Local and global estimates for solutions of systems involving the p-laplacian in unbounded domains

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1 Electroic Joural of Differetial Euatios, Vol , No 19, 1 14 ISSN: UR: htt://ejdemathswtedu or htt://ejdemathutedu ft ejdemathswtedu logi: ft ocal global estimates for solutios of systems ivolvig the -alacia i ubouded domais A echah Abstract I this aer, we study the local global behavior of solutios of systems ivolvig the -alacia oerator i ubouded domais We exted some Serri-tye estimates which are kow for simle euatios to systems of euatios 1 Itroductio We cosider the system u = fx, u, v x, 11 v = gx, u, v x, 12 u = v = 0 x 13 where R N is a exterior domai, f, g are a give fuctios deedig of the variables x, u, v is the -alacia oerator; for 1 < < + is defied by u = div u 2 u Here, we study the local global behavior of solutios of System we follow the work of Serri [4] cocerig the uasiliear euatio div Ax, u, u x = x, u, u x, 14 where A are a give fuctios deedig of the variables x, u, u x u x = u x 1,, u x I articular, 14 geeralizes the euatio u = fx, u x 15 I [4], Serri roves that if the fuctio f is bouded by the term a u 1 + g, where > 1 is a fixed exoet, a is a ositive costat g is a measurable fuctio, the for each y R > 0 we have the estimate su ux cr N R y u 2R y + R N R ɛ g N 1 1 ɛ 2R y 16 Mathematics Subject Classificatios: 35J20, 35J45, 35J50, 35J70 Key words: uasiliear systems, -alacia oerator, ubouded domai, Serri estimate c 2001 Southwest Texas State Uiversity Submitted November 23, 2001 Published March 23,

2 2 ocal global estimates for solutios EJDE 2001/19 for all 0 < ɛ 1 I may cases, esecially for ubouded domai, whe we wish to show that the solutio decay at ifiity, the estimate 16 reuires that the fuctio f belogs to α with α > N/, which is ot trivial to rove i some cases To avoid this difficulty Yu [5], Egell [1] others have roved that the solutio of 15 have a regularity for each, this for all fuctio f bouded by a subliear, suerliear or a homogeeous terms We ote that i the case of a mixed terms this last techiue caot be adated For the case of a homogeeous system see the aer of Fleckiger, Maàsevich, Stavrakakis de Théli [2] The first art of this aer is devoted to the local behavior of solutios of System We obtai a estimate of Serri tye i the followig cases: 1 f g are bouded by a sum of homogeeous critical terms 2 f g are bouded by a sum of homogeeous costat terms Thus, we exted the results of [5], [1] cocerig Euatio those of [2] cocerig System I the secod art, we obtai a global estimates of solutios of System i the articular case f = A u α 1 u v β+1 g = u α+1 v β 1 v uder some coditios o α, β, Also we obtai aother global estimate whe f g satisfy 2 We recall that D 1, is the closure of C 0 with resect to the orm u D 1, = u = S by 1 N is the cojugate of, = N is the Sobolev exoet we defie 1 u = if S u u W 1, \0 2 ocal estimates for solutios of Theorem 21 et u, v D 1, R N D 1, R N be a solutio of τ = N N, τ = N N Assume that max, < N, fx, u, v C u 1 + u 1 + v / + v τ τ, 21 gx, u, v C v 1 + v τ 1 + u / + u τ τ, 22 where m is the cojugate of m C is a costat The 1 For ay R > 0 x R N satisfyig C max 2 S τ 1, 2 2 S 1 N R τ 1 u τ 1 2R x + v τ 1 τ 2R x < 1 23

