for x δ ad u W,px Γ θ Γ θ2. The variable expoets cosidered o our problem implied two itegrability coditios whe were seekig solutios still i W,px 0. This happeed because we were ot able to show the boudedess C dx θx ux C 2 dx θx a.e. x for θx = px βx/px + αx for x. For p, β ad α costats ad for some particular cases, this iequality is true see for istace Bougherara, Giacomoi ad Heradez [7]. I fact, we prove see Propositio that C dx θ C 2 dx θ 2 ux for x δ for c like that oe cosidered i [7]. For a geeral c, we proved a iequality with i the place of θ see Propositios ad This fact has great iuece i the al shape of the solutio, that is, i this case the solutio u belogs just to W,px Γ Γ θ2, where W,px Γ Γ θ2 = that is, W,px Γ Γ θ2 = W,px 0 if ad oly if W,px Γ if αx βx + αx 0, W,px Γ θ2 if αx βx + αx > 0, { αx < max 2 rx βx, + } θ 2 rx βx for all x. 8 More, as claimed i Theorem 0.0.5, if cx c δ i δ for some c δ > 0, the W,px Γ θ if αx, u W,px Γ θ Γ θ2 = W,px Γ θ2 if αx >, 9

15 that is, u W,px 0 if either αx mi {, + θ } rx βx or < αx < + θ 2 rx βx for all x. 9 Returig to costat expoets that were treated i literature up to ow, it follows from Theorem ad above iformatios that the solutio u still belogs to W,px 0 if either W,px Γ Γ θ2 = W,px 0 or W,px Γ θ Γ θ2 = W,px 0. I particular, Theorem ii icludes the mai results foud i the literature about this issue up to ow: i Lazer ad McKea - 99 [5] Let px 2, αx = α, βx 0 ad cx C with c > 0 i. Thus, it follows from 8 ad 9, that W,px Γ θ Γ θ2 = W,px 0 0 < α < 3, ii Zhag ad Cheg [72] Let px 2, αx = α, βx = β 0, 2 ad cx = c with c > 0 i. Thus, W,px Γ θ Γ θ2 = W,px 0 0 < α < 3 2β, iii Mohammed [55] Let px p, αx = α, βx 0 ad cx L with c > 0 i. Thus, W,px Γ θ Γ θ2 = W,px 0 0 < α < 2p p, iv Giacomoi, Bougherara ad Heradez [7] Let px p, αx = α, βx < p ad cx L with c > 0 i. Thus, W,px Γ θ Γ θ2 = W,px 0 α < 2p p p p β, v Yijig ad Duazhi [66] The problem px u = cxdx qx u qx, u = 0, 0 possesses a solutio i W,px 0 for ay 0 < c L ad 0 < q C 0, due to 8. It icludes the example preseted i [66] that cosidered px = qx = 2 ad cx = c > 0 for some real costat c > 0. 0

16 The proof of Theorem relies o a geeralized Galerki Method, which cosists i dig oe solutio to a "regularized problem" ad the to perform a apriori uiform estimates. From the hypothesis H, the term u αx may ot be mootoe i u > 0 aymore, this makes it more dicult to show uiform positivity estimates for a approximate sequece i the iterior of coditio ii i Deitio The importace of our ext result is pricipally because it may be applied to differet types of problems, amely: purely sigular with weak sigularity i.e., fx, t 0 ad 0 < αx, purely sigular with strog sigularity i.e., fx, t 0 ad αx >, sigular-subliear αx > 0, purely subliear αx < 0 ad to oscillated problems i.e. αx chagig its sigal. Moreover, sice cxdx βx may ot lie i L, we emphasize that the aalysis of the behavior of the trio c, α, β oly ear the boudary is very essetial. The third result deals with regularity of solutios for Problem. It is stated as a combiatio of Theorem 3..4 ad the Corollary 3..5 of the Chapter 3. Theorem Assume H H 4. Let u W,px Γ Γ θ2 be the solutio for give by Theorem The there exists a 0 < λ λ is possibly less tha λ give i Theorem such that for all 0 λ < λ, we have: i u L if rx > N/p, ii u C 0,γ U for all ope set U such that U = Γ Γ θ2 if rx > N/p, Nr p +α iii u L Nr p if βx + αx > > 0 i δ ad r < N/p with ad { β Np + α + max N pp + α + p p + α +, Np + + } θ 2 + α + N pp + α + p p + β+ r, θ 2 + α + Nr p +α iv u L Nr p if βx + αx > = 0 i δ ad Np Np N p α r < N p. for δ > 0 as i Theorem I additio, if q + < p i H 2, the λ =. Besides this, the same coclusios hold true if we chage Γ for Γ θ. The boudedess ad regularity of the solutios depedig o the trio c, α, β have bee cosidered i [6, 7, 26] for particular cases. We establishes similar results

