Existence of Weak Solutions for Elliptic Nonlinear System in R N

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1 Inernaional Journal of Parial Differenial Equaions and Alicaions, 4, Vol., o., 3-37 Available online a h://ubs.scieub.co/idea///3 Science and Educaion Publishing DOI:.69/idea---3 Exisence of Wea Soluions for Elliic onlinear Syse in ahar Bouali *, afi Guefaifia Dearen of Maheaics, Universiy ebessa, ebessa, Algeria *Corresonding auhor: boualiahar@yahoo.fr eceived May 6, 4; evised May 8, 4; Acceed May 8, 4 ( h u f( xuv ( h v g( xuv div =,, in Absrac We sudy he nonuniforly elliic, nonlinear syse Under div =,, in groh and regulariy condiions on he nonlineariies f and g, e obain ea soluions in a subsace of he Sobolev sace (, by alying a varian of he Mounain Pass heore. Keyords: nonuniforly elliic, nonlinear syses, ounain ass heore, ealy coninuously differeniable funcional Cie his Aricle: ahar Bouali, and afi Guefaifia, Exisence of Wea Soluions for Elliic onlinear Syse in. Inernaional Journal of Parial Differenial Equaions and Alicaions, vol., no. (4: doi:.69/idea Inroducion We sudy he nonuniforly elliic, nonlinear syse ( h u f( xuv ( h v g( xuv div =,, in div =,, in here ( ( 3, h L, h x, i =,. i loc i Syse (., here, under aroriae groh and regulariy condiions on he funcions f( xuv,, and g( xuv,,, he ea soluions are exacly he criical oins of a funcional defined on a ilber sace of funcions uv, in (. In he scalar case, he roble α div( x u + b u = f ( x, u in ih 3 and αε (,, has been sudied by Mihailescu and adulescu []. In his siuaion, he auhors overcoe he lac of coacness of he roble by using he he Caffarelli-Kohn-irenberg inequaliy In his aer, e consider (. hich ay be a nonuniforly elliic syse. We shall reduce (. o a uniforly elliic syse by using aroriae eighed Sobolev saces. hen alying a varian of he Mounain ass heore in [9], e rove he exisence of ea soluions of syse (. in a subsace of (,. o rove our ain resuls, e inroduce he folloing soe hyoheses: (. here exiss a funcion F( x, (, C such ha F u ( x, f( x,, G( x, g( x, v = ( uv, (. = =, for all x, ( ( ( f x,, g x, C,, ( ( f x,, = g x,, =, for all x, here exiss a osiive consan α such ha for all for all (, (, f x + g x α x, ( = uv,. (.3 here exiss a consan β > such ha ( ( < β F x, < F x, { x, ( Le (, nor?,. be he usual Sobolev sace under he ( = u + v + u + v, (, (, = uv Consider he subsace {(, (, : ( E = u v u + v <

2 Inernaional Journal of Parial Differenial Equaions and Alicaions 33 hen E is a ilber sace ih he nor I is clear ha ( = E + u v (, γ, E, γ >. E and he ebeddings q E? (,? L (,, q oreover, he ebedding? (, ( see [8]. e no inroduce he sace are coninuous. E L is coac {(, : ( ( ( = u v E h x u + h x v < endoed ih he nor.. ear ( ( ( = + h x u h x v Since h h E, for all x e have ih.. Proosiion and C (,. he se is a ilber sace ih he inner roduc ( ( ( = h x u u + h x v v, = u, u, = v, v. for all ( ( Proof. I suffices o chec ha any Cauchy sequences in converges o. Indeed, le { { {( u, v = be a Cauchy sequence in. hen ; ( h x u u li = + h v v and { is bounded. Moreover, by ear., { is also a Cauchy sequence in E. ence he sequence { converges o = ( uv, E ; i.e, ( ( ( li