3 EJDE 2001/19 A echah 3 where S S are the Sobolev costats, we have u R2 x c 1 + R NN 3 max R N u R x, R N / v R x v R2 x c 1 + R NN 2 max R N v R x, R N witch c ideedet of u, v, x R 2 Moreover, lim ux = lim vx = 0 x + x + u R x Remark 22 There exists a R 0 such that for all R < R 0, 23 is satisfied uiformly for all x This follows from the absolute cotiuity of the fuctioals A A u dx A A v τ dx To be more secific, for each ɛ > 0 there exists η > 0 such that for all R > 0 x R N satisfyig R x η, we have R x u dx < ɛ R x v τ dx < ɛ Proof et x R N be fixed For y 2R x ay fuctio h defied o 2R x we defie ht = hy, t = y x R Sice u, v is a solutio for 11 13, the ũ, ṽ satisfies ũ = R fy, ũ, ṽ, 24 ṽ = R gy, ũ, ṽ 25 I this roof c deotes a ositive costat ideedet of u, v, x R For ay ball 2 0, we have w W 1, 0 w τ S w, w W 1, 0 w τ N S w 26 S S are the Sobelev costats et m be a seuece of ositive umbers satisfyig σ < where σ is defied below r a decreasig seuece defied by r 0 = 2, r = mi + σ i=0 1/,

4 4 ocal global estimates for solutios EJDE 2001/19 where R is ositive σ = mi + i=0 1/ We deote by = 0, r we defie η C0 R N so that 0 η 1, η = 1 i +1, suη 1/ m + η c 27 We multily 24 by ũ m ũη, itegrate over Usig 21, we obtai I 1 + I 2 R I 3 + I 4 + I 5 + I 6, 28 where I 1 = 1 + m η ũ m ũ dx, I 2 = η 1 η ũ ũ 2 ũ m ũdx, I 3 = C ũ +m η dx, I 4 = C ũ +m η dx, I 5 = C ũ m ũ ṽ / η dx, I 6 = C ũ m ũ ṽ τ τ η dx Sice 1+m = 1m + m+, we deduce from Youg ieuality the facts, η 1, that for ay s > 0 I 2 s Choosig s such that s + s We deduce from m + m + η ũ ũ m dx 1 2, usig 27, we have η ũ m+ dx I I 1 + c ũ m+ dx 29 I 1 2R 6 i=3 I i + c ũ m+ dx 210

5 EJDE 2001/19 A echah 5 Usig Sobolev ieuality observig that for ay a 0 b 0 a + b 2 1 a + b, we have η / ũ m+ τ 2 1 S I 7 + I 8, 211 where I 7 = 1 η η ũ m+ m + dx c ũ m+ dx, 1 m + I 8 = η ũ m ũ m + dx I 1, thus we deduce from 210 that η / ũ m+ 1 m + c ũ m+ dx + 2 S R τ First ste We costruct the seueces by = τ, = τ, 6 I i i=3 212 we set m = τ 1, l = τ 1 We show that if the coditio C max 2 S R τ 1, N S R τ 1 ũ τ 1 + ṽ τ 1 2 τ 2 < 1, is satisfied, the solutio ũ, ṽ belogs to First, we start by estimatig the itegrals I i, i = 3,, 6 We have I 3 = C ũ +m η dx c ũ 213 Remarkig that m+1 + = 1, we deduce from Hölder ieuality that I 5 = C ũ m ũ ṽ / η dx c ũ m+1 ṽ / 214 We write m + = τ 1 + m +, = τ m+ +1 Observig that m τ 1 = 1, we deduce from Hölder ieuality I 4 =C ũ +m η dx C ũ τ 1 ũ +m η τ m+ +1 dx 215 C ũ τ 1 η/ ũ τ τ