17 ad prove the Hölder cotiuity adoptig the method of De Giorgi developed by Ladyzheskaya ad Ural'tseva ad derived the suitable Caccioppoli type iequality [3]. To coclude, we preset a suciet coditio for uiqueess of solutio for. Theorem Assume that H H 4 holds true with rx > N/p. If βx < o, the there exists a 0 < λ λ is possibly less tha λ give i Theorem such that for all 0 λ < λ the problem admits a oly solutio i W,px loc i sese of Deitio Beside this, λ = + if p = q +. Theorem will be fudametal to prove uiqueess of W,px loc -solutios for. I the last years, some papers have proved uiqueess of solutios for purelysigular problems for dieret operators. The semiliear case px 2 ad gx, t = t α, for α > 0 beig a real costat, was proved i [4]. This result was recetly exteded for gx, t = axt α with 0 ax L for px = 2 i [3], for px p i [5], while i [6] the fractioal plaplacia operator was cosidered. Our result geeralizes ad complemets these results, sice we cosider the variable expoet px-laplacia ad the variable power αx, which ca oscillates from egative to positive values. That is, we do ot require gx, t beig mootoe ad do ot impose gx, t = fxht. As a ovelty i this chapter, we poit out that we took advatage as most as possible of the variability of the expoets ad powers. As a cosequece of this, we have show that the diculty i aswerig the pricipal issues about this kid of problem is cocetrated i uderstadig the behaviors of the powers ad expoets just ear to the boudary of the domai where the sigularity is really triggered for Dirichlet boudary coditios problems. For istace, the itegrability coditio of trio c, α, β oly ear the boudary of the domai is suciet to obtai existece of solutios still i W,px 0. We cojecture that the coverse claim is true as well. I Chapter 4, we study the problem px u = axu αx + λfx, u i, u > 0 i, u = 0 o, where is a bouded domai i R N with smooth boudary, 0 < a L rx for some r C, p C, λ > 0 is a real parameter ad fx, t has a superliear local behavior at t = + to be preseted below. 2

18 Coclite ad Palamieri [2] cosider the problem u = u α + λu p = gx, u i, u > 0 i, u = 0 o, 2 with α, p > 0 ad λ > 0 ad showed that there exists λ > 0 such that 2 has a solutio for all 0 < λ < λ ad o classical solutio for λ > λ. Log, Su ad Wu [53] studied 2 with 0 < α < ad < p < 2 to obtai the existece of a λ > 0 such that 2 has at least two weak solutios for all 0 < λ < λ. Later, Su ad Wu [54] retured to the problem 2 ad obtaied a exact result value for λ > 0. After these works, a broad literature has bee accumulatig i relatio to Laplacia operator with gx, s i dieret kids of hypotheses, see for istace [, 44, 46, 47] ad their refereces. More geeral operators have bee cosidered recetly, as well. For the plaplacia p u = axu α + λfx, u i, u > 0 i, u = 0 o. 3 Perera ad Zhag [62] obtaied multiplicity of solutios for 3 combiig a cuto argumet, variatioal methods, results relatig to W,p versus C miimizers, with p 2, α > 0 ad fx, t satisfyig the classical Ambrosetti-Rabiowitz coditio ad H α there are φ 0 0 i C 0 ad q > such that aφ α 0 L q. Later, Perera ad Silva [60] improved above result by cosiderig stroger hypotheses. For istace, they did ot assume p 2 or ay stroger regularity assumptio o f. Giacomoi, Schidler ad Takac [4] established the global multiplicity results of the above problem for a certai rage of λ by cosiderig 0 < α < ad fx, t = t q. Still related to problem 3, we may also cite [39, 59, 6] ad their refereces. I cotext of px Laplacia, to our kowledge, there exists few works treatig the problem like. Byu ad Ko [9] ad Ghami ad Saoudi [40] px u = λu αx + fx, u i, u > 0 i, u = 0, o, 4 3

19 improved the pricipal result i [4] by cosiderig variable 0 < αx < ad fx, t superliear at + throughout the domai. Their proof is variatioal ad the fudametal tool used i their approach is a extesio for the pxlaplacia cotext of the local miimizatio C versus W,px 0. To state ours results, let us rst dee a solutio to Problem. Deitio A positive fuctio u W,px 0 is a solutio to 4. if u px2 u φdx = axu αx φdx + λ fx, uφdx for all φ W,px 0. Besides this, let us remid that δ := {x / dx < δ}, for each δ > 0, stads for the iterior δ-strip aroud the boudary of the domai, Γ t = {x / [t αx] /rx + > 0}, for t {, θ, θ 2 }, is a subset of the boudary of the domai ad the umbers θ ad θ 2 are deed by px max if αx >, θ = x δ px + αx px ad θ 2 = mi x if αx, δ px + αx. Related to the fuctios αx, ax ad fx, t, we make the followig geeral assumptios. Assume that there exists a δ > 0 such that: H α : R is a C 0, -fuctio that satises α > p, H 2 0 < a L rx with rx > N/p ad oe of the items below: i a L δ ad Γ Γ θ2 =, ii ax a δ > 0 i δ, a L δ ad Γ θ Γ θ2 =, H 3 ax αx Lrx {αx }. f f : [0, [0, is a Caratheodory fuctio such that for each M > 0 give there exists c = c M > 0 satisfyig 0 fx, s c for every 0 s M ad a.e. x i, 4

