Existence and Nonexistence of Weak Positive Solution for Classes of 3 3 P-Laplacian Elliptic Systems

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1 Intenational Jounal of Patial Diffeential Euations Alications 03 Vol. No. 3-7 Aailable online at htt://ubs.scieub.co/ijdea///3 Science Education Publishing DOI:0.69/ijdea---3 Existence Nonexistence of Weak Positie Solution fo Classes of 3 3 P-Lalacian Ellitic Systes Guefaifia Rafik Akout Kael * Saifia Wada LAMIS Laboatoy Tebessa Uniesity Tebessa Algeia LANOS Laboatoy Badji Mokhta Uniesity Annaba Algeia *Coesonding autho: akoutkael@gail.co Receied Noebe 03; Reised Noebe 9 03; Acceted Decebe 3 03 Abstact In this ok e ae inteested to obtain soe esult of existence nonexistence of lage ositie u = α x f u in u = µβ ( x g ( u in eak solution fo the folloing -lalacian syste hee = νγ ( x h ( u in u = = = 0 on σ σ z = di z z σ µ ν ae a ositie aaete is a bounded doain in N ith sooth bounday. The oof of the ain esults is based to the sub-suesolutions ethod. Keyods: Positie solutions Sub-suesolutions -Lalacian systes Cite This Aticle: Guefaifia Rafik Akout Kael Saifia Wada Existence Nonexistence of Weak Positie Solution fo Classes of 3 3 P-Lalacian Ellitic Systes. Intenational Jounal of Patial Diffeential Euations Alications no. (03: 3-7. doi: 0.69/ijdea Intoduction Pobles inoling the -Lalacian aise fo any banches of ue athe-atics as in the theoy of uasiegula uasiconfoal aing as ell as fo aious obles in atheatical hysics notably the flo of non-netonian fluids. Hai Shiaji [9] studied the existence of ositie solution fo the -Lalacian syste u = f in = g u in u = = 0 on (. hich f(s; g(s ae the inceasing functions in [0 satisfy M( gs li f = 0 M > 0 s + S the authos shoed that the oble (. has at least one ositie solution oided that > 0 is lage enough. In [6] the autho studied the existence nonexistence of ositie eak solution to the folloing uasilinea ellitic syste α γ u = u in δ β u = u in u = = 0 on (. The fist eigenfunction is used to constuct the subsolution of oble (.3 the ain esults ae as follos: (i If α β 0 γ δ > 0 θ = ( α (( β γδ > 0 then oble (.3 has a ositie eak solution fo each k > 0; (ii If = 0 γ= ( α then thee exists 0 > 0 such that fo 0 < < 0 then oble (.3 has no nontiial nonnegatie eak solution. In this ae e ae concened ith the existence nonexistence of ositie eak solution to the uasilinea ellitic syste u = α x f u in = µβ x g u in = νγ x h u in u = = = 0 on σ hee z di ( z σ z (.3 = σ µ ν ae a ositie aaete is a bounded doain in

2 4 Intenational Jounal of Patial Diffeential Euations Alications N ith sooth bounday. We oe the existence of a lage ositie eak solution fo µ ν lage hen ( ( ( f ttt gttt httt li 0 t t = t = + t =. Definitions Notations Definition. We called ositie eak solution (u; ; of (.3 such that satisfies u u. = α( x f ( u u. ω dx = µ β( x g ( u u3. = γ ( x h( u fo all ω Η 0 ( ith ω 0 Definition. We called ositie eak subsolution ( 3 suesolution (z z z 3 of (.3 such that i zi i= 3 satisfies 3. α x f 3. ω dx µ β x g γ x h z z. α( x f ( z z z3 z z. ω dx µ β( x g ( z z z3 z3 z3. γ ( x h( z z z3 fo all Η 0 ( ith 0. We suose that α β γ f g h eify the folloing assutions; (H f gh : ([ 0 [ ae C onotone functions such that = li f t t t3 li g t t t3 t t t3 + t t t3 + = li ht ( t t3 = + t t t3 + k0 > 0: f yz g yz h yz k0 fo all yz 0 ( x ( x ( x α0 α α α0 β0 γ0 α β γ > 0: β0 β β γ0 γ γ ( ( ( f ttt gttt httt (H li 0 t t = t = + t = ξ ξ ξ3 η η η3 3 > 0: f ( t t t3 ξt ηt t (H3 g( t t t3 ξt ηt t ht ( t t3 ξ3t + η3t + 33 t Let µ ν be the fist eigenalue of ith Diichlet bounday conditions the coesonding ositie eigenfunction ith = = = δ > 0 such that on δ = x d x µ { : ( δ} We denote by ( ( ( ( ( ( µ ξ η ξ ξ 0 = ( µ 0 = + ( µ η ξ η η µ 0 = + + ( ξ3 + η Existence Results Theoe. Let (H (H hold. Then fo lage the syste (.3 has a lage ositie solution (u : Poof. We shall eify that α = µβ = γ 3 = is a subsolution of (.3 fo lage: Let ω ω 0.A calculation shos that ω dx = H 0 ith k0 = dx σ σ σ ω k 0 σ σ ( σ ω dx = σ k0 = ( σ σ.