6 6 ocal global estimates for solutios EJDE 2001/19 Remark that τ τ = τ 1, τ 1 + m = 1, 216 τ τ m the from Hölder ieuality, we have I 6 =C ũ m ũ ṽ τ τ η dx + τ = 1, +1 C ṽ τ 1 η τ m+1 +1 ũ m +1 η τ +1 ṽ / dx C ṽ τ 1 1+m τ η/ ũ τ τ ηṽτ Substitutig m by τ 1 i 212, we obtai τ 217 η / ũ τ τ τ 1 2 S R I 4 + I 6 τ c 1 ũ dx + 2 S R I 3 + I 5 It follows from the fact that η / ũ τ τ C2 S R τ 1 ũ τ 1 η/ ũ τ τ + ṽ τ 1 τ η/ ũ τ 1+m τ ηṽτ τ c1 + R τ 1 ũ + ũ m+1 ṽ / Similarly, we have ηṽ τ τ C22 S 1 N R τ 1 ṽ τ 1 τ ηṽτ τ + ũ τ 1 τ ηṽτ 1+l τ η/ ũ τ τ c1 + R τ 1 ṽ + cr τ 1 ṽ l+1 ũ / 220 Next, we defie θ +1 = max η / ũ τ τ, ηṽτ τ, E = max ũ ṽ, ieuality the defiitio of 1/ E θ, show that θ +1 C max 2 S R τ 1, N ũ τ 1 + ṽ τ 1 τ Simle comutatios usig Hölder S R τ 1 θ +1 c1 + R τ 1 E 221

7 EJDE 2001/19 A echah 7 We kow that there exists R 0 > 0 such that for ay R < R 0 C max 2 S R τ 1, N S R τ 1 ũ τ 1 + ṽ τ 1 2 τ 2 < Also, remark that θ +1 max ũ +1 +1, ṽ max ũ , ṽ /τ = E +1 Therefore, from , the fact 223 E +1 c1 + R τ 1 E So This imlies that E +1 c1 + R 1/ τ 1 E ũ E +1 c1 + R i=0 1 i 1 τ i i=0 τ τ i E 0 Sice i=0 1 τ = N i 2 Similarly, we have ṽ / i= ṽ E c1 + R i 1 τ i <, we deduce that ũ i=0 1 τ i i=0 τ i 1 τ i E 0, therefore v Secod ste We remark that hyothesis 23 is euivalet to C max 2 S R τ 1, N S R τ 1 ũ τ 1 + ṽ τ < 1 We assume that R, u v satisfy 23, which by the first ste imlies that ũ, ṽ τ 2 1 τ 2 1 We let δ = τ 2 τ 2 τ+1 χ = τ δ It is clear that 1 < δ < τ, so χ > 1 We costruct a seueces s t by s = χ, t = χ I this ste m r are defied by χ m = δ 1, r 0 = 1, r = 1 1 2σ 1 mi + i=0 1/,

8 8 ocal global estimates for solutios EJDE 2001/19 which imlies m + = s /δ Now, we estimate the itegrals I i i=3,,6 We have I 3 c ũ s/δ s δ c ũ s/δ 224 Remarkig that m+1 s /δ We have τ 1 τ 2 I 5 c ũ m+1 s δ + m+ s Observig that τ 1 τ 2 + / t /δ = 1, it follows from Hölder ieuality that ṽ / t δ ũ m+1 s ṽ / = 1, thus from Hölder ieuality we have s 225 I 4 c ũ τ 1 τ2 ũ s/δ c ũ s/δ m+1 s + / t = 1, it follows from Hölder ieuality that I 6 c ṽ τ 1 ũ m+1 ṽ / dx c ṽ τ 1 τ2 ũ m+1 c ũ m+1 s ṽ / t s ṽ / t We deduce from 212, the fact that η / ũ χ /δ τ cχ R s + ũ m+1 Similarly, we have ηṽ χ /δ τ cχ R ũ s/δ ṽ t/δ t + ṽ l s ṽ / t 228 t ũ / s 1/s As i the first ste, we let Λ = max ũ s s ṽ t, t Γ = max η / ũ χ /δ τ, /δ ηṽχ τ 1 Υ = max ũ s t s ṽ t, t Simle comutatios show that ũ m+1 s ṽ / t mi ṽ l+1 t ũ / s mi Λ s/δ Λ s/δ Also, remark that Γ max ũ s/δ s +1, ṽ t/δ +1 t , Υ t/δ, 230, Υ t/δ 231 = Λ s/δ +1 = Υt/δ 232