20 where the last hypothesis was ispired o a hypotheses i [60]. Our rst result is. Theorem Suppose H, H 2 ad f are satised. The there exist λ 0 > 0 such that the problem has a positive weak solutio u λ W,px 0 L for each 0 < λ < λ 0 give satisfyig u λ m 0 dx i for some m 0 > 0. I additio, there exists M 0, M, m > 0 such that: i m 0 dx u λ M 0 dx θ 2 for x δ if H 2 i holds, ii m dx θ u λ M dx θ 2 for x δ if H 2 ii holds. We ca also cosider a settig i what fx, s is allowed to chage its sigal ad is bouded from below by itegrable fuctios o bouded itervals of the variable s > 0, that is: f 2 f : [0, R is a Caratheodory fuctio such that for each M > 0 give there exists c 2 = c 2 M > 0 ad 0 h = h M L satisfyig hx fx, s c 2 for every 0 s M ad a.e. x, f 3 there are ζ > 0 ad c 3 > 0 such that fx, s c 3 ax for every 0 s ζ ad a.e. x. Our secod result is. Theorem Suppose H, H 2, f 2 ad f 3 are satised. If αx 0 i with αx < o, the there exist λ > 0 such that the problem has a positive weak solutio u λ W,px 0 L for each 0 < λ < λ give satisfyig u λ Cdx i for some C > 0. The proofs of Theorem ad relied upo dig a sub ad a supersolutio, say u, u, for i W,px 0 L, miimizig a appropriated C eergy fuctioal i W,px 0 ad showig that this miimum belogs to the coe [u, u] ad is a weak solutio for. To do this, the results obtaied i Chapter 3 were determiats, sice u ad u are solutios for a appropriated sigular-subliear problems. More, the Compariso priciple demostrated i Chapter 2 was a fudametal tool used to show that u u. However, whe fx, t is allowed to chage its sigal, the 5

21 restrictio αx 0 was ecessary i order to obtai a compariso betwee the sub ad supersolutio. I order to establish the existece of at least two solutios for the problem, we also assume: f 4 there exists C > 0 such that fx, t C + t qx for t > 0 ad a.e. x, with < q C ad p + < q + < p, f 5 there exists a subdomai D such that F x, t lim t t p + = + uiformly o x D, where F x, t = t fx, sds for t > 0 ad x, 0 f 6 there exist t 0, β 0 0 ad τ C with < τx < p, x such that p + F x, t fx, tt β 0 t τx for all t > t 0 ad a.e. x. This set of hypotheses was ispired i [37]. Note that f 4 does ot impose qx > px i ad f 5 implies that fx, s is just locally p + superliear at iity just i D, that is, lim fx, t + t/tp + = + uiformly i D. The hypothesis f 6, as poited out i [37] for costats fuctios sx ad qx, is a weaker form of the classical coditio of Ambrosetti-Rabiowitz. For istace, the fuctio F x, t = bxt sx + cxt qx with < s s + < p p + < q + < p, b 0, c R, satises f 4 f 6, but do ot satisfy Ambrosetti-Rabiowitz coditio if b 0 i D ad c 0 i some K /D. Theorem 0.0. Suppose H H 3, f 4 f 6 are satised. There exists λ > 0 such that the problem has at least two dieret solutios u λ, v λ W,px 0 for each 0 < λ < λ give. I additio, u λ v λ ad u λ has egative eergy while v λ is a positive eergy solutio. 6

22 The proof of Theorem 0.0. rely heavily o perturbatio argumets ad o the variatioal method employed by Perera ad Silva [60]. We verify that the cuto fuctioal associated with the problem satisfy the geometric hypotheses of the Moutai Pass Theorem ad the Cerami coditio. However, the chagig of sigal αx provides a obstacle i estimates ad, with our set of hypotheses, we do ot kow that the solutios belogs to C, which prevet us use results relatig of W,px versus C miimizers. The importace of our result, related to other works ivolvig variable expoets, is pricipally due to the fact that we do ot impose 0 < αx < i whe fx, t is oegative ad we just demad αx < o if fx, t is allowed to chage its sigal. Moreover, we do ot require i the hypothesis f 4 that q > p + i whole, as i the former works. We take advatage of the result obtaied i Chapter 3 to obtai the existece of solutios i W,px 0 with a dieret hypothesis that H α, as cosidered i [39, 60, 62], amely, ad tαx L δ, for t = {, θ, θ 2 } give i Theorem Besides these, the results obtaied i Chapter 4 complemets the results i Chapter 3 i the sese that a local superliearity at iity of fx, t implies multiplicity of solutios, i cotrast to uiqueess obtaied if fx, t is subliear i whole. I order to make the chapters self-suciet, we will state oce agai, i each chapter, the mai results as well as the problems ad hypotheses cosidered i the itroductio. 7

23 Notatio ad Termiology C ad C i deote positive costats. R N deote the N-dimesioal Euclidea Space. B r x is the ope ball cetered i x radius r > 0. If R N is Lebesgue mesurable, the deote the Lebesgue measure of. The otatio x x mea strogly covergece. The otatio x x mea weakly covergece. X Y deote that X is cotiuously embedded i Y. X Y deote that X is compactly embedded i Y. If u : R is mesurable, the u = mi{ux, 0} ad u + = max{ux, 0} deote the egative ad positive part, respectively. If u : R is mesurable, the u = ess if ux ad u + = ess sup ux. dx = if x y, for x, the stadard distace fuctio to, y s = {x / dx s}, for s > 0, the strip aroud of the boudary of Space of Fuctios C deote the space of cotiuous fuctios i ad C 0 the cotiuous fuctios with compact support i.