h x u u + h x v v = I follos ha { ( u, v = ( u, v and { L (,. herefore { ( x { and { ( x converges o ( x everyhere x = converges o converges o in converges o for alos. Alying Faou's lea e ge ( h ( ( x u + h x v ( ( ( li inf h x u + h x v < ence ( uv ( ( ( + li. =,. Alying again Faou's lea h x u u h x v v h u u li li inf = + h v v We conclude ha { ( x converges o ( uv,.3. Definiion We say ha ( uv, syse (. if = in = is a ea soluion of ( ( φ + ( ψ ( ( φ ( ψ h x u h x v f xuv,, + g xuv,, = =,. for all ϕ ( φψ Our ain resul is saed as follos..4. heore Le (. and (.3 are saisfied, he syse (. has a leas one non-rivial ea soluion in. his heore ill be roved by using variaional echniques based on a varian of he Mounain ass heore in [9]. Le us define he funcional J : given by ( J( = ( ( h x u + h x v (,, ( ( F x u v = P = uv,, for ( here ( = (,, ( ( = ( ( h x u + h x v P F x u v. Exisence of ea soluions In general, due o h L lo ( ay be no belong o (, he funcional J C, (in his or, e do no coleely care heher he funcional J belongs o C ( or no.his eans ha e canno aly direcly he Mounain ass heore by Abrosei-abinoiz (see [4], e recall he folloing conce of ealy coninuous differeniabiliy. Our aroach is based on a ea version of he Mounain ass heore by Duc (see [9]... Definiion Le J be a funcional fro a Banach sace Y in o. We say ha J is ealy coninuously differeniable on

3 34 Inernaional Journal of Parial Differenial Equaions and Alicaions Y if and only if he folloing condiions are saisfied ( i J is coninuous on Y. ( ii For any u Y, here exiss a DJ ( u linear a fro Y ino such ha J( u+ v J( u li = DJ ( u, v, v Y. ( iii For any v Y u DJ, he a ( u, v, vi is coninuous on Y. We denoe by C ( Y he se of ealy coninuously differeniable funcionals on Y ( (, here ( C Y C Y. I is clear ha C Y is he se of all coninuously Freche differeniable funcionals on Y. he folloing roosiion concerns he soohness of he funcional J.. Proosiion Under he assuions of heore.4, he funcional J(, given by (.3 is ealy coninuously differeniable on and (, ϕ ( ( ( φ ψ ( f( xuv,, φ g( xuv,, ψ DJ = h x u + h x v + for all = ( uv,, ϕ= ( φψ,. Proof. By condiions ( -- ( 3 and he ebedding? E is coninuous, i can be shon (cf. [[5], heore A.VI] ha he funcional P is ell-defined and C. Moreover, e have of class ( DP(, ϕ = ( (,, (,, f xuvφ + g xuvψ for all = ( uv,, ϕ= ( φψ,. ex, e rove ha is coninuous on. Le { be a sequence converging o in, here = ( u, v, =,,...., = ( uv, hen li ( ( ( h x u u + h x v v = and { is bounded. Observe furher ha ( h ( ( x u h x u ( ( h x u u ( ( h x u u u u ( h x u u u ( h x u u u ( ( ( = + + h x u u ( h x u ( h u u ( h u ( + +. Siilarly, e obain ( h ( ( x v h x v ( + Fro he above inequaliies, e obain ( ( ( + as. hus is coninuous on. ex e rove ha for all = uv,, ϕ= φψ,. ( ( (, ϕ = ( ( ( φ + ψ Indeed, for any, ( uv ϕ ( φψ (,?{ and x e have DJ h x u h x v ( φ ( ( ( φ h ( u φ ( ( =,, =,, any + h x u h x u = h x u + s φds + φ φ h x u + 3 h x. Since, h x u L h x φ L, ( ( ( ( ( = ( + 3 ( φ (. g x h x u h x L Alying Lebesgue's Doinaed convergence heore e ge h u+ φ h u li = h x u φ ( Siilarly, e have ( ψ ( + h x v h x v li = h x v ψ ( Cobining (. - (.3, e deduce ha ( + ϕ ( D (, ϕ = li = ( ( φ + ( ψ h x u h x v hus is ealy differeniable on. Le ϕ= ( φψ, be fixed. We no rove ha he a D (, ϕ is coninuous on. Le { be a sequence converging o in. We have D, ϕ D, ϕ ( ( ( ( h x u u φ + h x v v ψ