9 EJDE 2001/19 A echah 9 Thus, we deduce from that Λ s/δ +1 cχ R Λ s/δ, so Which imlies that Λ +1 c δ/s χ 1δ s 1 + R δ/s Λ ũ s Λ c i=0 δ s i χ i=0 Sice i=0 δ s i = δτ τ δ, i 1δ i=0 s i <, the ũ 12 Similarly, we have lim su ũ s + c 1 + R As teds to ifiity, we obtai y the imbeddigs the fact we have δτ τ δ max i 1δ s i 1 + R i=0 δ s i Λ 0 ũ 1, ṽ / 1 Υ +1 c δ t χ 1δ t 1 + R δ t Υ ṽ 12 lim su ṽ t + c 1 + R δτ τ δ max ṽ 1, ũ , δτ τ δ = τ NN = τ 1 2 2, ũ 12 c 1 + R NN 3 1 max ũ 1, ṽ / 1, ṽ 12 c 1 + R NN 2 max ṽ 1, ũ 1 Comig back to u, v by a simle chage of variables, we fid u R2 x c 1 + R NN 3 max R N u R x, R N / v R x

10 10 ocal global estimates for solutios EJDE 2001/19 v R2 x c 1 + R NN 2 max R N v R x, R N The roof of 2 follows from 1 Remark 22 u R x Proositio 23 et u, v D 1, R N D 1, R N a solutio of We assume, fx, u, v C u 1 + v / + 1, 233 where m is the cojugate of m The u 1 c 1 + R N 2 max v 1 c 1 + R N max gx, u, v C v 1 + u / + 1, 234 1, R N 1, R N u 2, R N / v 2, 235 u, R N 2 v Proof We use the same chage of variables as i the roof of Theorem 21 Thus, we obtai that ũ, ṽ satisfies Also we kee the same seueces m, r, the same fuctio η We multily Euatio 24 by ũ m ũη, itegrate over Usig 233, we have where I 1 + I 2 R I 3 + I 4 + I 5, 237 I 1 = 1 + m η ũ m ũ dx, I 2 = η 1 η ũ ũ 2 ũ m ũdx, I 3 = C ũ +m η dx, I 4 = C ũ m ũ ṽ / η dx, I 5 = C ũ m ũη dx The itegrals I 1, I 2, I 3 I 4 are the same to those obtaied i Theorem 21 Simle comutatios used before show that η / ũ m+ 1 m + 5 c ũ m+ dx + 2 S R I i τ i=3 238

11 EJDE 2001/19 A echah 11 Now, we defie by = τ, = τ, let m = τ 1,, l = τ 1 The we estimate the itegrals I i, i = 3,, 5 It is clear from that I 3 c ũ I 4 c ũ m+1 ṽ / O the other h I 5 C ũ m+1 dx = c ũ m+1 m+1 c 1 m 1 c 2 1 τ ũ m m+1 ũ m+1 c ũ m We deduce from that ũ η/ ũ τ τ Similarly, we have cτ 1 ũ + R ũ + ũ m+1 ṽ ηṽτ τ cτ 1 ṽ + R ṽ + ṽ l+1 ṽ / ũ / Followig the roof of Theorem 21 we let E = max 1, ũ, ṽ 1/ 1 F = 1, ũ, ṽ We obtai + ũ m ṽ l ũ 1 lim + su ũ E c 1 + R N 2 E 0 = c 1 + R N 2 max 1, ũ 2, ṽ / 2 243