24 C k cosists of those fuctios o havig cotiuous derivatives up to order k ad C = k C k. C 0 = C C 0. L p = { u : R is mesurable / ux p dx < }, edowed with the orm L = orm L + = u p = if ux p dx p. { } u : R is mesurable / ess sup ux <, edowed with the x u = ess sup ux. x { } p L / ess if p >. x If p L +, we dee the the variable expoet space by L px = { u : R is mesurable / } ux px dx <, edowed with the Luxemburg orm { u px = if λ > 0 / ux λ px dx }, If p L +, we dee the space W,px = { u L px / u L px } edowed with the orm u,px = u px + u px. The space W,px 0 is deed as the closure of C 0 i W,px with respect of the orm,px edowed with the orm u = u px. 9

25 Chapter About variable expoet spaces I this chapter, let us preset some properties ad results about the spaces L px, W,px ad W,px 0, where R N is a bouded ope set. For the iterested reader i more iformatio about these spaces, icludig the proofs omitted i this chapter, we refer the papers Fa ad Zhao [34], Kovácik ad Rákosík [49] ad the book of Dieig et al. [28].. Lebesgue spaces with variable expoets Let us deote by L + the set { } L + = u L / ess if u x ad by L px the variable expoet space deed by { } L px = u : R is mesurable / ux px dx <, edowed with the Luxemburg orm { u px = if λ > 0 / ux λ px dx for each p L + give. This space is well-kow as the variable expoet Lebesgue space. It is well-kow that whe px = p is costat the Luxemburg orm coicides with the usual orm i L p, that is, the variable expoet Lebesgue space turs ito the classical Lebesgue space. },

26 Now, give a p L +, let us we deote by p ad p + the followig real umbers p = ess if px ad p + = ess sup px. x ad dee the modular fuctio ρ : L px R by ρu = ux px. Propositio.. [34], Theorems.2 ad.3 Let u L px. The:. u px < =, > ρu < =, >, 2. u px > u p px ρu u p + px, 3. u px < u p + px ρu u p px, 4. u px = a if, ad oly if, px u dx =. a Propositio..2 [34], Theorem.4 Let u L px. The,. lim u px = 0 if, ad oly if, lim ρu = 0. x 2. lim u px = + if, ad oly if, lim ρu = +. I special, for some u L px, lim u u px = 0 if, ad oly if, As a Corollary of the above result, we have. lim ρu u = 0. Corollary..3 [34] Let u L px with u u i L px. The there exists a subsequece u k such that. u k x ux a.e. i, 2. u k x hx for all k ad a.e. i with h L px. A importat estimate that will be frequetly used i this work is give i the ext propositio. Propositio..4 [34] Let h, p L + with hx px a.e i, ad u L px. The, u hx L px hx ad u hx px u h + px + u h px hx or u hx px hx { } max u h + px, u h px. Reciprocally, if u hx L px hx with hx px, the u L px, ad there is a umber h 0 [h, h + ] such that u hx px hx 2 = u h 0 px.

27 Propositio..5 [34], Theorems.6 ad.0 Let p L +. The the space L px, px is separable. I addictio, if p >, the L px is uiform covex ad thus is reexive. Give p L +, we deote by p x the cojugate fuctio of px, that is, px + p x = with the covetio that / = 0. Now we preset the geeralizatio of Hölder's Iequality. Propositio..6 [49], Theorem 2. Let p L + with p >. For ay u L px ad v L p x we have uvdx + u px v p p x 2 u px v p x. p I additio, if /px + /p x + /p x = holds true, the uvwdx + + u p p p px v p x w p x 3 u px v p x w p x for all u L px, v L p x ad w L p x. To ed, we preset the atural iclusio of variable expoet Lebesgue's spaces. Propositio..7 [34], Theorem. Let h, p L + with hx px a.e i. The L px is cotiuously embeddig ito L hx..2 Sobolev spaces with variable expoets I this sectio, we cosider oly the space of Sobolev W,px. The deitio ad properties of the spaces W k,px, with k >, ca be foud i the refereces quoted above i this chapter. The variable expoet Sobolev space W,px is deed by W,px = { u L px / u L px } edowed with the orm u,px = u px + u px.. 22