4 Inernaional Journal of Parial Differenial Equaions and Alicaions 35 I follos by alying Cauchy's inequaliy ha ( D ( D, ϕ, ϕ ϕ as. hus he a D (, ϕ, i is coninuous on and e conclude ha funcional is ealy coninuously differeniable on. Finally, J is ealy coninuously differeniable on..3. ear Fro Proosiion. e observe ha he ea soluions of syse (. corresond o he criical oins of he funcional J(, given by (.3 hus our idea is o aly a varian of he Mounain ass heore in [9] for obaining non-rivial criical oins of J and hus hey are also he non-rivial ea soluions of syse (...4. Proosiion he funcional J(, he Palais-Sale condiion. ha Proof Le { {( u, v given by (.3 saisfies = be a sequence in such ( ( li J = c, li DJ = Firs, e rove ha { is bounded in. We assue by conradicion ha { is no bounded in. hen here exiss a subsequence { ha follos ha of { such as. By assuion (.3 ( DJ ( J, = (, P ( + DP γ here γ =, his yields ( DJ ( DJ ( J γ +, γ ( = γ DJ i Leing, since (, DJ, e deduce ha (, J hich is a conradicion. ence { bounded in. is Since is a ilber sace and { is bounded in of { ealy, here exiss a subsequence { converging o in. Moreover, since he ebedding? E is coninuous, { is ealy convergen o in E. We shall rove ha ( li inf ( since { converges ealy o in E ; i.e, ( φ ( li u u + v v ψ = for all ϕ= ( φψ, E, his ilies ha { converges ealy o in L ( Ω,. Alying [[6], heore.6], e obain ( li inf ( hus (.8 is roved. We no rove ha ( F ( x ( li DP, = li,. = Indeed, by (., e have here Se (,.( (,.( (,. ( f ( x, θ u u g( x, θ v v F x = f x u u + g x v v + A u u + A v v 3 θ θ A, <, < A (,, 3 i i = are osiive consans. 3 =, =, = =. We have >, <, < and 3 herefore, li F x,. + + = 3 ( ( A3 A3 L L 3 L On he oher hand, using he coninuous ebeddings q (? E? L, q ogeher ih he inerolaion inequaliy ( here δ δ = +, i follos ha

5 36 Inernaional Journal of Parial Differenial Equaions and Alicaions δ δ. L L L ( ( Since he ebedding? ( ( L E L is coac e have as. ence as and (. is roved. ( L On he oher hand, by (. and (. i follos ( li D, = ence, by he convex roery of he funcional e deduce ha ( li su ( = li inf ( ( li < D (, >= elaions (.8 and (. ily ( = ( li Finally, e rove ha { converges srongly o in. Indeed, e assue by conradicion { ha is no srongly convergen o in. hen here exis a consan of { such ε > and a subsequence { ha ε > for any =,,3... ence + ( + = 4 ε 4 Wih he sae arguens as in he roof of (,8, and rear ha he sequence in E, e have ( li inf + converges ealy o + ence leing, fro (.3 and (.4 e infer ha ( li inf ε ( elaions (.5 and (.6 ily 4 ε >, hich is a conradicion. herefore, e conclude ha { converges srongly o in and J saisfies he Palais -Sale condiion on. o aly he Mounain ass heore e shall rove he folloing roosiion hich shos ha he funcional J has he Mounain ass geoery..5. Proosiion ( i here exis α > and r > such ha J( α, ih = r. ( for all such ha Proof. ( i Fro (.3 ii here exiss > r and J( < r ( ( s =., i is easy o see ha F x, z in F x, s. z > x and z, z. ( ( < F x, z ax F x, s. z x and < z. s = here ax s = F( xs, Cin vie of (. fro (.8 ha uniforly for x. (, F xz li =. z z By using he ebeddings?? (,. I follos E L, ih sile calculaions e infer fro (.9 ha inf J( α r > sall enough his ilies ( = r = > for i. ( ii By ( exiss η η(.7, for each coac se Ω here = Ω such ha ( F xz, η. z forall x Ω, z>. Le ϕ ( φψ, C (, = having coac suor, for >, large enough fro (. e have J ( ϕ = ϕ (, F x ϕ ϕ η ϕ. Ω here η = η( Ω, Ω= ( suφ su ψ. hen ( > ily ( ii.. and Proof of heore.4. I is clear ha J ( =. Furherore, he acceable se G = { γ C( [, ], : γ ( =, γ ( = here is given in Proosiion (.5 is no ey (i is easy o see ha he funcion γ ( = G. by Proosiion. and Proosiions (. - (.5, all assuions of he Mounain ass heore inroduced in [8] are saisfied. herefore here exiss such ha

6 Inernaional Journal of Parial Differenial Equaions and Alicaions 37 ( { ( ([ ] < α J = inf ax J γ, : γ G and DJ (, ϕ = for all ϕ,., ie is a ea soluion of syse (.. he soluion is a non-rivial soluion by J( α. eferences > he roof is colee. [] A. Abahi-Mchachi., J. Flecinger-Pellé; Exisence of osiive soluions for non-cooeraive seilinear elliic syses defined on an unbounded doain, Pian research oes in Mahs., 66 (99, 9-6. [] ]. A. Adas; Sobolev saces, Acadeic Press, e Yor - San Francisco - London, 975. [3] A. Abrosei., K.C. Chang., I. Eeland; onlinear funcional analysis and Alicaions o differenial equaions, Proceedings of he second school ICP, riese, Ialy, 997. [4] A. Abrosei., P.. abinoiz; Dual variaional ehods in criical oins heory and alicaions, J. Func. Anal. 4 (973, [5]. Beresyci., P. L. Lions; onlinear scalar field equaions, I.Exisence of a ground sae, Arch. a. Mech. Anal., 8 (984, [6]. Brezis; Analyse foncionnelle heori e alicaions, Massion, 99. [7] A. Delli., S. AS, Exisence of soluions for a class of elliic syses in n involving he - Lalacian, Elecronic Journal of Differenial Equaions, Vol. 3, o. 56,. -8. [8] D. G. Cosa; On a class of elliic syses in n, Elecronic Journal of Differenial Equaions 994 (94 o. 7, -4. [9] D. M. Duc; olinear singular elliic equaions, J. London. Mah. Soc. 4( (989, [] L. Gasnsi.,. S. Parageorgiou; onsooh criical oin heory and nonlinear boundary value robles, Vol. 8, A CC Press Coany, London - e Yor, 5. [] D. Gilbarg.,. C. rudinger; Elliic Parial Differenial Equaions of second order, Second Ediion, Sringer-Verlag, Berlin, 983. [] M. Mihàilescu., V. àdulescu; Ground sae soluions of nonlinear singular Schr odinger equaions ih lac coacness, Maheaical ehods in he Alied Sciences, 6 (3, [3] P. De àoli., M. C. Mariani; Mounain ass soluions o equaions of -Lalacian ye, onlinear Analysis, 54 (3, 5-9. [4] L. irenberg; Variaional and oology ehods in nonlinear robles, Bull. Aer. Mah. Soc. 4 (98, [5] P.. abinoiz; Miniax ehods and heir alicaions o Parial Differenial Equaions,onlinear analysis, a collecion of aers in honor of Erich ohe, Acadeic Press, e Yor, 978. [6] M. Srue; Variaional Mehods, Second Ediion, Sringer-Verlag,. [7].. Vu., D. M. Duc; onuniforly elliic equaions of - Lalacian ye, onlinear Analysis 6 (5, [8] E. Zeidler; onlinear Funcional Analysis and is alicaions, Vol. III, Sringer-Verlag, 985.