12 12 ocal global estimates for solutios EJDE 2001/19 ṽ 1 lim + su ṽ F c 1 + R N F 0 = c 1 + R N max 1, ũ, ṽ Usig a simle chage of variables i we obtai Global estimates for solutios of Proositio 31 et u, v D 1, D 1, a solutio of We assume that there exist a fuctios a, b 1 a costat C such that fx, u, v ax + C u 1 + v /, 31 gx, u, v bx + C v 1 + v /, 32 where > 1, > 1 The 1 u, v σ η for all σ, η [, + [, + 2 lim ux = lim vx = 0 x + x + Proof 1 et = τ, = τ, m = τ 1, t = τ 1, T k u = max k, mik, u w = T k u m T k u, with k > 0 Multilyig the euatio 11 by w itegratig over, we obtai m + 1 T k u T k u m dx = fx, u, vw dx Observig that 1 m + 1 T k u m+1 = T k u m T k u, 33 we deduce from Hölder Sobolev ieualities that for ay 0 < γ < 1, we have T k u τm+ 1/τ c a 1 γ a γ 1 u m+1 + u + v / u m+1 34 with c deedig from ettig k ted to ifiity i 34, we obtai u +1 c u m+1 + u + v / u m+1 35 We derive from 35 that u for all N Similarly, we rove that v for all N y iterolatio ieuality see [3] we rove that

13 EJDE 2001/19 A echah 13 u, v σ η, for all σ, η [, + [, + The roof of 2 follows from Serri ieuality [4] 1 Next, we study the sub-homogeeous system i a exterior domai or R N u = x u α 1 u v β+1, 36 Proositio 32 Assume that, C v = Cx u α+1 v β 1 v, 37 α β + 1 < 1, > 1, > 1 The each solutio u, v D 1, D 1, of the system 36, 37 satisfies 1 u, v σ η for all σ, η [, + [ [, + [ 2 lim x + ux = 0 lim x + vx = 0 Proof et τ = N N, τ = N α+1 N = 1 Assume, which imlies that τ τ We defie the seueces, f by β+1 f +1 = τf , f 0 = 1, = f, = f et T k u = max k, mik, u for k > 0 ω = T k u m T k u, with m = 1 α + 1 β + 1 = f Multilyig 36 by ω itegratig over, we obtai from 33 Sobolev ieuality 1 1 m + 1 S m + 1 T k u τm+ 1/τ u α v β+1 ωdx From the defiitio of m Hölder ieuality, we deduce that T k u m +1 1/τ m + 1 S m + 1 u α+1+m et k teds to ifiity, we have u m +1 1/τ m + 1 S m + 1 u α+1+m m + 1 = f +1 = +1, therefore u +1 +1, We cosider w = T k v t T k v, with t = 1 α + 1 β + 1 = τf + 1 v β+1 v β+1 To show that v 39

14 14 ocal global estimates for solutios EJDE 2001/19 Proceedig as above, we obtai v t +1 1 τ t + 1 S t + 1 C u α+1 et = t + 1 It is clear that v, sice = t + 1 = τf , v β+1+t the +1 y iterolatio ieuality see [3], we deduce that v +1 2 follows from Serri ieuality [4] 1 Refereces [1] E Egell Existece results for some uasiliear ellitic euatios Noliear Aal, 1989 [2] J Fleckiger, RF Maasevich, NM Stavrakakis, F de Theli Pricial eigevalues for some uasiliear ellitic euatios o R Adv Differetial Euatios, 26: , 1996 [3] D Gilbarg N S Trudiger Ellitic Partial Differetial Euatios of Secod Order Sriger-Verlag, 1983 [4] J Serri ocal behavior of solutios of uasiliear euatios Acta Math, 111: , 1964 [5] S Yu Noliear -alacia roblem o ubouded domais Proc Amer Math Soc, 1154: , 1992 Abdelhak echah aoratoire MIP UFR MIG, Uiversité Paul Sabatier Toulouse route de Narboe, Toulouse cedex 4, Frace bechah@mius-tlsefr or abechah@freefr web age: htt://mius-tlsefr/ bechah