28 Similarly to L px, the Baach space W,px is separable ad, if p >, it is also a reective space. The space W,px 0 is deed as the closure of C 0 i W,px with respect of the orm deed i.. It is also worth to poit out that ulike of the validity of the desity of C i W,p whe px = p >, i the cotext of the variable expoet space W,px 0 this will be true if we require additioal coditios o the domai ad expoet px, for istace, if is Lipschitz cotiuous ad px satises the log Hölder coditio, that is, log x y px py C for all x, y with 0 < x y < for some C > 0, the C is dese i W,px with respect of the orm deed i.. See for istace [28]. By usig similar argumets like those used i [48, Lemma.25], we obtai. Propositio.2. Let v W,px. i If v has compact support, the v W,px 0. ii If u W,px 0 ad 0 < v < u a.e. i, the v W,px 0. iii If u W,px 0 ad v < u a.e. i \ K, where K is a compact subset of, the v W,px 0. I this settig, the variable critical fuctio-expoet for embeddig of Sobolev to Lebesgue with variable expoets is deed by Npx if N > px, p x = N px + if N px, ad it is called as the critical fuctio with respect to px. Propositio.2.2 [34], Theorem 2.3 Let p L +. If p, q C with < p p + < N, the: i W,px L qx if qx p x a.e. i, ii W,px L qx if qx < p x a.e. i. More, there exists a costat C > 0 such that u px C u px for all u W,px 0. 23

29 The last iequality is well kow as Poicaré's iequality. As a cosequece of it, we ifer that u px dee o W,px 0 a equivalet orm to u,px. From ow o, we are goig to deote this orm by u ad we will use it for the whole paper. Propositio.2.3 Let u W,px 0 ad the modular fuctio ρ 0 : L px R deed by ρ 0 u = ux px. The the same coclusio of Propositio.. holds if we cosider ad ρ 0. The otio of a map of S + -type is useful to help us to prove that a sequece coverge strogly i W,px 0 uder appropriate assumptios. We say that a fuctio is S + -type if u u i W,px 0 ad lim sup Lu, u u 0, the u u i W,px 0. Propositio.2.4 [33], Theorem 3. The map L : W,px 0 W,px de- ed by Lu, v = u px2 u vdx is: i cotiuous, ii bouded, iii strictly mootoe, that is, iv S + -type. Lu Lv, u v > 0 for all u, v W,px 0, u v, Below, we preset two iequalities that will be useful i parts of this thesis. Lemma.2.5 [45], Hardy's Iequality Let be a ope ad bouded subset of R N. Assume that there exists a costat θ > 0 such that B r y c θ B r y for every y ad r > 0. The there exists positive costats c ad a 0 depedig oly o p, N ad θ such that the iequality u dx a c dx a u px, px holds for all u W,px 0 ad a [0, a 0. 24

30 Lemma.2.6 Simo's Iequality For all < p C there exists a positive costats C = Cp such that u v u px2 u v px2 v p if < px < 2, u + v 2px uv 2 3p + u v px if px 2 p + for all u, v W,px..3 Regularity results i variable expoets spaces I this sectio, we establish some results cocerig regularity of bouded fuctios satisfyig some relatios ivolvig the pxlaplace operator. The rst oe is a improvemet of a result of Fa ad Zhao [3] that was the rst result i this directio the cotext of variable expoets. We highlight that for our purposes, we eed of the below Propositio.3.5 istead of the classical Fa's result [3], because ours weights ax, bx i are ot i L. Despite of this geerality, we are to show that our solutio u of the Problem satises C dx ux C 2 dx θ 2 close to the boudary of for some C, C 2 > 0, where dx is the stadard distace fuctio to the boudary of. This makes possible to verify.3 ad apply Propositio.3.5. Deitio.3. Let M, γ, γ, δ, r, R be positive costats with δ 2, r > ad B R y. We say that a fuctio v belogs to class B p B R y, M, γ, γ, δ, /r if v W,px with max vx M ad the fuctios wx = ±vx satisfy the B R iequalities, px wx k w px dx γ dx + γ A k,t r.2 A k,τ t τ A k,t for arbitrary 0 < τ < t < R ad k such that k max B ty wx σm, where A k,ρ = {x B ρ : wx > k}. I a aalogue way, we say that a fuctio v belogs to class B p B R z, M, γ, γ, δ, /r if v W,px B R z with max vx < ad.2 holds for k max{ max B R z B R z wx δm, max max vx M, B R z B R z wx}. Deitio.3.2 We say that satises a exterior coe coditio at a poit x if there exists a ite right circular coe V x with vertex x such that V x = x, i particular, say that satisfes a uiform exterior coe coditio o if satisfes a exterior coe coditio at every x ad the coes V x are all cogruet to some xed coe V. 25

