An Existence Theorem for a Class of Nonuniformly Nonlinear Systems

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1 Australian Journal of Basic and Alied Sciences, 5(7): , 11 ISSN An Existence Theorem for a Class of Nonuniformly Nonlinear Systems G.A. Afrouzi and Z. Naghizadeh Deartment of Mathematics, Faculty of Mathematical Sciences, University of Mazandaran, Babolsar, Iran Abstract In this article, we discuss the existence of weak solution for the nonuniformly nonlinear ellitic system div( h1 ( u ) u u)= f ( x, u, v) in div( h ( v ) v v)= g( x, u, v) in u = v= on, where Ω is a bounded smooth oen set in R n,, $ and h 1, h C (R,R). Using variational methods, under suitable assumtions on the nonlinearities, we show the existence of weak solution. Key words: Weak solution; Nonuniformly ellitic system; Variational methods. INTRODUCTION In this aer, we study the existence of weak solution of the following Dirichlet system. div( h1 ( u ) u u)= f ( x, u, v) in div( h ( v ) v v)= g( x, u, v) in u = v= on, where Ω is a bounded smooth oen set in R n, # # and h 1, h C (R,R). Ellitic systems have several ractical alications. For examle they can describe the multilicative chemical reaction catalyzed by grains under constant or variant temerature, a corresondence of the stable station of dynamical system determined by the reaction-diffusion system. In recent years, many ublications have aeared concerning uasilinear ellitic systems which have been used in a great variety of alications, we refer the readers to (Brezis, 1983; Djellit, 3; Djellit, 4; Djellit, 7; Drabek, 3; D.D. ai, 7; Zhang, 4) and the references therein. J. Zhang and Z. Zhang (9) used variational methods to obtain weak solution of the nonlinear ellitic system (1) with = =. Motivated by (Zhang (9), in this aer, we will discuss roblem (1). Through this aer for (u,v) R, denote *(u,v)* =*u* +*v*. We assume that F : Ω R v R is of C 1 class such that F (X,, ) = for F F all x and ( f, g)=(, ), f and g are caratheodory functions satisfying the following growth u v conditions: f( x, u, v) g( x, u, v) i) lim u =, 1 lim v =. 1 u v ii) et h 1 and h C(R,R). We assume that h 1 and h are the continuous and nondecreasing satisfying the following growth conditions: (1) () Corresonding Author: G.A. Afrouzi, Deartment of Mathematics, Faculty of Mathematical Sciences, University of Mazandaran, Babolsar, Iran. afrouzi@umz.ac.ir 1313

2 There exist α 1, α, β 1 and β R such that < h() t, 1 < h () t. The main result of this aer is the following: Aust. J. Basic & Al. Sci., 5(7): , 11 Theorem 1: Assume that (i)-(ii) hold. Then system (1) has at least one weak solution. The lan of this aer is as follows. In section, we give some notations and recall some relevant lemmas. The main result is roved in section 3. Notations and Preliminary emmas: 1, 1, et the roduct sace = ( ) ( ) with the norm 1, 1, PuvP (, ) = PuP PvP =( u ) ( u ) 1 u 1 v huv (,)= h 1() s ds h () s ds (, )= ( J uv h u, v ). et us define the maings and J : by J ( uv, ),(, ) = [ h1( u ) u ( ) uh v v v] for any ( uv, ),(, ). et us define the maing. Wuv (, )= Fxuv (,, ) and W : by W( uv, ),(, ) = [ f( xuv,, ) g( xuv,, ) ] for any ( uv, ),(, ). As usual, a weak solution of system (1) is any (u,v) such that J ( u, v),(, ) = W ( u, v),(, ) for any (, ). We need certain roerties of functional J: 6R defined by 1 u 1 v Juv (,)= h1() sds h() sds for all (u,v) emma.1. Functional J is weakly lower semicontinuous. Proof. et (u,v) and ε > be fixed. Using the roerties of lower semicontinuous function (see [1, section I.3] ) is enough to rove that there exists δ > such that Juv (, ) Ju (, v) ( uv, ) Puv (, ) ( u, v) P<. (3) Using hyotheses (ii), it is easy to check that J is convex. ence we have J ( uv, ) Ju (, v) J( u, v),( uu, vv) ( uv, ). 1314