3 Intenational Jounal of Patial Diffeential Euations Alications 5 No on σ e hae α 3 α β γ ( x f ( ( α α( x 3 ( x f ( µ β ( β β( x ( γ ( x f ( 3 ( γ γ ( x Next on / σ e hae ρ fo soe ρ 0 theefoe fo µ ν lage f ( 3 g ( 3 µ µ ( 3 ( h Hence 3. f 3. ω dx µ g h i.e. ( 3 is a subsolution of (.3. Next let ae the solution of = in = in = in = 0 on. = 0 on. = 0 on. Let C z = z = µ g z3 = h Whee C > 0 is a lage nube to be chosen late: We shall eify that (z z z 3 is a suesolution of (.3 fo µ ν lage. To this end let ω H 0 ( ith ω 0. By (H (H e can choose C lage enough so that then g C C C µ µ g = z z h h = z3 then z3 hich ily that f C C C C f C C C f C C f ( z z z3 = Then e hae z z. C = C = f ( z z z3

4 6 Intenational Jounal of Patial Diffeential Euations Alications in anothe h z z. = µ g. C µ g ( µ g z z z siila z z. = h. C h ( h z z z 3 i.e (z z z 3 is a suesolution of (.3 ith z i i fo C lage i = 3. Thus thee exists a solution (u of (.3 ith u z z 3 ω z3. 4. Nonexistence Results Theoe. If f g h eify (H 3 the syste (.3 has not nontiial ositie solutions fo 0 < 0 < 0< µ 0 < µ 0 < 0 <. (4. Poof. Multilying the fist euation by u; e hae fo Young ineuality that u ( = f u udx ξ u + η + udx ( η ξ u u u dx η + ( ( ( ξ η + η + ξ + u + η + dx = + + u then e hae ( ( u ( ξ +η+ u + η + ( µ µ ξ u + ( ξ + η + µ + ( ( ( ξ 3 u + η Note that u = inf inf u inf = µ = ( ξ η Cobining (4. (4.3 e obtain ( ( µ µ 0 u (4. (4.3 hich is a contadiction if (4. hold. Thus (.3 has no nontiial nonnegatie eak solution. 5. Alications P Theoe 3. conside the folloing syste in W ( n l u = u in n l = µ u in 3 n3 l3 = u in u = = = 0 on the syste has a lage ositie solutions if In the case hee + n+ l < + n + l < 3 + n3 + l3 < ( + n+ l = + n + l = 3 + n3 + l3 = the syste has not non tiial ositie solutions if 0 (5. (5. (5.3 0 < < 0< µ < µ 0 < <. (5.4 Poof. (5. ily that (H. By using theoe the syste has a lage ositie solutions. The fist euation in (5.3 ily that + + = + + = 3 n l θ θ θ then the genealized Young ineuality gies (5.5

5 Intenational Jounal of Patial Diffeential Euations Alications 7 n l n lθ 3 f ( u = u θ u + θ + θ θ θ3 = u + + θ θ θ3 By the sae anne e conclude that the assution (H3 holds. Then he syste (5. has not nontiial ositie solutions if hich ily that 0 = ( ( < µ 0 = ( n+ ( n + + n3 < µ µ µ 0 = ( l+ µ l + l ( 3 + <. ( ( µ µ + + < (5.6 0 this ineuality hold if (5.4 hold. Theoe 4. The folloing oble has a lage ositie solution if lage 3 3 u x H u u u in = γ u = u = u = 0 on (5.7 hee is a bounded doain in N ith sooth bounday is a ositie aaete γ is a function of class L ( : ([ 0 [ 3 ( 3 ( H is of class C H t t t is inceasing ith esect to t t 3 H t t t 3 is deceasing ith esect to t ( H t t t li = 0 > + t t ([ [ k > 0: H t t t k t t t Poof. The oble (5.7 can be itten unde the folloing syste fo u = in = in = γ ( x H ( u in u = = = 0 on In this case the assutions of theoe (3. holds. Refeences [] S. Ala G. A. Afouzi Q. Zhang & A. Nikna Existence of ositie solutions fo aiable exonent ellitic systes Bounday Value Pobles 0 0:37. [] G. A. Afouzi & J. Vahidi On citical exonent fo the existence stability oeties of ositie eak solutions fo soe nonlinea ellitic systes inoling the ( -Lalacian indefinite eight function Poc. Indian Acad. Sci. (Math. Sci. Vol. No. Febuay [3] G. A. Afouzi & Z. Valinejad Nonexistence of esult fo soe - Lalacian Systes The Jounal of Matheatics Coute Science Vol. 3 No. (0-6. [4] J. Ali R. Shiaji Existence esults fo classes of Lalacian systes ith sign-changing eight Alied Matheatics Lettes 0 ( [5] J. Ali R. Shiaji Positie solutions fo a class of -Lalacian systes ith ultile aaetes J. Math. Anal. Al. 335 ( s [6] C. Chen On ositie eak solutions fo a class of uasilinea ellitic systes Nonlinea Analysis 6 ( [7] R. Dalasso Existence uniueness of ositie solutions of seilinea ellitic systes Nonlinea Analysis 39 ( [8] H. Dang S. Ouganti & R. Shiaji nonexistence of non ositie solutions fo a class of seilinea ellitic systes Rocky Mountain Jounal of atheatics Volue 36 Nube [9] D. D. Hai R. Shiaji An existence esult on ositie solutions fo a class of -Lalacian systes Nonlinea Analysis 56 ( [0] P. Dˆabek & J. Hen ez Existence uniueness of ositie solutions fo soe uasilinea ellitic obles Nonlinea Analysis 44 (00. [] S. Haghaieghi & G. A. Afouzi Sub-sue solutions fo (- Lalacian systes Bounday Value Pobles 0 0:5. [] R. Shiaji a & J. Ye Nonexistence esults fo classes of 3 3 ellitic systes Nonlinea Analysis 74 (