31 Lemma.3.3 Lemma 4.5, [68] Let p C, < p p + < ad be log- Hölder cotiuous i ad let R 0 0,, σ 0 > be umbers such that p 0 σ 0 > N, where p 0 = mi px ad x 0. Let B R y B R0 x 0 ad u W,px B R B R0 x 0 L B R. Suppose that, for arbitrary R R, there exists a umber r σ 0 such that u B p B R y, M, γ, γ, δ, /r, where M is a positive umber satisfyig u L B R M. The there exists a costat s = sn, p 0, σ 0, max BR0 x 0 px, M, γ, L > 2 such that, for arbitrary R R, sup ux if ux x B R y x B R y cr α R α, where c, α are costats idepedet of M. Lemma.3.4 Lemma 4.0, [68] Let p C, < p p + < ad be log- Hölder cotiuous i ad let R 0 0,, σ 0 > be umbers such that p 0 σ 0 > N, where p 0 = mi B R0 x 0 px ad x 0. Suppose that satisfes a exterior coe coditio at z. Let B R z B R0 x 0 ad u W,px B R z L B R z. Suppose that, for arbitrary R R, there exists a umber r σ 0 such that u B p B R yz, M, γ, γ, δ, /r ad satises sup ux if ux β 0R α 0, x B R z x B R z where β 0, α 0 are positive costats ad M is a positive umber satisfyig u L B R z M. The there exists a costat s = sn, p 0, σ 0, max BR0 x 0 px, M, γ, L, V z > 2 such that, for arbitrary R R, sup ux if ux x B R z x B R z cr α R α, where c, α are costats idepedet of M. As a cosequece of above Lemmas, we have the result. Propositio.3.5 If p C, < p p + < ad be log-hölder cotiuous i, the B ρ B R y, M, γ, γ, δ, /r C 0,β, where the costat β 0, ] is idepedet of M ad γ. I addictio, if satisfes a exterior coe coditio at z ad u B p B R z, M, γ, γ, δ, /r with u C 0,α satises sup ux if ux x B R z x B R z CRα.3 for some C, α > 0 costats, the u C 0,β 2, where the costat β 2 idepedet of M ad γ. 0, ] is 26

32 The followig Lemma is due to Ladyzheskaya ad Uraltseva [50] ad will be fudametal to apply the above Propositio. Lemma.3.6 [50], Lemma 4.7 Let x be a sequece such that x 0 λ η µ η 2 ad x + λµ x +η for ay N with λ, η ad µ beig positive costats ad µ >. The x coverges to 0 as. Aother applicatio of the above Lemma is the ext result. Lemma.3.7 [7], Lemma 2.4 Suppose 0 < b 0 b L αx with αx > N o. Let M > 0 ad u is the uique solutio of the problem { px u = Mbx i, u = 0 o. The, u C M p for M, ad u C 2 M p + for M <, where C, C 2 are positive costats depedig o p +, p, N, b L α ad. The ext C -regularity result is due to Fa. Theorem.3.8 Theorem.2, [29] If p is Hölder cotiuous o ad fx, t c + c 2 t qx for all x ad t R, where q C ad < qx < p x for x, the every solutio u W,px 0 of { px u = fx, u i, belogs to C,γ for some γ 0,. u = 0 o. To ed, we preset a strog maximum priciple for the pxlaplacia operator due to Fa, Zhag ad Zhao. Propositio.3.9 Theorems ad 2, [69] Suppose that p C, p >, u W,px, u 0 ad u 0 i. If px u + hxu qx 0 i, where h L, h 0 ad px qx p x, the u > 0 i, ad whe u C, the u η > 0 o, where η is the iward uit ormal o. 27

33 Chapter 2 A Compariso Priciple for a kid of px subliear problems 2. Itroductio I this chapter we preset a Compariso priciple for sub ad super solutios i W,px loc to a kid of px subliear problems, which will be so useful i several poits of this thesis. Cosider the problem px u = gx, u i, u > 0 i, u = 0 o, 2. where g : 0, [0, + is a fuctio satisfyig: g t gx, t is a cotiuous fuctio a.e. x ad for each t > 0 the fuctio x gx, t is mesurable, g 2 t gx, t t p is strictly decreasig o 0, for a.e. x. Note that uder the above hypotheses, we do ot impose growth restrictio just o g with respect to the variable t ad allow gx, t to be sigular at the origi, that is, gx, t + as t 0 + a.e. x. For istace, the fuctio gx, t = t αx + t βx, t > 0, with αx > p ad βx < p o satises g g 2.

34 From Lazer ad Mckea [5], the existece of weak solutios with zero-boudary value i the sese of the trace fuctio to sigular problems is possible just i some cases. For example, if px 2 ad gx, t = t α, t > 0, the there exists a solutio still i H 0 if, ad oly if, 0 < α < 3. Therefore, the way of uderstadig the boudary coditio will be the followig: Deitio 2.. Let u W,px loc. We say that u 0 o if uɛ + W,px 0 for every ɛ > 0. Furthermore u = 0 o if u is oegative ad u 0 o. It is readily see that if u W,px 0, the u = 0 o i the sese of the above deitio. Moreover, the fuctio σdx θ if dx < δ, dx 2δ t ux = σδ θ + σθδ θ δ δ 2δ 2δ t σδ θ + σθδ θ δ δ 2 p dt if δ dx < 2δ, 2 p dt if 2δ dx, where dx is the distace fuctio i, does ot belog to W,px 0 if θ > /p +, but u ɛ + W,px 0 for each ɛ > 0 give. Deitio 2..2 We say that u W,px loc is a subsolutio of 2. if u 0, gx, u L loc ad u px2 u φdx gx, uφdx 0 φ C0, φ 0. Aalogously, u W,px loc is a supersolutio of 2. if u 0, gx, u L loc ad u px2 u φdx gx, uφdx 0 φ C0, φ 0. The mai result of this chapter is the followig Compariso Priciple. Theorem 2..3 Assume that g g 2 hold ad suppose that for each h > 0 g 3 the fuctioal I h : R, deed by u px I h u = dx px is coercive ad weakly lower semicotiuous o with respect to W,px 0 -orm, where G h x, s := s 0 G h x, udx, K := {w W,px 0 / 0 w u} g h x, t + hdt, s 0 ad g h x, t := gx, t + h for t 0 29