3 Aust. J. Basic & Al. Sci., 5(7): , 11 Using condition (ii) and olders ineuality we deduce there exists a ositive constant C > such that. Juv (, ) Ju (, v) h( u ) u u uu 1 h( v ) v v vv J( u, v) PuP PuuP 1 1,, Pv P Pvv P J( u, v ) cp( uu, vv ) P ( u, v). 1,, It is clear that taking = c relation (3) holds true for all (u 1,v 1 ) with P (u,v)-(u 1,v 1 ) P <δ. Thus we roved that J is strongly lower semicontinuous. Taking into account the fact that J is convex then by [1, corollary III.8] we conclude that J is weakly lower semicontinuous and the roof of emma.1 is comlete. emma.. Functional W is weakly continuous. Proof. et {w n }={(u n, V n )} be a seuence converges weakly to w = (u, v) in. We will show that lim F( x, un, vn) = F( x, u, v). n (4) From (i) and the continuity of the otential F, for any ε >, there exists a ositive constant M = M (ε) such that (,, ) (,, ) f xuv u M g xuv v M for all ( x, uv, ) R. ence F( x, u, v ) F( x, u, v) = F( x ( w w)) ( w w) n n n n n = F ( xu, ( u u), v ( v v)) ( u u) u 1, n n, n n n F ( x, u ( u u), v ( v v)) ( v v) v 1, n n, n n n ( x), ( x) 1 n =( 1, n,, n) 1, n, n where and for all X Ω Now, using (5) and olders ineuality we conclude that (5) [ F( xu,, v) F( xuv,, )] F( xu, ( u u), v ( v v) u u n n u 1, n n, n n n F ( x, u ( u u), v ( v v) v v v 1, n n, n n n ( u ( u u) M ) u u ( v ( v v) M ) v v 1, n n n 1, n n n 1 M Pu up Pu ( u u) P Pu up 1 n ( ) 1, n n ( ) n ( ) (6) 1315

4 Aust. J. Basic & Al. Sci., 5(7): , 11 1 M Pv vp Pv ( v v) P Pv vp 1 n ( ), n n ( ) n ( ) j [, ) i j on the other hand, since ( ) ( ) is comact for all i [, ) and the seuence {w n } converges to w = (u, v) in the sace (Ω) (Ω), i.e., {w n } converges strongly to u in (Ω) and {w n } converges strongly to v in (Ω). ence, it is easy to see that the seuences are bounded. Thus, it follows from (6) that relation (4) holds true. { Pu ( u u ) P } and 1, n n ( ) 3. Proof of Main Theorem: In this section we give the roof of theorem 1. Proof of theorem 1. et by E( uv, )= Juv (, ) Fxuv (,, ) (, )= ( Juv h u, v ) as in section, and let the energy E:6R given for any (u,v). Then a weak solution of system (1) is a critical oint of E(u,v) in. emma.1 and. imly that E is weakly lower semicontinuous. By olders ineuality, (5), we have u F F( x, u, v)= ( x, s, v) ds F( x,, v) s uf vf = ( xsv,, ) ds ( x,, s) dsfx (,,) s s u v ( u M ) ds ( v M ) ds = u Mu v Mv so F( xuv,, ) Fxuv (,, ) 1 1 [ u v ] M [ u v] S1 u S v 1 M S ( u ) M S ( v ) 1316

5 Aust. J. Basic & Al. Sci., 5(7): , 11 1 S u S v A[ u ) v ) ] where S 1, S are the embedding constants of ( ) ( ), ( ) ( ) 1, 1, and 1 A =max{ M S, M S }. ence (, ) ( ) ( ) (, ). Euv 1 S 1 u S v APuvP etting = min{, }. Noting that S S 1 1 u v u v [( ) ( ) ] 1 since #, we obtain Euv (, ) min{ 1, } [ PuvP (, ) 1] APuvP (, ) it follows that E is coercive in. By (i), (ii), E is continuously differentiable on and E ( u, v),(, ) = [ h( u ) u uh ( v ) u v 1 f( xuv,, ) gxuv (,, ) ] = J( uv, ),(, ) W( uv, ),(, ) for any (u,v). Therefore E has a minimum at some oint (u,v) and E (u,v)=. Thus, this imlies that J ( u, v),(, ) = W ( u, v),(, ) for any (u,v), that is, (u,v) is a weak solution of system (1). This comletes the roof of theorem 1. REFERENCES Brezis,., Analyse Fonctionnelle. Theorie et Alications, Collection of Alied Mathematics for the Masters Degree, Masson, Paris. Djellit, A., S. Tas, 3. Existence of solutions for a class of ellitic systems in R N involving the - alacian, Electrinic J. Diff. Ens., 56: 1-8. Djellit, A., S. Tas, 4. On some nonlinear ellitic systems, Nonlinear Anal., 59: Djellit, A., S. Tas, 7. Quasilinear ellitic systems with critical Sobolev exonents in R N, Nonlinear Anal., 66: Drabek, P., N.M. Stavrakakis, N.B. Zograhooulos, 3. Multile nonsemitrivial solutions for uasilinear systems,differential Integral Euations, 16(1): ai, D.D.,. Wang, 7. Nontrivial solutions for -alacian systems, J. Math. Anal. Al., 33: Zhang, J., 4. Existence results for the ositive solutions of nonlinear ellitic systems, Al. Math. Com., 153: Zhang, J., Z. Zhang, 9. Existence results for some nonlinear ellitic systems, Nonlinear Anal. 71,