35 Let u, u W,px loc be a subsolutio ad a supersolutio of Problem 2., respectively. If u L loc with u 0 o ad ess if ux > 0 for each U, the u u a.e. i. x U The hypotheses g ad g 2 are used to derive a type of Diaz-Saá's Iequality see 2.2 below i variable expoets cotext. Due to the absece of growth coditio ad the lack of positivity of the rst eigevalue i the settig of W,px, we have to cosider the assumptio g 3 that will be used to obtai a fudametal estimate i our proof. More details are preseted i the ext sectio. The importace of our rst result is pricipally because it may be applied to subsolutios ad supersolutios just i W,px loc. To our kowledge, this result is ew eve for Laplacia operator. 2.2 Auxiliary results I this sectio we preset the results that will be useful i the proof of Theorem Ispired by the ideas i [43], let D = {u L p loc / u 0, u,px W loc }. Fixed φ C0, cosider the fuctioal J = J φ : L loc, ] give by u p px φdx if u D, Ju = px + otherwise. Lemma 2.2. Let J be the above fuctioal. The J is covex ad J +. Proof. Let us begi our proof showig that J +. To this ed, xed x 0, take R > 0 such that B R x 0 is the closed ball cetered i x 0 with radius R. Now, give θ > p let v θ R x x 0 if x B R x 0, vx = 0 otherwise. where v R : [0, R] R is deed by if t = 0, v R t = liear if 0 < t < R/2, 0 if R/2 t R. 30

36 Evidetly v D. Moreover, v p px θ p v θ p vr φdx = px B R x 0 px p+ θ p p p+ θ p p that is, Jv +. have px θ p v B R x 0 φdx p v R px φdx v R px φdx < +, B R x 0 Now we are goig to show that J is covex. As i proof of [27, Lemma ], we sw + sw 2 /p p s w /p p + s w /p 2 p for all s [0, ], where w, w 2 D. Sice the fuctio s s px/p the above iequality that Jsw + sw 2 = is covex o [0,, it follows from sw + sw 2 p px φdx px p s w p + s w s w p px px px = sjw + sjw 2, for each s [0, ] give. This shows the Lemma. p 2 p + s w p px The ext result will be fudametal for our purposes. px p 2 px φdx φdx Lemma Diaz-Saá's type Iequality Assume that w, w 2 L loc D. If w i /w j L loc, i j, the [ p w px2 p w w w 2 holds, for all φ C 0. w p /p p w 2 px2 w p 2 w w 2 w p /p 2 ] φdx Proof. To obtai the iequality 2.2 it suces to show J u, v = u p px2 p u p p u v φdx, 2.3 3

37 for all u, v L loc D with w /w 2 L loc ad apply Lemma I fact, admittig it by ow, it follows from Lemma 2.2., that 0 p J w J w 2, w w 2 [ p = w px2 p w w w 2 L loc w p /p p w 2 px2 w p 2 w w 2 w p /p 2 Now, we are goig to prove that 2.3 holds true. First, we otice that if u,px D, the u W. I fact, by deotig w = u /p, we have that loc u = w p = p w p p p w = p u u p px L loc. Let u, v L loc D. Thus, J Ju + tv Ju u, v = lim t 0 t where ad = lim t 0 hx, t tpx hx, t = φdx = lim t 0 = lim t 0 ] u + tv p px u p px φdx tpx dh x, tdx, 2.4 dt u + tv p px u p px φ, dh px x, t = u + tv p px2 u + tv p p p u + tv v φ dt p for x ad t > 0. The last equality at follows from Mea Value Theorem, that is, there exists a 0 < s = sx, t < t < such that hx, t/t = dh x, s for x. Sice dt dh p x, s dt px p u + sv u + sv p 2p p p u + sv p 2 p v u + sv + p u + sv v φ p px p u + sv u + s v p u + sv p u + sv p p u + sv v u + s v + p 2 u + sv 2 p u + sv v φ px p u + v u + v p u p u + v p v u + v + p u + v v φ, p 2 u 2 p u φdx. 32

38 where we used u, v > 0 i the last iequality. that So, it follows from the above iformatio ad hypotheses v, u, u/v, v/u L loc, dh x, s dt px p px u + v px p v u + v px u px u + φ v C φ u C 2 φ u px L supp φ p px v + u + u px px px p + v p u px px p L supp φ u px + v px u px + v px [ ] C 3 φ u px p u + v px p u px px p v v [ ] C 4 φ u p px + v p px L loc, 2.5 where C 3 = C 3 u/v L, u L, p +, p > 0 is a real costat. Thus, the Lebesgue Domiated Covergece theorem implies that J u, v = hx, t p tpx dx = u p px2 u p p u p vφdx holds. This eds our proof. I [5], the authors showig the compariso betwee a sub ad a supersolutio for the problem 2. with px = p ad gx, t = axt α, where α > 0 is a real costat, by trucatig the sigularity i a suitable way. Ispired i these ideas, let us dee g h x, t = gx, t + h for x, t 0,, for each h > 0 give, ad cosider the problem px u = g h x, u i, u 0 i, u = 0 o. 2.6 So, we have. Lemma Assume g ad g 3 hold. The there exists a w K deed at 2..3 such that w px2 w ψdx g h x, w + hψdx 0, ψ W,px 0, ψ

39 Proof. It follows by hypothesis g 3, that the fuctioal I h u = u px dx G h x, udx, u K px is coercive ad weakly lower semicotiuous o the closed ad covex set K. So, there exists a w = w h such that I h w = if u K I h u. Now, give 0 ψ C 0, set v t = mi{w + tψ, u} ad w t = w + tψ u +, for t > 0 such that tψ u. Sice v t, w t W,px, 0 v t w + tψ, 0 w t tψ ad 0 w t u with supp w t v t, w t W,px 0, that is, v t, w t K. For y K, let us dee supp ψ, it follows by Propositio.2. i that σt = I h ty + tw for t [0, ] ad deduce from I h w = mi K I h that σ0 σt for all t [0, ], that is, 0 σ 0 = I hw, y w. Now, by takig y = v t ad oticig that v t w = tψ w t, we obtai 0 w px2 w tψ w t dx g h x, w + htψ w t dx = t w px2 w ψdx g h x, w + hψdx w px2 w w t dx + g h x, w + hw t dx. 2.8 O the other had, sice 0 w t W,px 0, there exists a sequece ζ C 0 with ζ 0, supp ζ supp w t ad ζ w t i W,px 0 as. Now, by usig the fact that u W,px loc is a supersolutio of problem 2.6, with ζ as a test fuctio, we obtai u px2 u ζ dx g h x, uζ dx 0 for all N, that lead us to u px2 u w t dx g h x, uw t dx

40 by the usig of the Lebesgue's covergece theorem together with the fact that u px is itegrable o the support of w t ad 0 w t tψ for each x. So, it follows from 2.8 ad 2.9, that 0 t w px2 w ψdx g h x, w + hψdx u px2 u w px2 w w t dx + g h x, w + h g h x, u w t dx for all t > 0 eough small. Sice g h x, w + h g h x, u w t g h x, w + htψ o supp w t, it follows from 2.0, by dividig 2.0 by t > 0, that 0 w px2 w ψdx g h x, w + hψdx + u px2 u w px2 w ψ + {w+tψ u} {w+tψ u} g h x, w + hψdx, that is, by doig t 0, usig Propositio.2.6, ad applyig Lebesgue's Covergece Theorem, we coclude that 2.7 is true for all 0 ψ C 0. The result follows by a stadard desity of C 0 i W,px 0. Now we are able to prove the Theorem Proof of Theorem Completed Let us do the proof by a cotradictio argumet by combiig the above results together with a very e aalysis. Proof. Cosider the set h := {x / ux h > wx + h} ad assume, by cotradictio, that h > 0 for some h > 0. From compactess of h, there exists a x 0 h ad R > 0 such that K R = B R h > 0, where B R be the ball of radius R cetered i x 0. We ca assume, without loss of geerality, that B R. I fact, sice is smooth, the = 0. I particular, there exits δ > 0 such that the set δ = {x 35

41 / dx, < δ} satises δ < h /4. Moreover, by compactess of \ δ there exists a ite cover m i=0b ri x i with x i ad r i δ/4 such that db ri x i, 3δ/4, for all i =, 2,..., m. Thus, \ δ h 3 h /4. So there exists B R := B ri x i such that BR h > 0 for some i m. Fix 0 < t < R < δ/4 ad take 0 < s < t such that K s := B s h ad K t := B t h have positive Lebesgue measure. Dee φ s C 0 such that 0 φ s, φ s i B s, supp φ s B t ad φ s Ct s /2p +. Now deig φ = φ s [u h + p w + h p ] + u h + p ad φ 2 = φ s [u h + p w + h p + ], w + h p we obtai 0 φ u h + ad 0 φ 2 C s,h u h Moreover, [ p ] p w + h w + h φ = φ s + p u h + p w u h u h u h p w + h p + φ s u h p ad ad φ 2 = φ s [ + p p u h u h w p w + h w + h + φ s u h p w + h p w + h p. Sice u L loc ad w > h i K t, we obtai φ p φ s u + w + φ s u + w φ 2 p φ s u p L K R h p u + w + φ s u p p u h + ] L K R h p that is, φ, φ 2 W,px L with supp φ, supp φ 2 K t. Besides that we ifer that φ, φ 2 W,px 0 L, by the usig 2. ad Propositio.2.. By takig φ 2 as a test fuctio i Lemma 2.2.3, we obtai w px2 u h p w + h p w φ s K t w + h p dx g h x, w + h K t 36 p u h w + h p w + h p φ s dx 2.2,